Integrand size = 32, antiderivative size = 835 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\frac {f^2 x \sqrt {a+b x^2}}{4 e (b e-a f) (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )^2}+\frac {f^2 (b e (11 d e-6 c f)-a f (8 d e-3 c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f)^2 (d e-c f)^2 \sqrt {c+d x^2} \left (e+f x^2\right )}-\frac {\sqrt {d} \left (a^2 d f^2 \left (8 d^2 e^2+10 c d e f-3 c^2 f^2\right )+b^2 e \left (8 d^3 e^3+13 c^2 d e f^2-6 c^3 f^3\right )-a b f \left (16 d^3 e^3+13 c d^2 e^2 f+4 c^2 d e f^2-3 c^3 f^3\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{8 \sqrt {c} (b c-a d) e^2 (b e-a f)^2 (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \left (3 a^2 d f^2 \left (8 d^2 e^2-4 c d e f+c^2 f^2\right )-b^2 e \left (8 d^3 e^3-32 c d^2 e^2 f+13 c^2 d e f^2-4 c^3 f^3\right )-a b f \left (16 d^3 e^3+19 c d^2 e^2 f-8 c^2 d e f^2+3 c^3 f^3\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a (b c-a d) e^2 (b e-a f) (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^2 \left (3 a^2 f^2 \left (8 d^2 e^2-4 c d e f+c^2 f^2\right )-2 a b e f \left (28 d^2 e^2-17 c d e f+4 c^2 f^2\right )+b^2 e^2 \left (35 d^2 e^2-28 c d e f+8 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a \sqrt {d} e^3 (b e-a f)^2 (d e-c f)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/4*f^2*x*(b*x^2+a)^(1/2)/e/(-a*f+b*e)/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e )^2+1/8*f^2*(b*e*(-6*c*f+11*d*e)-a*f*(-3*c*f+8*d*e))*x*(b*x^2+a)^(1/2)/e^2 /(-a*f+b*e)^2/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(f*x^2+e)-1/8*d^(1/2)*(a^2*d*f^ 2*(-3*c^2*f^2+10*c*d*e*f+8*d^2*e^2)+b^2*e*(-6*c^3*f^3+13*c^2*d*e*f^2+8*d^3 *e^3)-a*b*f*(-3*c^3*f^3+4*c^2*d*e*f^2+13*c*d^2*e^2*f+16*d^3*e^3))*(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))/c ^(1/2)/(-a*d+b*c)/e^2/(-a*f+b*e)^2/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^ (1/2)/(d*x^2+c)^(1/2)-1/8*c^(1/2)*d^(1/2)*(3*a^2*d*f^2*(c^2*f^2-4*c*d*e*f+ 8*d^2*e^2)-b^2*e*(-4*c^3*f^3+13*c^2*d*e*f^2-32*c*d^2*e^2*f+8*d^3*e^3)-a*b* f*(3*c^3*f^3-8*c^2*d*e*f^2+19*c*d^2*e^2*f+16*d^3*e^3))*(b*x^2+a)^(1/2)*Inv erseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/(-a*d+b*c)/e^2 /(-a*f+b*e)/(-c*f+d*e)^4/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1 /8*c^(3/2)*f^2*(3*a^2*f^2*(c^2*f^2-4*c*d*e*f+8*d^2*e^2)-2*a*b*e*f*(4*c^2*f ^2-17*c*d*e*f+28*d^2*e^2)+b^2*e^2*(8*c^2*f^2-28*c*d*e*f+35*d^2*e^2))*(b*x^ 2+a)^(1/2)*EllipticPi(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^3/(-a*f+b*e)^2/(-c*f+d*e)^4/(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 = 8.08 (sec) , antiderivative size = 595, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\frac {-\sqrt {\frac {b}{a}} e x \left (a+b x^2\right ) \left (2 c (b c-a d) e f^3 (b e-a f) (d e-c f) \left (c+d x^2\right )+c (b c-a d) f^3 (b e (13 d e-6 c f)+a f (-10 d e+3 c f)) \left (c+d x^2\right ) \left (e+f x^2\right )+8 d^4 e^2 (b e-a f)^2 \left (e+f x^2\right )^2\right )-i c \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (b e \left (a^2 d f^2 \left (8 d^2 e^2+10 c d e f-3 c^2 f^2\right )+b^2 e \left (8 d^3 e^3+13 c^2 d e f^2-6 c^3 f^3\right )+a b f \left (-16 d^3 e^3-13 c d^2 e^2 f-4 c^2 d e f^2+3 c^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b c-a d) f \left (b e (d e-c f) (b e (11 d e-6 c f)+a f (-8 d e+3 c f)) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+\left (b^2 e^2 \left (-35 d^2 e^2+28 c d e f-8 c^2 f^2\right )-3 a^2 f^2 \left (8 d^2 e^2-4 c d e f+c^2 f^2\right )+2 a b e f \left (28 d^2 e^2-17 c d e f+4 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{8 \sqrt {\frac {b}{a}} c (b c-a d) e^3 (b e-a f)^2 (d e-c f)^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \] Input:
Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)^3),x]
Output:
(-(Sqrt[b/a]*e*x*(a + b*x^2)*(2*c*(b*c - a*d)*e*f^3*(b*e - a*f)*(d*e - c*f )*(c + d*x^2) + c*(b*c - a*d)*f^3*(b*e*(13*d*e - 6*c*f) + a*f*(-10*d*e + 3 *c*f))*(c + d*x^2)*(e + f*x^2) + 8*d^4*e^2*(b*e - a*f)^2*(e + f*x^2)^2)) - I*c*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)^2*(b*e*(a^2*d*f^2 *(8*d^2*e^2 + 10*c*d*e*f - 3*c^2*f^2) + b^2*e*(8*d^3*e^3 + 13*c^2*d*e*f^2 - 6*c^3*f^3) + a*b*f*(-16*d^3*e^3 - 13*c*d^2*e^2*f - 4*c^2*d*e*f^2 + 3*c^3 *f^3))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (b*c - a*d)*f*(b*e *(d*e - c*f)*(b*e*(11*d*e - 6*c*f) + a*f*(-8*d*e + 3*c*f))*EllipticF[I*Arc Sinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (b^2*e^2*(-35*d^2*e^2 + 28*c*d*e*f - 8*c ^2*f^2) - 3*a^2*f^2*(8*d^2*e^2 - 4*c*d*e*f + c^2*f^2) + 2*a*b*e*f*(28*d^2* e^2 - 17*c*d*e*f + 4*c^2*f^2))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a] *x], (a*d)/(b*c)])))/(8*Sqrt[b/a]*c*(b*c - a*d)*e^3*(b*e - a*f)^2*(d*e - c *f)^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)
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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )^2}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 426 |
