Integrand size = 32, antiderivative size = 429 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\frac {d (3 d e-c f) x \sqrt {a+b x^2}}{2 e f^2 \sqrt {c+d x^2}}-\frac {(d e-c f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e f \left (e+f x^2\right )}-\frac {\sqrt {c} \sqrt {d} (3 d e-c f) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{2 e f^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} \sqrt {d} (3 b e-a f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a e f^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} \left (3 b d e^2-a f (2 d e+c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{2 a \sqrt {d} e^2 f^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/2*d*(-c*f+3*d*e)*x*(b*x^2+a)^(1/2)/e/f^2/(d*x^2+c)^(1/2)-1/2*(-c*f+d*e)* x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/f/(f*x^2+e)-1/2*c^(1/2)*d^(1/2)*(-c*f+ 3*d*e)*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b* c/a/d)^(1/2))/e/f^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/2*c^ (3/2)*d^(1/2)*(-a*f+3*b*e)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)* x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/e/f^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x ^2+c)^(1/2)-1/2*c^(3/2)*(3*b*d*e^2-a*f*(c*f+2*d*e))*(b*x^2+a)^(1/2)*Ellipt icPi(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^ (1/2)/e^2/f^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 5.32 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\frac {i b c e f (-3 d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i e \left (-3 b d^2 e^2+2 a d^2 e f+b c^2 f^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+(-d e+c f) \left (\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )-i \left (-3 b d e^2+a f (2 d e+c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{2 \sqrt {\frac {b}{a}} e^2 f^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2)^2,x]
Output:
(I*b*c*e*f*(-3*d*e + c*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*e*(-3*b*d^2*e^2 + 2 *a*d^2*e*f + b*c^2*f^2)*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)] + (-(d*e) + c*f)*(Sqrt[b/ a]*e*f^2*x*(a + b*x^2)*(c + d*x^2) - I*(-3*b*d*e^2 + a*f*(2*d*e + c*f))*Sq rt[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))
Leaf count is larger than twice the leaf count of optimal. \(1030\) vs. \(2(429)=858\).
Time = 1.49 (sec) , antiderivative size = 1030, normalized size of antiderivative = 2.40, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.656, Rules used = {425, 420, 324, 320, 388, 313, 414, 425, 414, 425, 413, 413, 412, 424, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 420 |
| \(\displaystyle \frac {b \left (\frac {d \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a}}dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \frac {b \left (\frac {d \left (c \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+d \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {d \left (d \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {d \left (d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {d \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {(d e-c f) \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\) |
| \(\Big \downarrow \) 425 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {(d e-c f) \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {(d e-c f) \left (\frac {d \sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f \sqrt {a+b x^2}}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {(d e-c f) \left (\frac {d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{f \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{f}-\frac {c^{3/2} \sqrt {a+b x^2} (d e-c f) \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {(d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 424 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}-\frac {b d \int \frac {f x^2+e}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{2 e (b e-a f) (d e-c f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (e \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \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 {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \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 e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \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 {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \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 e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \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 {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \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 e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \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 {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \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 e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f) \sqrt {b x^2+a}}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \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 {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \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 e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \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 e (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{f}\right )}{f}\right )}{f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b \left (\frac {d \left (\frac {\sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) c^{3/2}}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+d \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )\right )}{f}-\frac {c^{3/2} (d e-c f) \sqrt {b x^2+a} \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 {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{f}-\frac {(b e-a f) \left (\frac {c^{3/2} \sqrt {d} \sqrt {b x^2+a} \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 e f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {\sqrt {-a} d \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {(d e-c f) \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {\sqrt {-a} (b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \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^2 (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{f}\right )}{f}\right )}{f}\) |
Input:
Int[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2)^2,x]
Output:
-(((b*e - a*f)*(-(((d*e - c*f)*((Sqrt[-a]*d*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + ( d*x^2)/c]*EllipticPi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a*d)/(b*c )])/(Sqrt[b]*e*f*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) - ((d*e - c*f)*((f^2*x*S qrt[a + b*x^2]*Sqrt[c + d*x^2])/(2*e*(b*e - a*f)*(d*e - c*f)*(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]*Sq rt[(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*(b*e - a* f)*(d*e - c*f)) + (Sqrt[-a]*(b*e*(3*d*e - 2*c*f) - a*f*(2*d*e - c*f))*Sqrt [1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), ArcSin[(Sqrt[b ]*x)/Sqrt[-a]], (a*d)/(b*c)])/(2*Sqrt[b]*e^2*(b*e - a*f)*(d*e - c*f)*Sqrt[ a + b*x^2]*Sqrt[c + d*x^2])))/f))/f) + (c^(3/2)*Sqrt[d]*Sqrt[a + b*x^2]*El lipticPi[1 - (c*f)/(d*e), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/( a*e*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/f) + (b*((d *(d*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*El lipticE[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)*(d*e -...
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[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[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[d/b Int[(c + d*x^2)^(q - 1)*(e + f*x^2)^r, x], x] + Simp[(b*c - a*d)/b Int[(c + d*x^2)^(q - 1)*((e + f*x^2)^r/(a + b*x^2 )), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && GtQ[q, 1]
Int[1/(((a_) + (b_.)*(x_)^2)^2*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* (x_)^2]), x_Symbol] :> Simp[b^2*x*Sqrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(b* c - a*d)*(b*e - a*f)*(a + b*x^2))), x] + (Simp[(b^2*c*e + 3*a^2*d*f - 2*a*b *(d*e + c*f))/(2*a*(b*c - a*d)*(b*e - a*f)) Int[1/((a + b*x^2)*Sqrt[c + d *x^2]*Sqrt[e + f*x^2]), x], x] - Simp[d*(f/(2*a*(b*c - a*d)*(b*e - a*f))) 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. \(1074\) vs. \(2(397)=794\).
Time = 5.10 (sec) , antiderivative size = 1075, normalized size of antiderivative = 2.51
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1075\) |
| default | \(\text {Expression too large to display}\) | \(1572\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(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*(c*f-d*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/(-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)*Elliptic F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*d^2/f^2*a+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)*El lipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c^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*d^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)*d*b/f^2*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-3 /2/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*c*d+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*d^2-1/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))*a*d^2+1/2*c^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)/f*b/e*Elliptic F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/2*c^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)/f*b/e*Ell...
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)/(f*x**2+e)**2,x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)/(e + f*x**2)**2, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/(f*x^2 + e)^2, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/(f*x^2 + e)^2, x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2)^2,x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2)^2, x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{\left (e+f x^2\right )^2} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e)^2,x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x **2)*b*c*x - 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 - 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*f** 2*x**2 - 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 - 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*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...