\(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^2 (e+f x^2)^3} \, dx\) [356]

Optimal result

Integrand size = 30, antiderivative size = 668 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {b \left (8 b^4 c e^2 (d e-c f)^3-a^4 d^2 f^3 \left (4 d^2 e^2+11 c d e f-3 c^2 f^2\right )+6 a^3 b d f^2 \left (2 d^3 e^3+3 c d^2 e^2 f+2 c^2 d e f^2-c^3 f^3\right )+2 a b^3 e \left (2 d^4 e^4+9 c^3 d e f^3-5 c^4 f^4\right )-3 a^2 b^2 f \left (4 d^4 e^4+12 c^2 d^2 e^2 f^2-3 c^3 d e f^3-c^4 f^4\right )\right ) x}{8 a c (b c-a d)^2 e^2 (b e-a f)^3 (d e-c f)^3 \sqrt {a+b x^2}}-\frac {d^4 x}{2 c (b c-a d) (d e-c f)^3 \sqrt {a+b x^2} \left (c+d x^2\right )}-\frac {f^3 x}{4 e (b e-a f) (d e-c f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {f^3 (a f (11 d e-3 c f)-8 b e (2 d e-c f)) x}{8 e^2 (b e-a f)^2 (d e-c f)^3 \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {d^4 (a d (d e-7 c f)-2 b c (2 d e-5 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{5/2} (d e-c f)^4}+\frac {f^3 \left (4 a b e f \left (25 d^2 e^2-16 c d e f+3 c^2 f^2\right )-a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )-8 b^2 e^2 \left (10 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{7/2} (d e-c f)^4} \] Output:

1/8*b*(8*b^4*c*e^2*(-c*f+d*e)^3-a^4*d^2*f^3*(-3*c^2*f^2+11*c*d*e*f+4*d^2*e 
^2)+6*a^3*b*d*f^2*(-c^3*f^3+2*c^2*d*e*f^2+3*c*d^2*e^2*f+2*d^3*e^3)+2*a*b^3 
*e*(-5*c^4*f^4+9*c^3*d*e*f^3+2*d^4*e^4)-3*a^2*b^2*f*(-c^4*f^4-3*c^3*d*e*f^ 
3+12*c^2*d^2*e^2*f^2+4*d^4*e^4))*x/a/c/(-a*d+b*c)^2/e^2/(-a*f+b*e)^3/(-c*f 
+d*e)^3/(b*x^2+a)^(1/2)-1/2*d^4*x/c/(-a*d+b*c)/(-c*f+d*e)^3/(b*x^2+a)^(1/2 
)/(d*x^2+c)-1/4*f^3*x/e/(-a*f+b*e)/(-c*f+d*e)^2/(b*x^2+a)^(1/2)/(f*x^2+e)^ 
2+1/8*f^3*(a*f*(-3*c*f+11*d*e)-8*b*e*(-c*f+2*d*e))*x/e^2/(-a*f+b*e)^2/(-c* 
f+d*e)^3/(b*x^2+a)^(1/2)/(f*x^2+e)+1/2*d^4*(a*d*(-7*c*f+d*e)-2*b*c*(-5*c*f 
+2*d*e))*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(3/2)/(-a*d 
+b*c)^(5/2)/(-c*f+d*e)^4+1/8*f^3*(4*a*b*e*f*(3*c^2*f^2-16*c*d*e*f+25*d^2*e 
^2)-a^2*f^2*(3*c^2*f^2-14*c*d*e*f+35*d^2*e^2)-8*b^2*e^2*(3*c^2*f^2-10*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)^(7/2)/(-c*f+d*e)^4
 

Mathematica [A] (verified)

