Integrand size = 30, antiderivative size = 720 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=-\frac {b \left (12 a b^2 c e f^2 (4 d e-3 c f)+3 a^3 d f^3 (7 d e-3 c f)-8 b^3 e^2 (d e-c f)^2-3 a^2 b f^2 \left (16 d^2 e^2-5 c d e f-3 c^2 f^2\right )\right ) x}{24 a (b c-a d) e^2 (b e-a f)^3 (d e-c f)^2 \left (a+b x^2\right )^{3/2}}-\frac {b \left (3 a^5 d^2 f^4 (7 d e-3 c f)-16 b^5 c e^3 (d e-c f)^2+8 a b^4 e^2 (d e-c f)^2 (5 d e+11 c f)-18 a^4 b d f^3 \left (3 d^2 e^2-c^2 f^2\right )+9 a^3 b^2 c f^3 \left (12 d^2 e^2-7 c d e f-c^2 f^2\right )-2 a^2 b^3 e f \left (56 d^3 e^3-112 c d^2 e^2 f+83 c^2 d e f^2-21 c^3 f^3\right )\right ) x}{24 a^2 (b c-a d)^2 e^2 (b e-a f)^4 (d e-c f)^2 \sqrt {a+b x^2}}+\frac {f^2 x}{4 e (b e-a f) (d e-c f) \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}+\frac {f^2 (2 b e (7 d e-5 c f)-a f (7 d e-3 c f)) x}{8 e^2 (b e-a f)^2 (d e-c f)^2 \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}+\frac {d^5 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (b c-a d)^{5/2} (d e-c f)^3}-\frac {f^3 \left (a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )-4 a b e f \left (15 d^2 e^2-15 c d e f+4 c^2 f^2\right )+8 b^2 e^2 \left (10 d^2 e^2-15 c d e f+6 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{9/2} (d e-c f)^3} \] Output:
-1/24*b*(12*a*b^2*c*e*f^2*(-3*c*f+4*d*e)+3*a^3*d*f^3*(-3*c*f+7*d*e)-8*b^3* e^2*(-c*f+d*e)^2-3*a^2*b*f^2*(-3*c^2*f^2-5*c*d*e*f+16*d^2*e^2))*x/a/(-a*d+ b*c)/e^2/(-a*f+b*e)^3/(-c*f+d*e)^2/(b*x^2+a)^(3/2)-1/24*b*(3*a^5*d^2*f^4*( -3*c*f+7*d*e)-16*b^5*c*e^3*(-c*f+d*e)^2+8*a*b^4*e^2*(-c*f+d*e)^2*(11*c*f+5 *d*e)-18*a^4*b*d*f^3*(-c^2*f^2+3*d^2*e^2)+9*a^3*b^2*c*f^3*(-c^2*f^2-7*c*d* e*f+12*d^2*e^2)-2*a^2*b^3*e*f*(-21*c^3*f^3+83*c^2*d*e*f^2-112*c*d^2*e^2*f+ 56*d^3*e^3))*x/a^2/(-a*d+b*c)^2/e^2/(-a*f+b*e)^4/(-c*f+d*e)^2/(b*x^2+a)^(1 /2)+1/4*f^2*x/e/(-a*f+b*e)/(-c*f+d*e)/(b*x^2+a)^(3/2)/(f*x^2+e)^2+1/8*f^2* (2*b*e*(-5*c*f+7*d*e)-a*f*(-3*c*f+7*d*e))*x/e^2/(-a*f+b*e)^2/(-c*f+d*e)^2/ (b*x^2+a)^(3/2)/(f*x^2+e)+d^5*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a) ^(1/2))/c^(1/2)/(-a*d+b*c)^(5/2)/(-c*f+d*e)^3-1/8*f^3*(a^2*f^2*(3*c^2*f^2- 10*c*d*e*f+15*d^2*e^2)-4*a*b*e*f*(4*c^2*f^2-15*c*d*e*f+15*d^2*e^2)+8*b^2*e ^2*(6*c^2*f^2-15*c*d*e*f+10*d^2*e^2))*arctanh((-a*f+b*e)^(1/2)*x/e^(1/2)/( b*x^2+a)^(1/2))/e^(5/2)/(-a*f+b*e)^(9/2)/(-c*f+d*e)^3
Time = 17.76 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\frac {1}{24} x \sqrt {a+b x^2} \left (\frac {8 b^4}{a (-b c+a d) (-b e+a f)^3 \left (a+b x^2\right )^2}+\frac {8 b^4 \left (2 b^2 c e+14 a^2 d f-a b (5 d e+11 c f)\right )}{a^2 (b c-a d)^2 (b e-a f)^4 \left (a+b x^2\right )}+\frac {6 f^4}{e (b e-a f)^3 (d e-c f) \left (e+f x^2\right )^2}+\frac {3 f^4 (2 b e (9 d e-7 c f)+a f (-7 d e+3 c f))}{e^2 (b e-a f)^4 (d e-c f)^2 \left (e+f x^2\right )}\right )-\frac {d^5 \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} (-b c+a d)^{5/2} (-d e+c f)^3}-\frac {f^3 \left (a^2 f^2 \left (15 d^2 e^2-10 c d e f+3 c^2 f^2\right )-4 a b e f \left (15 d^2 e^2-15 c d e f+4 c^2 f^2\right )+8 b^2 e^2 \left (10 d^2 e^2-15 c d e f+6 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (-b e+a f)^{9/2} (d e-c f)^3} \] Input:
Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)*(e + f*x^2)^3),x]
Output:
(x*Sqrt[a + b*x^2]*((8*b^4)/(a*(-(b*c) + a*d)*(-(b*e) + a*f)^3*(a + b*x^2) ^2) + (8*b^4*(2*b^2*c*e + 14*a^2*d*f - a*b*(5*d*e + 11*c*f)))/(a^2*(b*c - a*d)^2*(b*e - a*f)^4*(a + b*x^2)) + (6*f^4)/(e*(b*e - a*f)^3*(d*e - c*f)*( e + f*x^2)^2) + (3*f^4*(2*b*e*(9*d*e - 7*c*f) + a*f*(-7*d*e + 3*c*f)))/(e^ 2*(b*e - a*f)^4*(d*e - c*f)^2*(e + f*x^2))))/24 - (d^5*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*(-(b*c) + a*d)^(5/2)*(-(d* e) + c*f)^3) - (f^3*(a^2*f^2*(15*d^2*e^2 - 10*c*d*e*f + 3*c^2*f^2) - 4*a*b *e*f*(15*d^2*e^2 - 15*c*d*e*f + 4*c^2*f^2) + 8*b^2*e^2*(10*d^2*e^2 - 15*c* d*e*f + 6*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)^3)
Time = 1.43 (sec) , antiderivative size = 887, normalized size of antiderivative = 1.23, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.767, Rules used = {421, 25, 402, 25, 402, 25, 27, 402, 402, 27, 291, 221, 421, 402, 25, 402, 25, 27, 291, 221, 407, 291, 221}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{5/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{5/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {6 d f a^2-b (5 d e+3 c f) a+6 b (b c-a d) f x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{3 a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {6 d f a^2-5 b d e a-3 b c f a+6 b (b c-a d) f x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int -\frac {f \left (4 b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2-b (d e-3 c f) a+4 b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\int \frac {f \left (4 b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2-b (d e-3 c f) a+4 b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {f \int \frac {4 b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2-b (d e-3 c f) a+4 b^2 c e\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {f \left (\frac {\int \frac {2 b \left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) x^2+a \left (18 d f^2 a^3-3 b f (23 d e+3 c f) a^2+4 b^2 e (4 d e+9 c f) a+8 b^3 c e^2\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {f \left (\frac {\frac {\int -\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \left (\frac {\frac {f \left (\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}+\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{a (b e-a f)}+\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {d f x^2+2 d e-c f}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {\int -\frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}+\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\int \frac {8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\frac {f x \sqrt {a+b x^2} (2 b e (5 d e-3 c f)-a f (7 d e-3 c f))}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {\left (a^2 f^2 (7 d e-3 c f)-4 a b e f (5 d e-2 c f)+8 b^2 e^2 (2 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {\frac {b x \left (12 a^2 d f-9 a b c f-5 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}+\frac {f \left (\frac {x \sqrt {a+b x^2} \left (6 a^3 d f^2+a^2 b f (49 d e-3 c f)-20 a b^2 e (2 c f+d e)+8 b^3 c e^2\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}+\frac {\frac {x \sqrt {a+b x^2} \left (-18 a^4 d f^3+9 a^3 b f^2 (c f+9 d e)+2 a^2 b^2 e f (41 d e-21 c f)-8 a b^3 e^2 (11 c f+5 d e)+16 b^4 c e^3\right )}{2 e \left (e+f x^2\right ) (b e-a f)}-\frac {3 a^2 \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+31 d e)+4 a b^2 e f (4 c f+21 d e)-24 b^3 e^2 (2 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}+\frac {b x (b c-a d)}{3 a \left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2 (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 407 |
| \(\displaystyle \frac {\left (\frac {d^2 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\left (\frac {d^2 \left (\frac {d \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}-\frac {f \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {d^2 \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} \sqrt {b c-a d} (d e-c f)}-\frac {f \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} \sqrt {b e-a f} (d e-c f)}\right )}{(d e-c f)^2}-\frac {f \left (-\frac {f (d e-c f) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}-\frac {\frac {f (2 b e (5 d e-3 c f)-a f (7 d e-3 c f)) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {\left (8 b^2 (2 d e-c f) e^2-4 a b f (5 d e-2 c f) e+a^2 f^2 (7 d e-3 c f)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{3 a (b e-a f) \left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^2}+\frac {\frac {b \left (12 d f a^2-5 b d e a-9 b c f a+2 b^2 c e\right ) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}+\frac {f \left (\frac {\left (6 d f^2 a^3+b f (49 d e-3 c f) a^2-20 b^2 e (d e+2 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{4 e (b e-a f) \left (f x^2+e\right )^2}+\frac {\frac {\left (-18 d f^3 a^4+9 b f^2 (9 d e+c f) a^3+2 b^2 e f (41 d e-21 c f) a^2-8 b^3 e^2 (5 d e+11 c f) a+16 b^4 c e^3\right ) x \sqrt {b x^2+a}}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {3 a^2 \left (-24 e^2 (d e+2 c f) b^3+4 a e f (21 d e+4 c f) b^2-a^2 f^2 (31 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}\right )}{a (b e-a f)}}{3 a (b e-a f)}\right )}{(b c-a d)^2}\) |
Input:
Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)*(e + f*x^2)^3),x]
Output:
