Integrand size = 30, antiderivative size = 1441 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Output:
1/24*b*(8*b^5*c^2*e^2*(-c*f+d*e)^4+27*a^3*b^2*d*f*(c*f+d*e)^3*(c^2*f^2-4*c *d*e*f+d^2*e^2)+9*a^5*d^3*f^3*(c^3*f^3-5*c^2*d*e*f^2-5*c*d^2*e^2*f+d^3*e^3 )-9*a^4*b*d^2*f^2*(3*c^4*f^4-11*c^3*d*e*f^3-16*c^2*d^2*e^2*f^2-11*c*d^3*e^ 3*f+3*d^4*e^4)+36*a*b^4*c*e*(c^5*f^5-2*c^4*d*e*f^4-2*c*d^4*e^4*f+d^5*e^5)- 9*a^2*b^3*(c^6*f^6+7*c^5*d*e*f^5-24*c^4*d^2*e^2*f^4-24*c^2*d^4*e^4*f^2+7*c *d^5*e^5*f+d^6*e^6))*x/a/c^2/(-a*d+b*c)^3/e^2/(-a*f+b*e)^3/(-c*f+d*e)^4/(b *x^2+a)^(3/2)+1/24*b*(16*b^7*c^3*e^3*(-c*f+d*e)^4-88*a*b^6*c^2*e^2*(-c*f+d *e)^4*(c*f+d*e)+9*a^7*d^4*f^4*(c^3*f^3-5*c^2*d*e*f^2-5*c*d^2*e^2*f+d^3*e^3 )-6*a^6*b*d^3*f^3*(6*c^4*f^4-23*c^3*d*e*f^3-26*c^2*d^2*e^2*f^2-23*c*d^3*e^ 3*f+6*d^4*e^4)+6*a^5*b^2*d^2*f^2*(9*c^5*f^5-17*c^4*d*e*f^4-52*c^3*d^2*e^2* f^3-52*c^2*d^3*e^3*f^2-17*c*d^4*e^4*f+9*d^5*e^5)-36*a^4*b^3*d*f*(c^6*f^6+2 *c^5*d*e*f^5-13*c^4*d^2*e^2*f^4-13*c^2*d^4*e^4*f^2+2*c*d^5*e^5*f+d^6*e^6)- 2*a^2*b^5*c*e*(21*c^6*f^6-119*c^5*d*e*f^5+320*c^4*d^2*e^2*f^4-480*c^3*d^3* e^3*f^3+320*c^2*d^4*e^4*f^2-119*c*d^5*e^5*f+21*d^6*e^6)+3*a^3*b^4*(3*c^7*f ^7+41*c^6*d*e*f^6-104*c^5*d^2*e^2*f^5-104*c^2*d^5*e^5*f^2+41*c*d^6*e^6*f+3 *d^7*e^7))*x/a^2/c^2/(-a*d+b*c)^4/e^2/(-a*f+b*e)^4/(-c*f+d*e)^4/(b*x^2+a)^ (1/2)-1/4*d^4*x/c/(-a*d+b*c)/(-c*f+d*e)^3/(b*x^2+a)^(3/2)/(d*x^2+c)^2-1/8* d^4*(2*b*c*(-11*c*f+5*d*e)-3*a*d*(-5*c*f+d*e))*x/c^2/(-a*d+b*c)^2/(-c*f+d* e)^4/(b*x^2+a)^(3/2)/(d*x^2+c)+1/4*f^4*x/e/(-a*f+b*e)/(-c*f+d*e)^3/(b*x^2+ a)^(3/2)/(f*x^2+e)^2+1/8*f^4*(2*b*e*(-5*c*f+11*d*e)-3*a*f*(-c*f+5*d*e))...