Time = 19.89 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\frac {1}{8} \left (x \sqrt {a+b x^2} \left (-\frac {8 b^5}{a (b c-a d)^2 (-b e+a f)^3 \left (a+b x^2\right )}-\frac {4 d^5}{c (b c-a d)^2 (-d e+c f)^3 \left (c+d x^2\right )}+\frac {2 f^4}{e (b e-a f)^2 (d e-c f)^2 \left (e+f x^2\right )^2}+\frac {f^4 (2 b e (9 d e-5 c f)+a f (-11 d e+3 c f))}{e^2 (b e-a f)^3 (d e-c f)^3 \left (e+f x^2\right )}\right )+\frac {4 d^4 (a d (d e-7 c f)+2 b c (-2 d e+5 c f)) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{c^{3/2} (-b c+a d)^{5/2} (d e-c f)^4}+\frac {f^3 \left (-4 a b e f \left (25 d^2 e^2-16 c d e f+3 c^2 f^2\right )+a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )+8 b^2 e^2 \left (10 d^2 e^2-10 c d e f+3 c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {-b e+a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{e^{5/2} (-b e+a f)^{7/2} (d e-c f)^4}\right ) \] Input:

Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)^2*(e + f*x^2)^3),x]
 

Output:

(x*Sqrt[a + b*x^2]*((-8*b^5)/(a*(b*c - a*d)^2*(-(b*e) + a*f)^3*(a + b*x^2) 
) - (4*d^5)/(c*(b*c - a*d)^2*(-(d*e) + c*f)^3*(c + d*x^2)) + (2*f^4)/(e*(b 
*e - a*f)^2*(d*e - c*f)^2*(e + f*x^2)^2) + (f^4*(2*b*e*(9*d*e - 5*c*f) + a 
*f*(-11*d*e + 3*c*f)))/(e^2*(b*e - a*f)^3*(d*e - c*f)^3*(e + f*x^2))) + (4 
*d^4*(a*d*(d*e - 7*c*f) + 2*b*c*(-2*d*e + 5*c*f))*ArcTan[(Sqrt[-(b*c) + a* 
d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(c^(3/2)*(-(b*c) + a*d)^(5/2)*(d*e - c*f 
)^4) + (f^3*(-4*a*b*e*f*(25*d^2*e^2 - 16*c*d*e*f + 3*c^2*f^2) + a^2*f^2*(3 
5*d^2*e^2 - 14*c*d*e*f + 3*c^2*f^2) + 8*b^2*e^2*(10*d^2*e^2 - 10*c*d*e*f + 
 3*c^2*f^2))*ArcTan[(Sqrt[-(b*e) + a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(e^ 
(5/2)*(-(b*e) + a*f)^(7/2)*(d*e - c*f)^4))/8
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx\)

\(\Big \downarrow \) 426

\(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 421

\(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}-\frac {b \int -\frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \int \frac {-b d x^2+b c-2 a d}{\left (b x^2+a\right )^{3/2} \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\int \frac {a (b d e+b c f-2 a d f)-4 b (b c-a d) f x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^3}dx}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {\int \frac {a \left (4 e (d e+2 c f) b^2-3 a f (5 d e+c f) b+6 a^2 d f^2\right )-2 b f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x^2}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {\frac {\int -\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {\int \frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right ) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}+\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 421

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \int \frac {\sqrt {b x^2+a} \left (d f x^2+2 d e-c f\right )}{\left (f x^2+e\right )^3}dx}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}-\frac {\int -\frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {f \left (2 b (3 d e-c f) x^2+a (7 d e-3 c f)\right )}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e f}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int \frac {2 b (3 d e-c f) x^2+a (7 d e-3 c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\frac {\int -\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {\int \frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 e (b e-a f)}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 422

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \left (\frac {d \int \frac {\sqrt {b x^2+a}}{d x^2+c}dx}{d e-c f}-\frac {f \int \frac {\sqrt {b x^2+a}}{f x^2+e}dx}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 301

