Integrand size = 32, antiderivative size = 983 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {b^2 x}{3 a (b c-a d) (b e-a f) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (a b d^2 e-a^2 d^2 f+b^2 c (d e-c f)\right ) x}{3 a c (b c-a d)^2 (b e-a f) (d e-c f) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {b \left (2 a^3 b d^3 e f-a^4 d^3 f^2+2 b^4 c^2 e (d e-c f)-a b^3 c \left (9 d^2 e^2-4 c d e f-5 c^2 f^2\right )-a^2 b^2 d \left (d^2 e^2-12 c d e f+12 c^2 f^2\right )\right ) x}{3 a^2 c (b c-a d)^3 (b e-a f)^2 (d e-c f) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (a^5 d^4 f^2 (2 d e-5 c f)+2 b^5 c^3 e (d e-c f)^2-5 a b^4 c^2 (d e-c f)^2 (2 d e+c f)+a^3 b^2 d^3 e \left (2 d^2 e^2+15 c d e f-26 c^2 f^2\right )-a^4 b d^3 f \left (4 d^2 e^2-13 c^2 f^2\right )-a^2 b^3 c d \left (10 d^3 e^3-26 c d^2 e^2 f+26 c^2 d e f^2-13 c^3 f^3\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} (b c-a d)^4 (b e-a f)^2 (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {d} \left (3 a^4 c d^4 f^3-a^3 b d^3 f (d e-4 c f)^2+b^4 c^2 (d e-c f)^3-3 a b^3 c d^2 e \left (6 d^2 e^2-15 c d e f+10 c^2 f^2\right )+a^2 b^2 d^2 \left (d^3 e^3+7 c d^2 e^2 f-29 c^2 d e f^2+30 c^3 f^3\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^4 (b e-a f) (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f^5 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (b e-a f)^2 (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/3*b^2*x/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)+1/3*d*(a *b*d^2*e-a^2*d^2*f+b^2*c*(-c*f+d*e))*x/a/c/(-a*d+b*c)^2/(-a*f+b*e)/(-c*f+d *e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*b*(2*a^3*b*d^3*e*f-a^4*d^3*f^2+2*b ^4*c^2*e*(-c*f+d*e)-a*b^3*c*(-5*c^2*f^2-4*c*d*e*f+9*d^2*e^2)-a^2*b^2*d*(12 *c^2*f^2-12*c*d*e*f+d^2*e^2))*x/a^2/c/(-a*d+b*c)^3/(-a*f+b*e)^2/(-c*f+d*e) /(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/3*d^(1/2)*(a^5*d^4*f^2*(-5*c*f+2*d*e)+2 *b^5*c^3*e*(-c*f+d*e)^2-5*a*b^4*c^2*(-c*f+d*e)^2*(c*f+2*d*e)+a^3*b^2*d^3*e *(-26*c^2*f^2+15*c*d*e*f+2*d^2*e^2)-a^4*b*d^3*f*(-13*c^2*f^2+4*d^2*e^2)-a^ 2*b^3*c*d*(-13*c^3*f^3+26*c^2*d*e*f^2-26*c*d^2*e^2*f+10*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))/a ^2/c^(3/2)/(-a*d+b*c)^4/(-a*f+b*e)^2/(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c) )^(1/2)/(d*x^2+c)^(1/2)-1/3*d^(1/2)*(3*a^4*c*d^4*f^3-a^3*b*d^3*f*(-4*c*f+d *e)^2+b^4*c^2*(-c*f+d*e)^3-3*a*b^3*c*d^2*e*(10*c^2*f^2-15*c*d*e*f+6*d^2*e^ 2)+a^2*b^2*d^2*(30*c^3*f^3-29*c^2*d*e*f^2+7*c*d^2*e^2*f+d^3*e^3))*(b*x^2+a )^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a^2/c ^(1/2)/(-a*d+b*c)^4/(-a*f+b*e)/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2 )/(d*x^2+c)^(1/2)-c^(3/2)*f^5*(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/(-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)
Result contains complex when optimal does not.
