Integrand size = 33, antiderivative size = 363 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {b^2 x \sqrt {c+d x^2}}{a (b c+a d) (b e+a f) \sqrt {a-b x^2}}-\frac {b^{3/2} c \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{a^{3/2} (b c+a d) (b e+a f) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}}+\frac {\sqrt {b} \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{a^{3/2} (b e+a f) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}}+\frac {\sqrt {a} f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
b^2*x*(d*x^2+c)^(1/2)/a/(a*d+b*c)/(a*f+b*e)/(-b*x^2+a)^(1/2)-b^(3/2)*c*(-b *x^2+a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/ 2))/a^(3/2)/(a*d+b*c)/(a*f+b*e)/(1-b*x^2/a)^(1/2)/(d*x^2+c)^(1/2)+b^(1/2)* (-b*x^2+a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^ (1/2))/a^(3/2)/(a*f+b*e)/(1-b*x^2/a)^(1/2)/(d*x^2+c)^(1/2)+a^(1/2)*f*(1-b* x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d /b/c)^(1/2))/b^(1/2)/e/(a*f+b*e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 4.66 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {b^2 \sqrt {-\frac {b}{a}} c e x+b^2 \sqrt {-\frac {b}{a}} d e x^3+i b^2 c e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i b (b c+a d) e \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i a b c f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-i a^2 d f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{a \sqrt {-\frac {b}{a}} (b c+a d) e (b e+a f) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[1/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(b^2*Sqrt[-(b/a)]*c*e*x + b^2*Sqrt[-(b/a)]*d*e*x^3 + I*b^2*c*e*Sqrt[1 - (b *x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/ (b*c))] - I*b*(b*c + a*d)*e*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipt icF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - I*a*b*c*f*Sqrt[1 - (b*x^2 )/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)] *x], -((a*d)/(b*c))] - I*a^2*d*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E llipticPi[-((a*f)/(b*e)), I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(a*S qrt[-(b/a)]*(b*c + a*d)*e*(b*e + a*f)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
Time = 0.85 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.33, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {421, 402, 399, 323, 323, 321, 331, 330, 327, 415, 323, 323, 321, 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.
| \(\displaystyle \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \int \frac {-b f x^2+b e+2 a f}{\left (a-b x^2\right )^{3/2} \sqrt {d x^2+c}}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \left (\frac {\int \frac {a (b d e+b c f+2 a d f)-b d (b e+a f) x^2}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \left (\frac {(a d+b c) (2 a f+b e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx-b (a f+b e) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{\sqrt {c+d x^2}}-b (a f+b e) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{\sqrt {a-b x^2} \sqrt {c+d x^2}}-b (a f+b e) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-b (a f+b e) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {b \sqrt {1-\frac {b x^2}{a}} (a f+b e) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {b \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}+\frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {a-b x^2}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 415 |
| \(\displaystyle \frac {f^2 \left (\frac {(a f+b e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}-\frac {b \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{f}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f^2 \left (\frac {(a f+b e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}-\frac {b \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{f \sqrt {c+d x^2}}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f^2 \left (\frac {(a f+b e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}-\frac {b \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{f \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {f^2 \left (\frac {(a f+b e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{f \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {f^2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} (a f+b e) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f \sqrt {a-b x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{f \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {f^2 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a f+b e) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{f \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{f \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {f^2 \left (\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a f+b e) \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} e f \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{f \sqrt {a-b x^2} \sqrt {c+d x^2}}\right )}{(a f+b e)^2}+\frac {b \left (\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a f+b e) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{a (a d+b c)}+\frac {b x \sqrt {c+d x^2} (a f+b e)}{a \sqrt {a-b x^2} (a d+b c)}\right )}{(a f+b e)^2}\) |
Input:
Int[1/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(b*((b*(b*e + a*f)*x*Sqrt[c + d*x^2])/(a*(b*c + a*d)*Sqrt[a - b*x^2]) + (- ((Sqrt[a]*Sqrt[b]*(b*e + a*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*Elliptic E[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c])) + (Sqrt[a]*(b*c + a*d)*(b*e + 2*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))])/(S qrt[b]*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]))/(a*(b*c + a*d))))/(b*e + a*f)^2 + (f^2*(-((Sqrt[a]*Sqrt[b]*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic F[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(f*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])) + (Sqrt[a]*(b*e + a*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))])/( Sqrt[b]*e*f*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])))/(b*e + a*f)^2
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])
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]
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]
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]
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]
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])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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[d/b Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Simp[(b*c - a*d)/b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2 ]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Time = 8.68 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(\frac {\left (\sqrt {\frac {b}{a}}\, b^{2} d e \,x^{3}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b d e +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c e -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c e +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a^{2} d f +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right ) a b c f +\sqrt {\frac {b}{a}}\, b^{2} c e x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{e \sqrt {\frac {b}{a}}\, a \left (a f +b e \right ) \left (a d +b c \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\) | \(411\) |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) b x}{a \left (a d +b c \right ) \left (a f +b e \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right ) b}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, a \left (a f +b e \right )}-\frac {b^{2} c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{a \left (a d +b c \right ) \left (a f +b e \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {f \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{\left (a f +b e \right ) e \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(427\) |
Input:
int(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
((b/a)^(1/2)*b^2*d*e*x^3+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elliptic F(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*a*b*d*e+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/ c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b^2*c*e-((-b*x^2+a)/a)^ (1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b^2*c* e+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b /e,(-1/c*d)^(1/2)/(b/a)^(1/2))*a^2*d*f+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^ (1/2)*EllipticPi(x*(b/a)^(1/2),-a*f/b/e,(-1/c*d)^(1/2)/(b/a)^(1/2))*a*b*c* f+(b/a)^(1/2)*b^2*c*e*x)*(d*x^2+c)^(1/2)*(-b*x^2+a)^(1/2)/e/(b/a)^(1/2)/a/ (a*f+b*e)/(a*d+b*c)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="frica s")
Output:
Timed out
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(1/((a - b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxim a")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac" )
Output:
integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(1/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {1}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d f \,x^{8}-2 a b d f \,x^{6}+b^{2} c f \,x^{6}+b^{2} d e \,x^{6}+a^{2} d f \,x^{4}-2 a b c f \,x^{4}-2 a b d e \,x^{4}+b^{2} c e \,x^{4}+a^{2} c f \,x^{2}+a^{2} d e \,x^{2}-2 a b c e \,x^{2}+a^{2} c e}d x \] Input:
int(1/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(a**2*c*e + a**2*c*f*x**2 + a**2*d *e*x**2 + a**2*d*f*x**4 - 2*a*b*c*e*x**2 - 2*a*b*c*f*x**4 - 2*a*b*d*e*x**4 - 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d *f*x**8),x)