Time = 27.26 (sec) , antiderivative size = 630, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\frac {1}{24} x \sqrt {a+b x^2} \left (\frac {8 b^6}{a (b c-a d)^3 (b e-a f)^3 \left (a+b x^2\right )^2}+\frac {8 b^6 \left (2 b^2 c e+20 a^2 d f-11 a b (d e+c f)\right )}{a^2 (b c-a d)^4 (b e-a f)^4 \left (a+b x^2\right )}+\frac {6 d^6}{c (b c-a d)^3 (-d e+c f)^3 \left (c+d x^2\right )^2}+\frac {3 d^6 (3 a d (d e-5 c f)+2 b c (-7 d e+13 c f))}{c^2 (b c-a d)^4 (d e-c f)^4 \left (c+d x^2\right )}+\frac {6 f^6}{e (b e-a f)^3 (d e-c f)^3 \left (e+f x^2\right )^2}+\frac {3 f^6 (2 b e (13 d e-7 c f)+3 a f (-5 d e+c f))}{e^2 (b e-a f)^4 (d e-c f)^4 \left (e+f x^2\right )}\right )-\frac {d^5 \left (24 b^2 c^2 \left (2 d^2 e^2-7 c d e f+7 c^2 f^2\right )+3 a^2 d^2 \left (d^2 e^2-6 c d e f+21 c^2 f^2\right )-4 a b c d \left (4 d^2 e^2-29 c d e f+49 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} (-b c+a d)^{9/2} (-d e+c f)^5}-\frac {f^5 \left (3 a^2 f^2 \left (21 d^2 e^2-6 c d e f+c^2 f^2\right )+24 b^2 e^2 \left (7 d^2 e^2-7 c d e f+2 c^2 f^2\right )-4 a b e f \left (49 d^2 e^2-29 c d e f+4 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (-b e+a f)^{9/2} (d e-c f)^5} \] Input:
Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3*(e + f*x^2)^3),x]
Output:
(x*Sqrt[a + b*x^2]*((8*b^6)/(a*(b*c - a*d)^3*(b*e - a*f)^3*(a + b*x^2)^2) + (8*b^6*(2*b^2*c*e + 20*a^2*d*f - 11*a*b*(d*e + c*f)))/(a^2*(b*c - a*d)^4 *(b*e - a*f)^4*(a + b*x^2)) + (6*d^6)/(c*(b*c - a*d)^3*(-(d*e) + c*f)^3*(c + d*x^2)^2) + (3*d^6*(3*a*d*(d*e - 5*c*f) + 2*b*c*(-7*d*e + 13*c*f)))/(c^ 2*(b*c - a*d)^4*(d*e - c*f)^4*(c + d*x^2)) + (6*f^6)/(e*(b*e - a*f)^3*(d*e - c*f)^3*(e + f*x^2)^2) + (3*f^6*(2*b*e*(13*d*e - 7*c*f) + 3*a*f*(-5*d*e + c*f)))/(e^2*(b*e - a*f)^4*(d*e - c*f)^4*(e + f*x^2))))/24 - (d^5*(24*b^2 *c^2*(2*d^2*e^2 - 7*c*d*e*f + 7*c^2*f^2) + 3*a^2*d^2*(d^2*e^2 - 6*c*d*e*f + 21*c^2*f^2) - 4*a*b*c*d*(4*d^2*e^2 - 29*c*d*e*f + 49*c^2*f^2))*ArcTan[(S qrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(-(b*c) + a*d) ^(9/2)*(-(d*e) + c*f)^5) - (f^5*(3*a^2*f^2*(21*d^2*e^2 - 6*c*d*e*f + c^2*f ^2) + 24*b^2*e^2*(7*d^2*e^2 - 7*c*d*e*f + 2*c^2*f^2) - 4*a*b*e*f*(49*d^2*e ^2 - 29*c*d*e*f + 4*c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[ a + b*x^2])])/(8*e^(5/2)*(-(b*e) + a*f)^(9/2)*(d*e - c*f)^5)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{5/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{5/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {6 d f a^2-b (5 d e+3 c f) a+6 b (b c-a d) f x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{3 a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\int \frac {6 d f a^2-5 b d e a-3 b c f a+6 b (b c-a d) f x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {f \left (4 b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2-b (d e-3 c f) a+4 b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\frac {\int \frac {f \left (4 b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2-b (d e-3 c f) a+4 b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\frac {f \int \frac {4 b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2-b (d e-3 c f) a+4 b^2 c e\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\frac {f \left (\frac {\int \frac {2 b \left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) x^2+a \left (18 d f^2 a^3-3 b f (23 d e+3 c f) a^2+4 b^2 e (4 d e+9 c f) a+8 b^3 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\frac {f \left (\frac {\frac {\int -\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {b \left (\frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f (d e-c f) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}+\frac {\int -\frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\int \frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 407 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx d^2}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx d^2}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\int \frac {a (b d e+b c f-2 a d f)-4 b (b c-a d) f x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\int \frac {a (b d e+b c f-2 a d f)-4 b (b c-a d) f x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {\int \frac {a \left (4 e (d e+2 c f) b^2-3 a f (5 d e+c f) b+6 a^2 d f^2\right )-2 b f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {b \left (\frac {b \left (\frac {\int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {\int \frac {a \left (4 e (d e+2 c f) b^2-3 a f (5 d e+c f) b+6 a^2 d f^2\right )-2 b f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^3 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\right )}{b c-a d}\) |
Input:
Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^3*(e + f*x^2)^3),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2 ]), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/((a + b*x^2)*Sqrt[e + f*x^2]), x], x] - Simp[d/(b*c - a*d) Int[1/((c + d*x^2)*Sqrt[e + f*x^2]), x], x] / ; FreeQ[{a, b, c, d, e, f}, x]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[b/(b*c - a*d) Int[(a + b*x^2)^p*(c + d*x^2)^ (q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d) Int[(a + b*x^2)^(p + 1 )*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && ILtQ[p, 0] && LeQ[q, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(3368\) vs. \(2(1393)=2786\).
