\(\int \frac {(e+f x^2)^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx\) [208]

Optimal result

Integrand size = 36, antiderivative size = 326 \[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=-\frac {e^2 \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 a c x^3}-\frac {2 e (b c e-a d e+3 a c f) \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 a^2 c^2 x}-\frac {2 \sqrt {b} e (b c e-a d e+3 a c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\left (2 b^2 c e^2+3 a^2 c f^2-a b e (d e-6 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} c \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

-1/3*e^2*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/x^3-2/3*e*(3*a*c*f-a*d*e+b*c 
*e)*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/x-2/3*b^(1/2)*e*(3*a*c*f-a*d* 
e+b*c*e)*(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a 
*d/b/c)^(1/2))/a^(3/2)/c^2/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+1/3*(2*b^2*c 
*e^2+3*a^2*c*f^2-a*b*e*(-6*c*f+d*e))*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*E 
llipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/a^(3/2)/b^(1/2)/c/(-b*x^2+a)^ 
(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.67 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.87 \[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {-\frac {b}{a}} e \left (-a+b x^2\right ) \left (c+d x^2\right ) \left (2 b c e x^2-2 a d e x^2+a c \left (e+6 f x^2\right )\right )+2 i b c e (b c e-a d e+3 a c f) x^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c \left (2 b^2 c e^2+3 a^2 c f^2+a b e (-d e+6 c f)\right ) x^3 \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{3 a^2 \sqrt {-\frac {b}{a}} c^2 x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(e + f*x^2)^2/(x^4*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

(Sqrt[-(b/a)]*e*(-a + b*x^2)*(c + d*x^2)*(2*b*c*e*x^2 - 2*a*d*e*x^2 + a*c* 
(e + 6*f*x^2)) + (2*I)*b*c*e*(b*c*e - a*d*e + 3*a*c*f)*x^3*Sqrt[1 - (b*x^2 
)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c 
))] - I*c*(2*b^2*c*e^2 + 3*a^2*c*f^2 + a*b*e*(-(d*e) + 6*c*f))*x^3*Sqrt[1 
- (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a 
*d)/(b*c))])/(3*a^2*Sqrt[-(b/a)]*c^2*x^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 1.22 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.71, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {448, 445, 25, 399, 323, 323, 321, 331, 330, 327, 445, 25, 27, 399, 323, 323, 321, 331, 330, 327}

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.

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

\(\Big \downarrow \) 448

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {f \left (\frac {\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} e \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a c}-\frac {e \sqrt {a-b x^2} \sqrt {c+d x^2}}{a c x}\right )}{e^2}+e \left (\frac {\frac {b d \left (\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (3 a c f-a d e+2 b c e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (3 a c f-2 a d e+2 b c e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}\right )}{a c}-\frac {\sqrt {a-b x^2} \sqrt {c+d x^2} (3 a c f-2 a d e+2 b c e)}{a c x}}{3 a c}-\frac {e \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 a c x^3}\right )\)

Input:

Int[(e + f*x^2)^2/(x^4*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

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

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 448

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e   Int[(g*x)^m*(a + b*x 
^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2   Int[(g*x)^(m 
 + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, 
b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
 

Maple [A] (verified)

Time = 8.59 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.18

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e^{2} \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 a c \,x^{3}}-\frac {2 e \left (3 a c f -a d e +b c e \right ) \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 a^{2} c^{2} x}+\frac {\left (f^{2}+\frac {b d \,e^{2}}{3 a c}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {2 b e \left (3 a c f -a d e +b c e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{3 a^{2} c \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(385\)
risch	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, e \left (6 a c f \,x^{2}-2 a d e \,x^{2}+2 b c e \,x^{2}+a c e \right )}{3 a^{2} c^{2} x^{3}}+\frac {\left (\frac {2 b e \left (3 a c f -a d e +b c e \right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {3 a^{2} c^{2} f^{2} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {a b c d \,e^{2} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a^{2} c^{2} \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(444\)
default	\(\frac {\left (6 \sqrt {\frac {b}{a}}\, a b c d e f \,x^{6}-2 \sqrt {\frac {b}{a}}\, a b \,d^{2} e^{2} x^{6}+2 \sqrt {\frac {b}{a}}\, b^{2} c d \,e^{2} x^{6}+3 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a^{2} c^{2} f^{2} x^{3}+6 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} e f \,x^{3}-\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b c d \,e^{2} x^{3}+2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{2} e^{2} x^{3}-6 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} e f \,x^{3}+2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b c d \,e^{2} x^{3}-2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{2} e^{2} x^{3}-6 \sqrt {\frac {b}{a}}\, a^{2} c d e f \,x^{4}+2 \sqrt {\frac {b}{a}}\, a^{2} d^{2} e^{2} x^{4}+6 \sqrt {\frac {b}{a}}\, a b \,c^{2} e f \,x^{4}-3 \sqrt {\frac {b}{a}}\, a b c d \,e^{2} x^{4}+2 \sqrt {\frac {b}{a}}\, b^{2} c^{2} e^{2} x^{4}-6 \sqrt {\frac {b}{a}}\, a^{2} c^{2} e f \,x^{2}+\sqrt {\frac {b}{a}}\, a^{2} c d \,e^{2} x^{2}-\sqrt {\frac {b}{a}}\, a b \,c^{2} e^{2} x^{2}-\sqrt {\frac {b}{a}}\, a^{2} c^{2} e^{2}\right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{3 \sqrt {\frac {b}{a}}\, x^{3} c^{2} a^{2} \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(732\)