| \(\displaystyle \frac {d \left (\frac {d \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {d (d e-c f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^2}dx}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 424 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}-\frac {b d \int \frac {f x^2+e}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (-\frac {b d \left (e \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (-\frac {b d \left (f \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (-\frac {b d \left (f \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} e \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{2 e \left (e+f x^2\right ) (b e-a f) (d e-c f)}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f)}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f) \sqrt {b x^2+a}}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} \sqrt {b x^2+a} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+a d f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{(d e-c f)^2}\right )}{d e-c f}-\frac {f \left (\frac {x \sqrt {b x^2+a} \sqrt {d x^2+c} f^2}{2 e (b e-a f) (d e-c f) \left (f x^2+e\right )}-\frac {b d \left (f \left (\frac {x \sqrt {b x^2+a}}{b \sqrt {d x^2+c}}-\frac {\sqrt {c} \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )+\frac {\sqrt {c} e \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\right )}{2 e (b e-a f) (d e-c f)}+\frac {(b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx}{2 e (b e-a f) (d e-c f) \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+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}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 )}{(d e-c f)^2}\right )}{d e-c 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 )}{d e-c f}\right )}{d e-c f}-\frac {f \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )^3}dx}{d e-c f}\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+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}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 )}{(d e-c f)^2}\right )}{d e-c 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 )}{d e-c f}\right )}{d e-c f}-\frac {f \int \left (-\frac {f^{3/2}}{8 (-e)^{3/2} \left (\sqrt {-e} \sqrt {f}-f x\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {f^{3/2}}{8 (-e)^{3/2} \left (f x+\sqrt {-e} \sqrt {f}\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {3 f}{16 e^2 \left (\sqrt {-e} \sqrt {f}-f x\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {3 f}{16 e^2 \left (f x+\sqrt {-e} \sqrt {f}\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}-\frac {3 f}{8 e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} \left (-f^2 x^2-e f\right )}\right )dx}{d e-c f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {c^{3/2} \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 ) f^2}{a \sqrt {d} e (d e-c f)^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (\frac {\sqrt {c} (b d e-2 b c f+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}}-\frac {\sqrt {d} (d e-c f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (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 )}{(d e-c f)^2}\right )}{d e-c 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 )}{d e-c f}\right )}{d e-c f}-\frac {f \left (-\frac {\int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f^{3/2}}{8 (-e)^{3/2}}-\frac {\int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f^{3/2}}{8 (-e)^{3/2}}-\frac {3 \int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f}{16 e^2}-\frac {3 \int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx f}{16 e^2}+\frac {3 \sqrt {-a} \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 )}{8 \sqrt {b} e^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}\right )}{d e-c f}\) |
Input:
Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2)^3),x]
Output:
$Aborted
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]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) ^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(3336\) vs. \(2(803)=1606\).
Time = 20.31 (sec) , antiderivative size = 3337, normalized size of antiderivative = 4.00
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(3337\) |
| default | \(\text {Expression too large to display}\) | \(8681\) |
Input:
int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^3,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)*(1/4*f^3/(a*c* f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x/(c*f-d*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^( 1/2)/(f*x^2+e)^2+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*d^4/c/(a*d-b*c)/(c*f-d*e)^3-11/8/(-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^3/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/ e/(c*f-d*e)*a*c+3/8*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^4*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2) ^2/e^2/(c*f-d*e)*a*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-3/8* 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^4*b/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/e^2/(c*f-d*e)*a*E llipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+5/4*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*f^3*b /(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/e/(c*f-d*e)*a*EllipticE(x*(-b/a)^(1/2 ),(-1+(a*d+b*c)/c/b)^(1/2))-3/4*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^3*b^2/(a*c*f^2-a*d*e*f-b* c*e*f+b*d*e^2)^2/e/(c*f-d*e)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^( 1/2))+3/4*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^3*b^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/e/...
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^3,x, algorithm="fric as")
Output:
Timed out
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x**2+a)**(1/2)/(d*x**2+c)**(3/2)/(f*x**2+e)**3,x)
Output:
Timed out
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^3,x, algorithm="maxi ma")
Output:
integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^3), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{3}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^3,x, algorithm="giac ")
Output:
integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e)^3), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:
int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^3),x)
Output:
int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2)^3), x)
\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\sqrt {b \,x^{2}+a}\, \left (d \,x^{2}+c \right )^{\frac {3}{2}} \left (f \,x^{2}+e \right )^{3}}d x \] Input:
int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^3,x)
Output:
int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)/(f*x^2+e)^3,x)