Time = 400.98 (sec) , antiderivative size = 3369, normalized size of antiderivative = 2.34
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(3369\) |
| default | \(\text {Expression too large to display}\) | \(14878\) |
Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
5/8*(63/5*((a*f-b*e)*e)^(1/2)*a^2*d^5*(a*f-b*e)^4*(d*x^2+c)^2*(b*x^2+a)^(3 /2)*(f*x^2+e)^2*(1/21*a^2*e^2*d^4-2/7*a*e*(a*f+8/9*b*e)*c*d^3+c^2*(a^2*f^2 +116/63*a*b*f*e+16/21*b^2*e^2)*d^2-28/9*b*f*c^3*(a*f+6/7*b*e)*d+8/3*b^2*c^ 4*f^2)*e^2*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+((a*d-b*c)*c)^( 1/2)*(-3/5*(a*d-b*c)^4*a^2*c^2*((21*a^2*e^2*f^2-196/3*a*b*e^3*f+56*b^2*e^4 )*d^2-6*c*f*e*(a^2*f^2-58/9*a*b*f*e+28/3*b^2*e^2)*d+c^2*f^2*(a^2*f^2-16/3* a*b*f*e+16*b^2*e^2))*(d*x^2+c)^2*(b*x^2+a)^(3/2)*(f*x^2+e)^2*f^5*arctan(e* (b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(c*f-d*e)*x*(3/ 5*a^3*e^3*x^2*(f*x^2+e)^2*(b*x^2+a)^2*(a*f-b*e)^4*d^9+a^2*c*(b*x^2+a)^2*(a *f-b*e)^4*(-3*a*f*x^2+e*(-14/5*b*x^2+a))*(f*x^2+e)^2*e^2*d^8-17/5*a*c^2*(1 5/17*a^6*x^6*(b*x^2+a)^2*f^8+2*a^5*(-26/17*b*x^2+a)*(b*x^2+a)^2*x^4*e*f^7+ 2*a^4*(b*x^2+a)^2*(52/17*b^2*x^4-66/17*a*b*x^2+a^2)*x^2*e^2*f^6+a^3*(b*x^2 +a)^2*(-156/17*b^3*x^6+246/17*a*b^2*x^4-130/17*a^2*b*x^2+a^3)*e^3*f^5-52/1 7*a^2*(b*x^2+a)^2*(-2*b^3*x^6+71/13*a*b^2*x^4-45/13*a^2*b*x^2+a^3)*b*e^4*f ^4+38/17*a*(-119/57*b^5*x^10+53/38*a*b^4*x^8+98/19*a^2*b^3*x^6-1/38*a^3*b^ 2*x^4-12/19*a^4*b*x^2+a^5)*b^2*e^5*f^3+28/17*(22/21*b^5*x^10-83/21*a*b^4*x ^8-87/14*a^2*b^3*x^6+3/7*a^3*b^2*x^4+33/14*a^4*b*x^2+a^5)*b^3*e^6*f^2-47/1 7*b^4*e^7*(-176/141*b^4*x^8-50/141*a*b^3*x^6+91/47*a^2*b^2*x^4+88/47*a^3*b *x^2+a^4)*f+16/17*(11/6*b^3*x^6+3*a*b^2*x^4+2*a^2*b*x^2+a^3)*b^5*e^8)*e*d^ 7-34/5*c^3*(-3/34*a^7*x^6*(b*x^2+a)^2*f^9+25/34*a^6*(b*x^2+a)^2*(-46/25...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**3/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{3} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^3*(f*x^2 + e)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 21752 vs. \(2 (1393) = 2786\).