Input:

int((f*x^2+e)^2/x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBO 
SE)
                                                                                    
                                                                                    
 

Output:

((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/3*e^2/a/ 
c*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)/x^3-2/3*e*(3*a*c*f-a*d*e+b*c*e)/a^2 
/c^2*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)/x+(f^2+1/3*b*d*e^2/a/c)/(b/a)^(1 
/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/ 
2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))+2/3*b*e*(3*a*c*f-a*d* 
e+b*c*e)/a^2/c/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a 
*d*x^2-b*c*x^2+a*c)^(1/2)*(EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2 
))-EllipticE(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.69 \[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (3 \, a b^{2} c e f + {\left (b^{3} c - a b^{2} d\right )} e^{2}\right )} \sqrt {a c} x^{3} \sqrt {\frac {b}{a}} E(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c}) - {\left (6 \, a b^{2} c e f + 3 \, a^{3} c f^{2} + {\left (2 \, b^{3} c + {\left (a^{2} b - 2 \, a b^{2}\right )} d\right )} e^{2}\right )} \sqrt {a c} x^{3} \sqrt {\frac {b}{a}} F(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c}) + {\left (a^{2} b c e^{2} + 2 \, {\left (3 \, a^{2} b c e f + {\left (a b^{2} c - a^{2} b d\right )} e^{2}\right )} x^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, a^{3} b c^{2} x^{3}} \] Input:

integrate((f*x^2+e)^2/x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="f 
ricas")
 

Output:

-1/3*(2*(3*a*b^2*c*e*f + (b^3*c - a*b^2*d)*e^2)*sqrt(a*c)*x^3*sqrt(b/a)*el 
liptic_e(arcsin(x*sqrt(b/a)), -a*d/(b*c)) - (6*a*b^2*c*e*f + 3*a^3*c*f^2 + 
 (2*b^3*c + (a^2*b - 2*a*b^2)*d)*e^2)*sqrt(a*c)*x^3*sqrt(b/a)*elliptic_f(a 
rcsin(x*sqrt(b/a)), -a*d/(b*c)) + (a^2*b*c*e^2 + 2*(3*a^2*b*c*e*f + (a*b^2 
*c - a^2*b*d)*e^2)*x^2)*sqrt(-b*x^2 + a)*sqrt(d*x^2 + c))/(a^3*b*c^2*x^3)
 

Sympy [F]

\[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{x^{4} \sqrt {a - b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate((f*x**2+e)**2/x**4/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral((e + f*x**2)**2/(x**4*sqrt(a - b*x**2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} x^{4}} \,d x } \] Input:

integrate((f*x^2+e)^2/x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="m 
axima")
 

Output:

integrate((f*x^2 + e)^2/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*x^4), x)
 

Giac [F]

\[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} x^{4}} \,d x } \] Input:

integrate((f*x^2+e)^2/x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="g 
iac")
 

Output:

integrate((f*x^2 + e)^2/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{x^4\,\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((e + f*x^2)^2/(x^4*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((e + f*x^2)^2/(x^4*(a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\left (e+f x^2\right )^2}{x^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

int((f*x^2+e)^2/x^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
 

Output:

( - sqrt(c + d*x**2)*sqrt(a - b*x**2)*e*f - 3*int((sqrt(c + d*x**2)*sqrt(a 
 - b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 - a*b*c**2*x**4 - 2*a*b*c*d*x* 
*6 - a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8),x)*a**2*c*d*e*f*x**3 
+ int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 
- a*b*c**2*x**4 - 2*a*b*c*d*x**6 - a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c 
*d*x**8),x)*a**2*d**2*e**2*x**3 + 3*int((sqrt(c + d*x**2)*sqrt(a - b*x**2) 
)/(a**2*c*d*x**4 + a**2*d**2*x**6 - a*b*c**2*x**4 - 2*a*b*c*d*x**6 - a*b*d 
**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8),x)*a*b*c**2*e*f*x**3 - 2*int((s 
qrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 - a*b*c* 
*2*x**4 - 2*a*b*c*d*x**6 - a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8) 
,x)*a*b*c*d*e**2*x**3 + int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c*d* 
x**4 + a**2*d**2*x**6 - a*b*c**2*x**4 - 2*a*b*c*d*x**6 - a*b*d**2*x**8 + b 
**2*c**2*x**6 + b**2*c*d*x**8),x)*b**2*c**2*e**2*x**3 + int((sqrt(c + d*x* 
*2)*sqrt(a - b*x**2))/(a**2*c*d + a**2*d**2*x**2 - a*b*c**2 - 2*a*b*c*d*x* 
*2 - a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a**2*d**2*f**2*x** 
3 - 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c*d + a**2*d**2*x**2 - 
 a*b*c**2 - 2*a*b*c*d*x**2 - a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x** 
4),x)*a*b*c*d*f**2*x**3 + int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c* 
d + a**2*d**2*x**2 - a*b*c**2 - 2*a*b*c*d*x**2 - a*b*d**2*x**4 + b**2*c**2 
*x**2 + b**2*c*d*x**4),x)*a*b*d**2*e*f*x**3 + int((sqrt(c + d*x**2)*sqr...