Integrand size = 32, antiderivative size = 429 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\frac {b (3 b e-a f) x \sqrt {c+d x^2}}{2 e f^2 \sqrt {a+b x^2}}-\frac {(b e-a f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e f \left (e+f x^2\right )}-\frac {\sqrt {a} \sqrt {b} (3 b e-a f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{2 e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \sqrt {b} (3 d e-c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (a c f^2-b e (3 d e-2 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c e^2 f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/2*b*(-a*f+3*b*e)*x*(d*x^2+c)^(1/2)/e/f^2/(b*x^2+a)^(1/2)-1/2*(-a*f+b*e)* x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/f/(f*x^2+e)-1/2*a^(1/2)*b^(1/2)*(-a*f+ 3*b*e)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a* d/b/c)^(1/2))/e/f^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/2*a^ (3/2)*b^(1/2)*(-c*f+3*d*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)* x/a^(1/2)),(1-a*d/b/c)^(1/2))/c/e/f^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^ 2+a))^(1/2)+1/2*a^(3/2)*(a*c*f^2-b*e*(-2*c*f+3*d*e))*(d*x^2+c)^(1/2)*Ellip ticPi(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^( 1/2)/c/e^2/f^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 5.54 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\frac {i b c e f (-3 b e+a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (b e-a f) \left (i \sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )+b e (3 d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+\left (a c f^2+b e (-3 d e+2 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{2 \sqrt {\frac {b}{a}} e^2 f^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(e + f*x^2)^2,x]
Output:
(I*b*c*e*f*(-3*b*e + a*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x ^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*(b*e - a*f)*(I*Sqrt [b/a]*e*f^2*x*(a + b*x^2)*(c + d*x^2) + b*e*(3*d*e + c*f)*Sqrt[1 + (b*x^2) /a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d )/(b*c)] + (a*c*f^2 + b*e*(-3*d*e + 2*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + ( d*x^2)/c]*(e + f*x^2)*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d )/(b*c)]))/(2*Sqrt[b/a]*e^2*f^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2 ))
Time = 1.03 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.69, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {425, 410, 324, 320, 388, 313, 414, 423, 406, 320, 388, 313, 413, 413, 412}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{f x^2+e}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 410 |
| \(\displaystyle \frac {b \left (\frac {b \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a}}dx}{f}-\frac {(b e-a f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \frac {b \left (\frac {b \left (c \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+d \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (d \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\sqrt {b x^2+a} \sqrt {d x^2+c}}{\left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 423 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b d \int \frac {e-f x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{2 e f^2}+\frac {1}{2} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b d \left (e \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e f^2}+\frac {1}{2} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e f^2}+\frac {1}{2} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )\right )}{2 e f^2}+\frac {1}{2} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {1}{2} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx+\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{2 e f^2}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {\sqrt {\frac {b x^2}{a}+1} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 \sqrt {a+b x^2}}+\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{2 e f^2}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{2 e f^2}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {\sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (\frac {a c}{e}-\frac {b d e}{f^2}\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{2 \sqrt {b} e \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{2 e f^2}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right )}\right )}{f}\) |
Input:
Int[((a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(e + f*x^2)^2,x]
Output:
-(((b*e - a*f)*((x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*e*(e + f*x^2)) + (b *d*(-(f*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2 ]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt [(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))) + (Sqrt[c]*e*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d ]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(2*e*f^2) + (Sq rt[-a]*((a*c)/e - (b*d*e)/f^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ell ipticPi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a*d)/(b*c)])/(2*Sqrt[b ]*e*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])))/f) + (b*((b*(d*((x*Sqrt[a + b*x^2]) /(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]* x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x ^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[ d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/f - (c^(3/2)*(b*e - a*f)*Sqrt[a + b*x^2]*Ellip ticPi[1 - (c*f)/(d*e), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*S qrt[d]*e*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/f
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_ )^2), x_Symbol] :> Simp[d/b Int[Sqrt[e + f*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*c - a*d)/b Int[Sqrt[e + f*x^2]/((a + b*x^2)*Sqrt[c + d*x^2]), x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && !SimplerSqrtQ[-f/e, -d/c]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
Int[(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_ )^2)^2, x_Symbol] :> Simp[x*Sqrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(a + b*x^ 2))), x] + (Simp[(b^2*c*e - a^2*d*f)/(2*a*b^2) Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Simp[d*(f/(2*a*b^2)) Int[(a - b*x^2)/ (Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> Simp[d/b Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(a + b*x^2)^p*(c + d*x ^2)^(q - 1)*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && ILt Q[p, 0] && GtQ[q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1167\) vs. \(2(397)=794\).
Time = 5.14 (sec) , antiderivative size = 1168, normalized size of antiderivative = 2.72
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1168\) |
| default | \(\text {Expression too large to display}\) | \(1689\) |
Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/2*(a*f-b*e) /e/f*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)-1/2/(-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)*Ellipt icF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*b^2/f^2*c+1/2/e^2/(-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)* EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a^2*c-1/f^2 /(-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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)) *b^2*c+3/2*e/f^3/(-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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/ 2)/(-b/a)^(1/2))*b^2*d+1/2*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)/f*b/e*a*EllipticF(x*(-b/a)^(1/2),( -1+(a*d+b*c)/c/b)^(1/2))-1/2*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)/f*b/e*a*EllipticE(x*(-b/a)^(1/2) ,(-1+(a*d+b*c)/c/b)^(1/2))+3/2/(-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))*b/f^2*a*d-3/2/(-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))*d*b^2/f^3*e+3/2*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)*b^2/f^2*EllipticE(x*(...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}{\left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral((a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(e + f*x**2)**2, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)/(f*x^2 + e)^2, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)/(f*x^2 + e)^2, x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}}{{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^2,x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2))/(e + f*x^2)^2, x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{\left (e+f x^2\right )^2} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
(2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*x + sqrt(c + d*x**2)*sqrt(a + b*x **2)*b*c*x - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2* a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x **6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b *d*e*f*x**6 + b*d*f**2*x**8),x)*a*b*d**2*e*f - 2*int((sqrt(c + d*x**2)*sqr t(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2* x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b *c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*a*b*d**2 *f**2*x**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a* c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x** 6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d *e*f*x**6 + b*d*f**2*x**8),x)*b**2*c*d*e*f - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f **2*x**6 + b*d*e**2*x**4 + 2*b*d*e*f*x**6 + b*d*f**2*x**8),x)*b**2*c*d*f** 2*x**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c* e*f*x**2 + a*c*f**2*x**4 + a*d*e**2*x**2 + 2*a*d*e*f*x**4 + a*d*f**2*x**6 + b*c*e**2*x**2 + 2*b*c*e*f*x**4 + b*c*f**2*x**6 + b*d*e**2*x**4 + 2*b*d*e *f*x**6 + b*d*f**2*x**8),x)*b**2*d**2*e**2 + 3*int((sqrt(c + d*x**2)*sqrt( a + b*x**2)*x**6)/(a*c*e**2 + 2*a*c*e*f*x**2 + a*c*f**2*x**4 + a*d*e**2...