\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^2 (e+f x^2)^3} \, dx\) [335]

Optimal result

Integrand size = 30, antiderivative size = 425 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {d^4 x \sqrt {a+b x^2}}{2 c (b c-a d) (d e-c f)^3 \left (c+d x^2\right )}-\frac {f^3 x \sqrt {a+b x^2}}{4 e (b e-a f) (d e-c f)^2 \left (e+f x^2\right )^2}-\frac {f^3 (2 b e (7 d e-3 c f)-a f (11 d e-3 c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f)^2 (d e-c f)^3 \left (e+f x^2\right )}-\frac {d^3 (a d (d e-7 c f)-2 b c (d e-4 c f)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{2 c^{3/2} (b c-a d)^{3/2} (d e-c f)^4}-\frac {f^2 \left (8 a b e f \left (10 d^2 e^2-5 c d e f+c^2 f^2\right )-8 b^2 e^2 \left (6 d^2 e^2-4 c d e f+c^2 f^2\right )-a^2 f^2 \left (35 d^2 e^2-14 c d e f+3 c^2 f^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{8 e^{5/2} (b e-a f)^{5/2} (d e-c f)^4} \] Output:

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

Mathematica [A] (verified)

Time = 14.91 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=-\frac {\frac {4 d^3 (d e-c f) x \left (d \left (a+b x^2\right )-\frac {(2 b c-a d) \left (c+d x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{c \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}\right )}{c (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )}+\frac {8 d f^2 (d e-c f) x \left (f \left (a+b x^2\right )-\frac {(2 b e-a f) \left (e+f x^2\right ) \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{e \sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e (b e-a f) \sqrt {a+b x^2} \left (e+f x^2\right )}+\frac {f^2 (d e-c f)^2 x \left (e f \left (a+b x^2\right ) \left (2 b e \left (4 e+3 f x^2\right )-a f \left (5 e+3 f x^2\right )\right )-\frac {\left (8 b^2 e^2-8 a b e f+3 a^2 f^2\right ) \left (e+f x^2\right )^2 \text {arctanh}\left (\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b e-a f) x^2}{e \left (a+b x^2\right )}}}\right )}{e^3 (b e-a f)^2 \sqrt {a+b x^2} \left (e+f x^2\right )^2}+\frac {24 d^3 f \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} \sqrt {b c-a d}}-\frac {24 d^2 f^2 \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt {e} \sqrt {a+b x^2}}\right )}{\sqrt {e} \sqrt {b e-a f}}}{8 (d e-c f)^4} \] Input:

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

Output:

-1/8*((4*d^3*(d*e - c*f)*x*(d*(a + b*x^2) - ((2*b*c - a*d)*(c + d*x^2)*Arc 
Tanh[Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]])/(c*Sqrt[((b*c - a*d)*x^2)/( 
c*(a + b*x^2))])))/(c*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)) + (8*d*f^2* 
(d*e - c*f)*x*(f*(a + b*x^2) - ((2*b*e - a*f)*(e + f*x^2)*ArcTanh[Sqrt[((b 
*e - a*f)*x^2)/(e*(a + b*x^2))]])/(e*Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2) 
)])))/(e*(b*e - a*f)*Sqrt[a + b*x^2]*(e + f*x^2)) + (f^2*(d*e - c*f)^2*x*( 
e*f*(a + b*x^2)*(2*b*e*(4*e + 3*f*x^2) - a*f*(5*e + 3*f*x^2)) - ((8*b^2*e^ 
2 - 8*a*b*e*f + 3*a^2*f^2)*(e + f*x^2)^2*ArcTanh[Sqrt[((b*e - a*f)*x^2)/(e 
*(a + b*x^2))]])/Sqrt[((b*e - a*f)*x^2)/(e*(a + b*x^2))]))/(e^3*(b*e - a*f 
)^2*Sqrt[a + b*x^2]*(e + f*x^2)^2) + (24*d^3*f*ArcTanh[(Sqrt[b*c - a*d]*x) 
/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt[c]*Sqrt[b*c - a*d]) - (24*d^2*f^2*ArcTa 
nh[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])])/(Sqrt[e]*Sqrt[b*e - a*f 
]))/(d*e - c*f)^4
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.97, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {426, 421, 402, 25, 402, 25, 27, 291, 221, 407, 291, 221, 426, 421, 25, 291, 221, 402, 25, 27, 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.