(d^2*((d^2*((d*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sq rt[c]*Sqrt[b*c - a*d]*(d*e - c*f)) - (f*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[ e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f]*(d*e - c*f))))/(d*e - c*f)^ 2 - (f*(-1/4*(f*(d*e - c*f)*x*Sqrt[a + b*x^2])/(e*(b*e - a*f)*(e + f*x^2)^ 2) - ((f*(2*b*e*(5*d*e - 3*c*f) - a*f*(7*d*e - 3*c*f))*x*Sqrt[a + b*x^2])/ (2*e*(b*e - a*f)*(e + f*x^2)) - ((a^2*f^2*(7*d*e - 3*c*f) - 4*a*b*e*f*(5*d *e - 2*c*f) + 8*b^2*e^2*(2*d*e - c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[e ]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f))))/(d *e - c*f)^2))/(b*c - a*d)^2 + (b*((b*(b*c - a*d)*x)/(3*a*(b*e - a*f)*(a + b*x^2)^(3/2)*(e + f*x^2)^2) + ((b*(2*b^2*c*e - 5*a*b*d*e - 9*a*b*c*f + 12* a^2*d*f)*x)/(a*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)^2) + (f*(((8*b^3*c* e^2 + 6*a^3*d*f^2 + a^2*b*f*(49*d*e - 3*c*f) - 20*a*b^2*e*(d*e + 2*c*f))*x *Sqrt[a + b*x^2])/(4*e*(b*e - a*f)*(e + f*x^2)^2) + (((16*b^4*c*e^3 - 18*a ^4*d*f^3 + 2*a^2*b^2*e*f*(41*d*e - 21*c*f) + 9*a^3*b*f^2*(9*d*e + c*f) - 8 *a*b^3*e^2*(5*d*e + 11*c*f))*x*Sqrt[a + b*x^2])/(2*e*(b*e - a*f)*(e + f*x^ 2)) - (3*a^2*(6*a^3*d*f^3 - 24*b^3*e^2*(d*e + 2*c*f) - a^2*b*f^2*(31*d*e + 3*c*f) + 4*a*b^2*e*f*(21*d*e + 4*c*f))*ArcTanh[(Sqrt[b*e - a*f]*x)/(Sqrt[ e]*Sqrt[a + b*x^2])])/(2*e^(3/2)*(b*e - a*f)^(3/2)))/(4*e*(b*e - a*f))))/( a*(b*e - a*f)))/(3*a*(b*e - a*f))))/(b*c - a*d)^2
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]
Time = 4.10 (sec) , antiderivative size = 1063, normalized size of antiderivative = 1.48
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1063\) |
| default | \(\text {Expression too large to display}\) | \(8901\) |
Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
5/8/((a*f-b*e)*e)^(1/2)*(-3/5*(a*d-b*c)^2*a^2*(b*x^2+a)^(3/2)*((a*d-b*c)*c )^(1/2)*(f*x^2+e)^2*f^3*(a^2*c^2*f^4-10/3*a*c*(a*d+8/5*b*c)*e*f^3+5*(a^2*d ^2+4*a*b*c*d+16/5*b^2*c^2)*e^2*f^2-20*b*d*e^3*(a*d+2*b*c)*f+80/3*b^2*d^2*e ^4)*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2))+((a*f-b*e)*e)^(1/2)*(8 /5*a^2*(b*x^2+a)^(3/2)*d^5*(a*f-b*e)^4*(f*x^2+e)^2*e^2*arctan(c*(b*x^2+a)^ (1/2)/x/((a*d-b*c)*c)^(1/2))+(3/5*a^3*c*x^2*(b*x^2+a)^2*(a*d-b*c)^2*f^7+(( -7/5*x^2*d+c)*a-14/5*x^2*b*c)*(a*d-b*c)^2*a^2*(b*x^2+a)^2*e*f^6-9/5*a*(a^6 *d^3-2/9*a^5*b*c*d^2-23/9*d*(27/23*d^2*x^4-32/23*c*d*x^2+c^2)*b^2*a^4+16/9 *(-9/8*d^3*x^6+35/8*c*d^2*x^4-4*c^2*d*x^2+c^3)*b^3*a^3+32/9*(9/8*d^2*x^4-9 9/32*c*d*x^2+c^2)*c*b^4*x^2*a^2+16/3*c^2*b^5*x^4*(-83/72*x^2*d+c)*a+88/27* b^6*x^6*c^3)*e^2*f^5+4*(a^6*d^3-2*b*d^2*(-d*x^2+c)*a^5+b^2*d*(d^2*x^4-4*c* d*x^2+c^2)*a^4+6*b^3*c*d*x^2*(-d*x^2+c)*a^3-16/5*c*(7/6*d^2*x^4-107/48*c*d *x^2+c^2)*b^4*x^2*a^2-38/15*c^2*(-17/19*x^2*d+c)*b^5*x^4*a+4/15*b^6*x^6*c^ 3)*b*e^3*f^4+8*(d*(d^2*x^4-4*c*d*x^2+c^2)*a^3-4/5*b*(-7/6*d^3*x^6+14/3*c*d ^2*x^4-25/6*c^2*d*x^2+c^3)*a^2-1/3*c*(1/5*d^2*x^4-28/5*c*d*x^2+c^2)*b^2*x^ 2*a+4/15*c^2*x^4*b^3*(-d*x^2+c))*b^4*e^4*f^3-16*(d^2*(-d*x^2+c)*a^3-3/5*d* b*(11/9*d^2*x^4-14/9*c*d*x^2+c^2)*a^2-1/10*(-5/3*d^3*x^6+1/3*c*d^2*x^4+5/3 *c^2*d*x^2+c^3)*b^2*a-1/15*b^3*c*x^2*(d^2*x^4-4*c*d*x^2+c^2))*b^4*e^5*f^2+ 8*d*(a^3*d^2+2/15*x^2*a^2*b*d^2-2/5*b^2*(5/3*d^2*x^4-5/6*c*d*x^2+c^2)*a-4/ 15*b^3*c*x^2*(-d*x^2+c))*b^4*e^6*f-16/5*d^2*(d*a^2-1/2*(-5/3*x^2*d+c)*b...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)*(f*x^2 + e)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 3998 vs. \(2 (682) = 1364\).
