Integrand size = 32, antiderivative size = 517 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {f^2 x \sqrt {c+d x^2}}{2 e (b e-a f) (d e-c f) \sqrt {a+b x^2} \left (e+f x^2\right )}-\frac {\sqrt {b} \left (a b c f^2-a^2 d f^2-2 b^2 e (d e-c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{2 \sqrt {a} (b c-a d) e (b e-a f)^2 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} \sqrt {b} \left (a^2 d f^2-a b f (6 d e+c f)+2 b^2 e (d e+2 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 c (b c-a d) e (b e-a f)^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} f^2 (b e (5 d e-4 c f)-a f (2 d e-c f)) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{2 \sqrt {b} c e^2 (b e-a f)^3 (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/2*f^2*x*(d*x^2+c)^(1/2)/e/(-a*f+b*e)/(-c*f+d*e)/(b*x^2+a)^(1/2)/(f*x^2+e )-1/2*b^(1/2)*(a*b*c*f^2-a^2*d*f^2-2*b^2*e*(-c*f+d*e))*(d*x^2+c)^(1/2)*Ell ipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(1/2)/(-a* d+b*c)/e/(-a*f+b*e)^2/(-c*f+d*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a)) ^(1/2)-1/2*a^(1/2)*b^(1/2)*(a^2*d*f^2-a*b*f*(c*f+6*d*e)+2*b^2*e*(2*c*f+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/(-a*d+b*c)/e/(-a*f+b*e)^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a) )^(1/2)+1/2*a^(3/2)*f^2*(b*e*(-4*c*f+5*d*e)-a*f*(-c*f+2*d*e))*(d*x^2+c)^(1 /2)*EllipticPi(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/(-a*f+b*e)^3/(-c*f+d*e)/(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 = 8.13 (sec) , antiderivative size = 1659, normalized size of antiderivative = 3.21 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
(Sqrt[b/a]*(2*b^3*Sqrt[b/a]*c*d*e^4*x - 2*b^3*Sqrt[b/a]*c^2*e^3*f*x - a^3* (b/a)^(3/2)*c^2*e*f^3*x + a^3*Sqrt[b/a]*c*d*e*f^3*x + 2*b^3*Sqrt[b/a]*d^2* e^4*x^3 - 2*b^3*Sqrt[b/a]*c^2*e^2*f^2*x^3 - a*b^2*Sqrt[b/a]*c^2*e*f^3*x^3 + a^3*Sqrt[b/a]*d^2*e*f^3*x^3 + 2*b^3*Sqrt[b/a]*d^2*e^3*f*x^5 - 2*b^3*Sqrt [b/a]*c*d*e^2*f^2*x^5 - a*b^2*Sqrt[b/a]*c*d*e*f^3*x^5 + a^3*(b/a)^(3/2)*d^ 2*e*f^3*x^5 + I*b*c*e*(-(a*b*c*f^2) + a^2*d*f^2 + 2*b^2*e*(d*e - c*f))*Sqr t[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*(-(b*c) + a*d)*e*(2*b*e + a*f)*(-(d*e) + c*f)* Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticF[I*ArcSinh[Sq rt[b/a]*x], (a*d)/(b*c)] + (5*I)*a*b^2*c*d*e^3*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) ] - (5*I)*a^2*b*d^2*e^3*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elliptic Pi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (4*I)*a*b^2*c^2*e^2 *f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*Arc Sinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*a^2*b*c*d*e^2*f^2*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)] + (2*I)*a^3*d^2*e^2*f^2*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*b*c^ 2*e*f^3*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^3*c*d*e*f^3*Sqrt[1 + (b*x^2)/a...
Time = 1.08 (sec) , antiderivative size = 817, normalized size of antiderivative = 1.58, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {426, 421, 25, 400, 313, 320, 414, 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.
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 426 |
\(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
\(\Big \downarrow \) 421 |
\(\displaystyle \frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
\(\Big \downarrow \) 400 |
\(\displaystyle \frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {(-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {b \left (\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {(-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {b \left (\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
\(\Big \downarrow \) 414 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{b e-a f}\) |
\(\Big \downarrow \) 424 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}-\frac {b d \int \frac {f x^2+e}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{b e-a f}\) |
\(\Big \downarrow \) 406 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (-\frac {b d \left (e \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{b e-a f}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (-\frac {b d \left (f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{b e-a f}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (-\frac {b d \left (f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{b e-a f}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}-\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{b e-a f}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {\sqrt {\frac {b x^2}{a}+1} (b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e \sqrt {a+b x^2} (b e-a f) (d e-c f)}-\frac {b d \left (\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{b e-a f}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{b e-a f}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {b \left (\frac {a^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right ) f^2}{\sqrt {b} c e (b e-a f)^2 \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {b \left (\frac {\sqrt {b} (b e-a f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} (b d e+b c f-2 a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(b e-a f)^2}\right )}{b e-a f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {\sqrt {-a} (b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{2 \sqrt {b} e^2 (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{b e-a f}\) |
Input:
Int[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)^2),x]
Output:
-((f*((f^2*x*Sqrt[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]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqr t[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])))/(b*e - a*f)) + (b*((b*((Sqrt[b ]*(b*e - a*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - ( a*d)/(b*c)])/(Sqrt[a]*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c* (a + b*x^2))]) - (Sqrt[c]*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[a + b*x^2]*Ellipt icF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a*d)* Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(b*e - a*f)^2 + ( a^(3/2)*f^2*Sqrt[c + d*x^2]*EllipticPi[1 - (a*f)/(b*e), ArcTan[(Sqrt[b]*x) /Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*(b*e - a*f)^2*Sqrt[a + b*x^2]*Sq rt[(a*(c + d*x^2))/(c*(a + b*x^2))])))/(b*e - a*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[(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[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[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[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]
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[b/(b*c - a*d) Int[(a + b*x^2)^p*(c + d*x^2)^ (q + 1)*(e + f*x^2)^r, x], x] - Simp[d/(b*c - a*d) Int[(a + b*x^2)^(p + 1 )*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && ILtQ[p, 0] && LeQ[q, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1338\) vs. \(2(491)=982\).
Time = 11.27 (sec) , antiderivative size = 1339, normalized size of antiderivative = 2.59
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1339\) |
default | \(\text {Expression too large to display}\) | \(2813\) |
Input:
int(1/(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)*(-(b*d*x^2+b*c )*b^2/a/(a*d-b*c)*x/(a*f-b*e)^2/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/2*f^3/(a *c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x/(a*f-b*e)*(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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b) ^(1/2))/a*b^2/(a*f-b*e)^2-1/2/(-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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d +b*c)/c/b)^(1/2))*b*d*f/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/(a*f-b*e)+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^3/(a*d-b*c)/a/(a*f-b*e)^2*EllipticE(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)*b*f^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e/(a* f-b*e)*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) *b*f^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e/(a*f-b*e)*EllipticE(x*(-b/a)^(1 /2),(-1+(a*d+b*c)/c/b)^(1/2))+1/2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e^2/(a *f-b*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))*a*c-1/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)/(a*f-b*e)*f^2/e/(-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...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**2,x)
Output:
Integral(1/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)**2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="maxi ma")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x, algorithm="giac ")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)^2), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}\, \left (f \,x^{2}+e \right )^{2}}d x \] Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)
Output:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^2,x)