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (\frac {\int -\frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^2 \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \left (\frac {d^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right ) \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {f \left (-\frac {\int \frac {2 b f (d e-c f) x^2+a f (7 d e-3 c f)-4 b e (2 d e-c f)}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{4 e (b e-a f)}-\frac {f x \sqrt {a+b x^2} (d e-c f)}{4 e \left (e+f x^2\right )^2 (b e-a f)}\right )}{(d e-c f)^2}\right )}{d e-c f}\)

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 407

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 426

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((f*((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))/(d*e - c*f)) + (d*((d*((d*(-1/2*(d*(d*e - c*f)*x*Sqrt[a + b* 
x^2])/(c*(b*c - a*d)*(c + d*x^2)) - ((a*d*(d*e - 3*c*f) - 2*b*c*(d*e - 2*c 
*f))*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(2*c^(3/2)*(b 
*c - a*d)^(3/2))))/(d*e - c*f)^2 + (f^2*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)^2)))/(d*e - c*f 
) - (f*((d^2*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(Sqrt 
[c]*Sqrt[b*c - a*d]*(d*e - c*f)^2) - (f*(-1/2*(f*(d*e - c*f)*x*Sqrt[a + b* 
x^2])/(e*(b*e - a*f)*(e + f*x^2)) + ((2*b*e*(2*d*e - c*f) - a*f*(3*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))))/(d*e - c*f)^2))/(d*e - c*f)))/(d*e - c*f)
 

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 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 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 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 407

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]
 

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 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 = 2.30 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.50

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (a d -b c \right ) \left (a^{2} c^{2} f^{4}-\frac {14 a c \left (a d +\frac {4 b c}{7}\right ) e \,f^{3}}{3}+\frac {35 \left (a^{2} d^{2}+\frac {8}{7} a b c d +\frac {8}{35} b^{2} c^{2}\right ) e^{2} f^{2}}{3}-\frac {80 \left (a d +\frac {2 b c}{5}\right ) d b \,e^{3} f}{3}+16 b^{2} d^{2} e^{4}\right ) c \left (x^{2} d +c \right ) \sqrt {\left (a d -b c \right ) c}\, \left (f \,x^{2}+e \right )^{2} f^{2} \arctan \left (\frac {e \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a f -b e \right ) e}}\right )-\frac {28 \sqrt {\left (a f -b e \right ) e}\, \left (d^{3} \left (x^{2} d +c \right ) \left (a f -b e \right )^{2} \left (f \,x^{2}+e \right )^{2} \left (c \left (a d -\frac {8 b c}{7}\right ) f -\frac {d e \left (a d -2 b c \right )}{7}\right ) e^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )+\frac {5 \left (-\frac {4 b^{2} d^{4} e^{6}}{5}+\frac {8 b \,d^{4} f \left (-b \,x^{2}+a \right ) e^{5}}{5}-\frac {4 d^{4} f^{2} \left (b^{2} x^{4}-4 a b \,x^{2}+a^{2}\right ) e^{4}}{5}-\frac {8 d \left (2 b^{2} c^{3}-2 b d \left (-b \,x^{2}+a \right ) c^{2}-2 a b c \,d^{2} x^{2}+a \,d^{3} x^{2} \left (-b \,x^{2}+a \right )\right ) f^{3} e^{3}}{5}-\frac {13 \left (-\frac {8 b^{2} c^{4}}{13}-\frac {5 \left (-\frac {6 b \,x^{2}}{5}+a \right ) d b \,c^{3}}{13}+d^{2} \left (\frac {14}{13} b^{2} x^{4}-\frac {19}{13} a b \,x^{2}+a^{2}\right ) c^{2}+a \,d^{3} x^{2} \left (-\frac {14 b \,x^{2}}{13}+a \right ) c +\frac {4 a^{2} d^{4} x^{4}}{13}\right ) f^{4} e^{2}}{5}+\left (a d -b c \right ) \left (\left (-\frac {6 b \,x^{2}}{5}+a \right ) c -\frac {11 a d \,x^{2}}{5}\right ) c \left (x^{2} d +c \right ) f^{5} e +\frac {3 a \,c^{2} f^{6} x^{2} \left (x^{2} d +c \right ) \left (a d -b c \right )}{5}\right ) \left (c f -d e \right ) \sqrt {\left (a d -b c \right ) c}\, \sqrt {b \,x^{2}+a}\, x}{28}\right )}{3}\right )}{8 \sqrt {\left (a d -b c \right ) c}\, \sqrt {\left (a f -b e \right ) e}\, \left (a f -b e \right )^{2} \left (f \,x^{2}+e \right )^{2} e^{2} \left (c f -d e \right )^{4} \left (a d -b c \right ) c \left (x^{2} d +c \right )}\)	\(636\)
default	\(\text {Expression too large to display}\)	\(3165\)

Input:

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

Output:

-3/8*((a*d-b*c)*(a^2*c^2*f^4-14/3*a*c*(a*d+4/7*b*c)*e*f^3+35/3*(a^2*d^2+8/ 
7*a*b*c*d+8/35*b^2*c^2)*e^2*f^2-80/3*(a*d+2/5*b*c)*d*b*e^3*f+16*b^2*d^2*e^ 
4)*c*(d*x^2+c)*((a*d-b*c)*c)^(1/2)*(f*x^2+e)^2*f^2*arctan(e*(b*x^2+a)^(1/2 
)/x/((a*f-b*e)*e)^(1/2))-28/3*((a*f-b*e)*e)^(1/2)*(d^3*(d*x^2+c)*(a*f-b*e) 
^2*(f*x^2+e)^2*(c*(a*d-8/7*b*c)*f-1/7*d*e*(a*d-2*b*c))*e^2*arctan(c*(b*x^2 
+a)^(1/2)/x/((a*d-b*c)*c)^(1/2))+5/28*(-4/5*b^2*d^4*e^6+8/5*b*d^4*f*(-b*x^ 
2+a)*e^5-4/5*d^4*f^2*(b^2*x^4-4*a*b*x^2+a^2)*e^4-8/5*d*(2*b^2*c^3-2*b*d*(- 
b*x^2+a)*c^2-2*a*b*c*d^2*x^2+a*d^3*x^2*(-b*x^2+a))*f^3*e^3-13/5*(-8/13*b^2 
*c^4-5/13*(-6/5*b*x^2+a)*d*b*c^3+d^2*(14/13*b^2*x^4-19/13*a*b*x^2+a^2)*c^2 
+a*d^3*x^2*(-14/13*b*x^2+a)*c+4/13*a^2*d^4*x^4)*f^4*e^2+(a*d-b*c)*((-6/5*b 
*x^2+a)*c-11/5*a*d*x^2)*c*(d*x^2+c)*f^5*e+3/5*a*c^2*f^6*x^2*(d*x^2+c)*(a*d 
-b*c))*(c*f-d*e)*((a*d-b*c)*c)^(1/2)*(b*x^2+a)^(1/2)*x))/((a*d-b*c)*c)^(1/ 
2)/((a*f-b*e)*e)^(1/2)/(a*f-b*e)^2/(f*x^2+e)^2/e^2/(c*f-d*e)^4/(a*d-b*c)/c 
/(d*x^2+c)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate(1/(sqrt(b*x^2 + a)*(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. 1833 vs. \(2 (389) = 778\).

Time = 3.67 (sec) , antiderivative size = 1833, normalized size of antiderivative = 4.31 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^2 \left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*b^(9/2)*(4*(2*b*c*d^4*e - a*d^5*e - 8*b*c^2*d^3*f + 7*a*c*d^4*f)*arct 
an(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^5*c^2*d^4*e^4 - a*b^4*c*d^5*e^4 - 4*b^5*c^3*d^3*e^3*f + 4*a*b 
^4*c^2*d^4*e^3*f + 6*b^5*c^4*d^2*e^2*f^2 - 6*a*b^4*c^3*d^3*e^2*f^2 - 4*b^5 
*c^5*d*e*f^3 + 4*a*b^4*c^4*d^2*e*f^3 + b^5*c^6*f^4 - a*b^4*c^5*d*f^4)*sqrt 
(-b^2*c^2 + a*b*c*d)) + (48*b^2*d^2*e^4*f^2 - 32*b^2*c*d*e^3*f^3 - 80*a*b* 
d^2*e^3*f^3 + 8*b^2*c^2*e^2*f^4 + 40*a*b*c*d*e^2*f^4 + 35*a^2*d^2*e^2*f^4 
- 8*a*b*c^2*e*f^5 - 14*a^2*c*d*e*f^5 + 3*a^2*c^2*f^6)*arctan(1/2*((sqrt(b) 
*x - sqrt(b*x^2 + a))^2*f + 2*b*e - a*f)/sqrt(-b^2*e^2 + a*b*e*f))/((b^6*d 
^4*e^8 - 4*b^6*c*d^3*e^7*f - 2*a*b^5*d^4*e^7*f + 6*b^6*c^2*d^2*e^6*f^2 + 8 
*a*b^5*c*d^3*e^6*f^2 + a^2*b^4*d^4*e^6*f^2 - 4*b^6*c^3*d*e^5*f^3 - 12*a*b^ 
5*c^2*d^2*e^5*f^3 - 4*a^2*b^4*c*d^3*e^5*f^3 + b^6*c^4*e^4*f^4 + 8*a*b^5*c^ 
3*d*e^4*f^4 + 6*a^2*b^4*c^2*d^2*e^4*f^4 - 2*a*b^5*c^4*e^3*f^5 - 4*a^2*b^4* 
c^3*d*e^3*f^5 + a^2*b^4*c^4*e^2*f^6)*sqrt(-b^2*e^2 + a*b*e*f)) + 8*(2*(sqr 
t(b)*x - sqrt(b*x^2 + a))^2*b*c*d^3 - (sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d^ 
4 + a^2*d^4)/((b^5*c^2*d^3*e^3 - a*b^4*c*d^4*e^3 - 3*b^5*c^3*d^2*e^2*f + 3 
*a*b^4*c^2*d^3*e^2*f + 3*b^5*c^4*d*e*f^2 - 3*a*b^4*c^3*d^2*e*f^2 - b^5*c^5 
*f^3 + a*b^4*c^4*d*f^3)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x 
- sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)) 
 + 2*(24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^2*d*e^3*f^3 - 8*(sqrt(b)*x -...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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