Time = 150.54 (sec) , antiderivative size = 21752, normalized size of antiderivative = 15.10 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3/(f*x^2+e)^3,x, algorithm="giac")
Output:
1/3*((2*b^18*c^5*e^5 - 19*a*b^17*c^4*d*e^5 + 56*a^2*b^16*c^3*d^2*e^5 - 74* a^3*b^15*c^2*d^3*e^5 + 46*a^4*b^14*c*d^4*e^5 - 11*a^5*b^13*d^5*e^5 - 19*a* b^17*c^5*e^4*f + 140*a^2*b^16*c^4*d*e^4*f - 370*a^3*b^15*c^3*d^2*e^4*f + 4 60*a^4*b^14*c^2*d^3*e^4*f - 275*a^5*b^13*c*d^4*e^4*f + 64*a^6*b^12*d^5*e^4 *f + 56*a^2*b^16*c^5*e^3*f^2 - 370*a^3*b^15*c^4*d*e^3*f^2 + 920*a^4*b^14*c ^3*d^2*e^3*f^2 - 1100*a^5*b^13*c^2*d^3*e^3*f^2 + 640*a^6*b^12*c*d^4*e^3*f^ 2 - 146*a^7*b^11*d^5*e^3*f^2 - 74*a^3*b^15*c^5*e^2*f^3 + 460*a^4*b^14*c^4* d*e^2*f^3 - 1100*a^5*b^13*c^3*d^2*e^2*f^3 + 1280*a^6*b^12*c^2*d^3*e^2*f^3 - 730*a^7*b^11*c*d^4*e^2*f^3 + 164*a^8*b^10*d^5*e^2*f^3 + 46*a^4*b^14*c^5* e*f^4 - 275*a^5*b^13*c^4*d*e*f^4 + 640*a^6*b^12*c^3*d^2*e*f^4 - 730*a^7*b^ 11*c^2*d^3*e*f^4 + 410*a^8*b^10*c*d^4*e*f^4 - 91*a^9*b^9*d^5*e*f^4 - 11*a^ 5*b^13*c^5*f^5 + 64*a^6*b^12*c^4*d*f^5 - 146*a^7*b^11*c^3*d^2*f^5 + 164*a^ 8*b^10*c^2*d^3*f^5 - 91*a^9*b^9*c*d^4*f^5 + 20*a^10*b^8*d^5*f^5)*x^2/(a^2* b^17*c^8*e^8 - 8*a^3*b^16*c^7*d*e^8 + 28*a^4*b^15*c^6*d^2*e^8 - 56*a^5*b^1 4*c^5*d^3*e^8 + 70*a^6*b^13*c^4*d^4*e^8 - 56*a^7*b^12*c^3*d^5*e^8 + 28*a^8 *b^11*c^2*d^6*e^8 - 8*a^9*b^10*c*d^7*e^8 + a^10*b^9*d^8*e^8 - 8*a^3*b^16*c ^8*e^7*f + 64*a^4*b^15*c^7*d*e^7*f - 224*a^5*b^14*c^6*d^2*e^7*f + 448*a^6* b^13*c^5*d^3*e^7*f - 560*a^7*b^12*c^4*d^4*e^7*f + 448*a^8*b^11*c^3*d^5*e^7 *f - 224*a^9*b^10*c^2*d^6*e^7*f + 64*a^10*b^9*c*d^7*e^7*f - 8*a^11*b^8*d^8 *e^7*f + 28*a^4*b^15*c^8*e^6*f^2 - 224*a^5*b^14*c^7*d*e^6*f^2 + 784*a^6...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^3\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^3*(e + f*x^2)^3),x)
Output:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^3*(e + f*x^2)^3), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{3} \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3/(f*x^2+e)^3,x)
Output:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^3/(f*x^2+e)^3,x)