Time = 17.59 (sec) , antiderivative size = 3998, normalized size of antiderivative = 5.55 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e)^3,x, algorithm="giac")
Output:
-sqrt(b)*d^5*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/ sqrt(-b^2*c^2 + a*b*c*d))/((b^2*c^2*d^3*e^3 - 2*a*b*c*d^4*e^3 + a^2*d^5*e^ 3 - 3*b^2*c^3*d^2*e^2*f + 6*a*b*c^2*d^3*e^2*f - 3*a^2*c*d^4*e^2*f + 3*b^2* c^4*d*e*f^2 - 6*a*b*c^3*d^2*e*f^2 + 3*a^2*c^2*d^3*e*f^2 - b^2*c^5*f^3 + 2* a*b*c^4*d*f^3 - a^2*c^3*d^2*f^3)*sqrt(-b^2*c^2 + a*b*c*d)) + 1/3*((2*b^14* c^3*e^5 - 9*a*b^13*c^2*d*e^5 + 12*a^2*b^12*c*d^2*e^5 - 5*a^3*b^11*d^3*e^5 - 19*a*b^13*c^3*e^4*f + 72*a^2*b^12*c^2*d*e^4*f - 87*a^3*b^11*c*d^2*e^4*f + 34*a^4*b^10*d^3*e^4*f + 56*a^2*b^12*c^3*e^3*f^2 - 198*a^3*b^11*c^2*d*e^3 *f^2 + 228*a^4*b^10*c*d^2*e^3*f^2 - 86*a^5*b^9*d^3*e^3*f^2 - 74*a^3*b^11*c ^3*e^2*f^3 + 252*a^4*b^10*c^2*d*e^2*f^3 - 282*a^5*b^9*c*d^2*e^2*f^3 + 104* a^6*b^8*d^3*e^2*f^3 + 46*a^4*b^10*c^3*e*f^4 - 153*a^5*b^9*c^2*d*e*f^4 + 16 8*a^6*b^8*c*d^2*e*f^4 - 61*a^7*b^7*d^3*e*f^4 - 11*a^5*b^9*c^3*f^5 + 36*a^6 *b^8*c^2*d*f^5 - 39*a^7*b^7*c*d^2*f^5 + 14*a^8*b^6*d^3*f^5)*x^2/(a^2*b^13* c^4*e^8 - 4*a^3*b^12*c^3*d*e^8 + 6*a^4*b^11*c^2*d^2*e^8 - 4*a^5*b^10*c*d^3 *e^8 + a^6*b^9*d^4*e^8 - 8*a^3*b^12*c^4*e^7*f + 32*a^4*b^11*c^3*d*e^7*f - 48*a^5*b^10*c^2*d^2*e^7*f + 32*a^6*b^9*c*d^3*e^7*f - 8*a^7*b^8*d^4*e^7*f + 28*a^4*b^11*c^4*e^6*f^2 - 112*a^5*b^10*c^3*d*e^6*f^2 + 168*a^6*b^9*c^2*d^ 2*e^6*f^2 - 112*a^7*b^8*c*d^3*e^6*f^2 + 28*a^8*b^7*d^4*e^6*f^2 - 56*a^5*b^ 10*c^4*e^5*f^3 + 224*a^6*b^9*c^3*d*e^5*f^3 - 336*a^7*b^8*c^2*d^2*e^5*f^3 + 224*a^8*b^7*c*d^3*e^5*f^3 - 56*a^9*b^6*d^4*e^5*f^3 + 70*a^6*b^9*c^4*e^...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,\left (d\,x^2+c\right )\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)*(e + f*x^2)^3),x)
Output:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)*(e + f*x^2)^3), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right ) \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e)^3,x)
Output:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)/(f*x^2+e)^3,x)