\(\displaystyle \frac {b \left (\frac {d^2 \left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) (a f (7 d e-3 c f)-4 b e (2 d e-c f))}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {x \sqrt {a+b x^2} (a f (7 d e-3 c f)-2 b e (3 d e-c f))}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e}+\frac {x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2}\right )}{(d e-c f)^2}\right )}{(b c-a d)^2}+\frac {b \left (\frac {b x (b c-a d)}{a \sqrt {a+b x^2} \left (e+f x^2\right )^2 (b e-a f)}-\frac {\frac {-\frac {a \text {arctanh}\left (\frac {x \sqrt {b e-a f}}{\sqrt {e} \sqrt {a+b x^2}}\right ) \left (6 a^3 d f^3-a^2 b f^2 (3 c f+23 d e)+4 a b^2 e f (3 c f+10 d e)-8 b^3 e^2 (3 c f+d e)\right )}{2 e^{3/2} (b e-a f)^{3/2}}-\frac {f x \sqrt {a+b x^2} \left (6 a^3 d f^2-a^2 b f (3 c f+19 d e)-2 a b^2 e (d e-5 c f)+8 b^3 c e^2\right )}{2 e \left (e+f x^2\right ) (b e-a f)}}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} \left (-2 a^2 d f-a b (3 d e-c f)+4 b^2 c e\right )}{4 e \left (e+f x^2\right )^2 (b e-a f)}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {b \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )}dx}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {(b c-a d) \int \frac {1}{c-\frac {(b c-a d) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {(b e-a f) \int \frac {1}{e-\frac {(b e-a f) x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^3}dx}{b c-a d}\)

\(\Big \downarrow \) 426

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e}\right )}{(d e-c f)^2}\right ) d^2}{(b c-a d)^2}+\frac {b \left (\frac {b (b c-a d) x}{a (b e-a f) \sqrt {b x^2+a} \left (f x^2+e\right )^2}-\frac {\frac {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 d e+3 c f) b+6 a^3 d f^3\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{2 e^{3/2} (b e-a f)^{3/2}}}{4 e (b e-a f)}-\frac {f \left (-2 d f a^2-b (3 d e-c f) a+4 b^2 c e\right ) x \sqrt {b x^2+a}}{4 e (b e-a f) \left (f x^2+e\right )^2}}{a (b e-a f)}\right )}{(b c-a d)^2}\right )}{b c-a d}-\frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )^3}dx}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 421

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \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}\right )}{(d e-c f)^2}\right ) 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 {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 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)}-\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 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \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 )}{d e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \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}\right )}{(d e-c f)^2}\right ) 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 {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 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)}-\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 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \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 e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \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}\right )}{(d e-c f)^2}\right ) 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 {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 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)}-\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 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \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 e-c f}\right )}{b c-a d}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {b \left (\frac {\left (\frac {d^2 \left (\frac {d \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {b x^2+a}}\right )}{\sqrt {c} d}\right )}{d e-c f}-\frac {f \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{f}-\frac {\sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {b x^2+a}}\right )}{\sqrt {e} f}\right )}{d e-c f}\right )}{(d e-c f)^2}-\frac {f \left (\frac {(d e-c f) \sqrt {b x^2+a} x}{4 e \left (f x^2+e\right )^2}+\frac {-\frac {(a f (7 d e-3 c f)-2 b e (3 d e-c f)) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a (a f (7 d e-3 c f)-4 b e (2 d e-c f)) \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}\right )}{(d e-c f)^2}\right ) 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 {-\frac {f \left (6 d f^2 a^3-b f (19 d e+3 c f) a^2-2 b^2 e (d e-5 c f) a+8 b^3 c e^2\right ) \sqrt {b x^2+a} x}{2 e (b e-a f) \left (f x^2+e\right )}-\frac {a \left (-8 e^2 (d e+3 c f) b^3+4 a e f (10 d e+3 c f) b^2-a^2 f^2 (23 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)}-\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 \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \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 e-c f}\right )}{b c-a d}\)

Input:

Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)^2*(e + f*x^2)^3),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

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]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

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]
 

rule 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 402

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]
 

rule 421

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]
 

rule 422

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[-d/(b*c - a*d)   Int[(c + d*x^2)^q*(e + f*x^2)^r, 
 x], x] + Simp[b/(b*c - a*d)   Int[(c + d*x^2)^(q + 1)*((e + f*x^2)^r/(a + 
b*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LeQ[q, -1]
 

rule 426

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]
 

Maple [A] (verified)

Time = 17.48 (sec) , antiderivative size = 1119, normalized size of antiderivative = 1.68

method	result	size
pseudoelliptic	\(\text {Expression too large to display}\)	\(1119\)
default	\(\text {Expression too large to display}\)	\(6453\)

Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^2/(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*(b*x^2+a)^(1/2)*c*(d*x^2+c)*(a 
^2*c^2*f^4-14/3*a*c*(a*d+6/7*b*c)*e*f^3+35/3*(a^2*d^2+64/35*a*b*c*d+24/35* 
b^2*c^2)*e^2*f^2-100/3*d*(a*d+4/5*b*c)*b*e^3*f+80/3*b^2*d^2*e^4)*((a*d-b*c 
)*c)^(1/2)*(f*x^2+e)^2*f^3*arctan(e*(b*x^2+a)^(1/2)/x/((a*f-b*e)*e)^(1/2)) 
+((a*f-b*e)*e)^(1/2)*(28/5*a*(b*x^2+a)^(1/2)*d^4*(d*x^2+c)*(a*f-b*e)^3*(f* 
x^2+e)^2*e^2*(c*(a*d-10/7*b*c)*f-1/7*d*e*(a*d-4*b*c))*arctan(c*(b*x^2+a)^( 
1/2)/x/((a*d-b*c)*c)^(1/2))+(3/5*a^2*c^2*x^2*(d*x^2+c)*(b*x^2+a)*(a*d-b*c) 
^2*f^7+(a*d-b*c)^2*a*c*(b*x^2+a)*(d*x^2+c)*(-11/5*a*d*x^2+(-2*b*x^2+a)*c)* 
e*f^6-13/5*(4/13*a^4*x^4*(b*x^2+a)*d^5+a^3*c*(b*x^2+a)*x^2*(-18/13*b*x^2+a 
)*d^4+a^2*(36/13*b^2*x^4-32/13*a*b*x^2+a^2)*c^2*(b*x^2+a)*d^3-14/13*a*(9/7 
*b^2*x^4-25/14*a*b*x^2+a^2)*c^3*(b*x^2+a)*b*d^2-11/13*c^4*(-8/11*b^3*x^6+6 
/11*a*b^2*x^4+17/11*a^2*b*x^2+a^3)*b^2*d+12/13*c^5*(2/3*b^2*x^4+a*b*x^2+a^ 
2)*b^3)*e^2*f^5-8/5*(a^3*(b*x^2+a)*x^2*(-3/2*b*x^2+a)*d^5-5/2*a^3*b*c*x^2* 
(b*x^2+a)*d^4-5/2*a^2*b*c^2*(b*x^2+a)*(-2*b*x^2+a)*d^3+5*(-3/5*b^3*x^6-1/2 
*a*b^2*x^4+1/2*a^2*b*x^2+a^3)*c^3*b^2*d^2-5/2*c^4*(2/5*b^2*x^4+a*b*x^2+a^2 
)*b^3*d+2*b^5*c^5*x^2)*e^3*f^4-4/5*((3*a^2*b^3*x^6-3*a^3*b^2*x^4-5*a^4*b*x 
^2+a^5)*d^5+6*b^5*c^2*d^3*x^6-6*b^5*c^3*d^2*x^4-10*b^5*c^4*d*x^2+2*c^5*b^5 
)*e^4*f^3+12/5*d*((1/3*b^2*x^4-2*a*b*x^2+a^2)*a*(b*x^2+a)*d^4+2/3*b^4*c*d^ 
3*x^6-10/3*b^4*c^2*d^2*x^4-2*b^4*c^3*d*x^2+2*c^4*b^4)*b*e^5*f^2-12/5*(a*(- 
2/3*b*x^2+a)*(b*x^2+a)*d^3-4/3*b^3*c*d^2*x^4+2/3*x^2*b^3*c^2*d+2*b^3*c^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**2/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{2} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)^2*(f*x^2 + e)^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2307 vs. \(2 (629) = 1258\).

Time = 24.10 (sec) , antiderivative size = 2307, normalized size of antiderivative = 3.45 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x, algorithm="giac")
 

Output:

b^5*x/((a*b^5*c^2*e^3 - 2*a^2*b^4*c*d*e^3 + a^3*b^3*d^2*e^3 - 3*a^2*b^4*c^ 
2*e^2*f + 6*a^3*b^3*c*d*e^2*f - 3*a^4*b^2*d^2*e^2*f + 3*a^3*b^3*c^2*e*f^2 
- 6*a^4*b^2*c*d*e*f^2 + 3*a^5*b*d^2*e*f^2 - a^4*b^2*c^2*f^3 + 2*a^5*b*c*d* 
f^3 - a^6*d^2*f^3)*sqrt(b*x^2 + a)) + 1/2*(4*b^(3/2)*c*d^5*e - a*sqrt(b)*d 
^6*e - 10*b^(3/2)*c^2*d^4*f + 7*a*sqrt(b)*c*d^5*f)*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^3* 
d^4*e^4 - 2*a*b*c^2*d^5*e^4 + a^2*c*d^6*e^4 - 4*b^2*c^4*d^3*e^3*f + 8*a*b* 
c^3*d^4*e^3*f - 4*a^2*c^2*d^5*e^3*f + 6*b^2*c^5*d^2*e^2*f^2 - 12*a*b*c^4*d 
^3*e^2*f^2 + 6*a^2*c^3*d^4*e^2*f^2 - 4*b^2*c^6*d*e*f^3 + 8*a*b*c^5*d^2*e*f 
^3 - 4*a^2*c^4*d^3*e*f^3 + b^2*c^7*f^4 - 2*a*b*c^6*d*f^4 + a^2*c^5*d^2*f^4 
)*sqrt(-b^2*c^2 + a*b*c*d)) + 1/8*(80*b^(5/2)*d^2*e^4*f^3 - 80*b^(5/2)*c*d 
*e^3*f^4 - 100*a*b^(3/2)*d^2*e^3*f^4 + 24*b^(5/2)*c^2*e^2*f^5 + 64*a*b^(3/ 
2)*c*d*e^2*f^5 + 35*a^2*sqrt(b)*d^2*e^2*f^5 - 12*a*b^(3/2)*c^2*e*f^6 - 14* 
a^2*sqrt(b)*c*d*e*f^6 + 3*a^2*sqrt(b)*c^2*f^7)*arctan(1/2*((sqrt(b)*x - sq 
rt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b^3*d^4*e^9 
- 4*b^3*c*d^3*e^8*f - 3*a*b^2*d^4*e^8*f + 6*b^3*c^2*d^2*e^7*f^2 + 12*a*b^2 
*c*d^3*e^7*f^2 + 3*a^2*b*d^4*e^7*f^2 - 4*b^3*c^3*d*e^6*f^3 - 18*a*b^2*c^2* 
d^2*e^6*f^3 - 12*a^2*b*c*d^3*e^6*f^3 - a^3*d^4*e^6*f^3 + b^3*c^4*e^5*f^4 + 
 12*a*b^2*c^3*d*e^5*f^4 + 18*a^2*b*c^2*d^2*e^5*f^4 + 4*a^3*c*d^3*e^5*f^4 - 
 3*a*b^2*c^4*e^4*f^5 - 12*a^2*b*c^3*d*e^4*f^5 - 6*a^3*c^2*d^2*e^4*f^5 +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^2\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^2*(e + f*x^2)^3),x)
 

Output:

int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^2*(e + f*x^2)^3), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{2} \left (f \,x^{2}+e \right )^{3}}d x \] Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x)
 

Output:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^2/(f*x^2+e)^3,x)