Time = 12.18 (sec) , antiderivative size = 744, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {\frac {b}{a}} e x \left (a^2 c d^4 (-b c+a d) (b e-a f)^2 (-d e+c f) \left (a+b x^2\right )^2+a^2 d^4 (b e-a f)^2 (b c (10 d e-13 c f)+a d (-2 d e+5 c f)) \left (a+b x^2\right )^2 \left (c+d x^2\right )-a b^4 c^2 (-b c+a d) (-b e+a f) (d e-c f)^2 \left (c+d x^2\right )^2-b^4 c^2 (d e-c f)^2 \left (2 b^2 c e+13 a^2 d f-5 a b (2 d e+c f)\right ) \left (a+b x^2\right ) \left (c+d x^2\right )^2\right )-i c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (-b e \left (a^5 d^4 f^2 (2 d e-5 c f)+2 b^5 c^3 e (d e-c f)^2-5 a b^4 c^2 (d e-c f)^2 (2 d e+c f)+a^3 b^2 d^3 e \left (2 d^2 e^2+15 c d e f-26 c^2 f^2\right )+a^4 b d^3 f \left (-4 d^2 e^2+13 c^2 f^2\right )+a^2 b^3 c d \left (-10 d^3 e^3+26 c d^2 e^2 f-26 c^2 d e f^2+13 c^3 f^3\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b c-a d) \left (b e (-d e+c f) \left (-2 a^3 b d^3 e f+a^4 d^3 f^2+2 b^4 c^2 e (-d e+c f)+a b^3 c \left (9 d^2 e^2-4 c d e f-5 c^2 f^2\right )+a^2 b^2 d \left (d^2 e^2-12 c d e f+12 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 a^2 c (-b c+a d)^3 f^4 \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c-a d)^4 e (b e-a f)^2 (d e-c f)^2 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \] Input:
Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
Output:
(-(Sqrt[b/a]*e*x*(a^2*c*d^4*(-(b*c) + a*d)*(b*e - a*f)^2*(-(d*e) + c*f)*(a + b*x^2)^2 + a^2*d^4*(b*e - a*f)^2*(b*c*(10*d*e - 13*c*f) + a*d*(-2*d*e + 5*c*f))*(a + b*x^2)^2*(c + d*x^2) - a*b^4*c^2*(-(b*c) + a*d)*(-(b*e) + a* f)*(d*e - c*f)^2*(c + d*x^2)^2 - b^4*c^2*(d*e - c*f)^2*(2*b^2*c*e + 13*a^2 *d*f - 5*a*b*(2*d*e + c*f))*(a + b*x^2)*(c + d*x^2)^2)) - I*c*(a + b*x^2)* Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*(-(b*e*(a^5*d^4*f^2*(2 *d*e - 5*c*f) + 2*b^5*c^3*e*(d*e - c*f)^2 - 5*a*b^4*c^2*(d*e - c*f)^2*(2*d *e + c*f) + a^3*b^2*d^3*e*(2*d^2*e^2 + 15*c*d*e*f - 26*c^2*f^2) + a^4*b*d^ 3*f*(-4*d^2*e^2 + 13*c^2*f^2) + a^2*b^3*c*d*(-10*d^3*e^3 + 26*c*d^2*e^2*f - 26*c^2*d*e*f^2 + 13*c^3*f^3))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b *c)]) + (b*c - a*d)*(b*e*(-(d*e) + c*f)*(-2*a^3*b*d^3*e*f + a^4*d^3*f^2 + 2*b^4*c^2*e*(-(d*e) + c*f) + a*b^3*c*(9*d^2*e^2 - 4*c*d*e*f - 5*c^2*f^2) + a^2*b^2*d*(d^2*e^2 - 12*c*d*e*f + 12*c^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b /a]*x], (a*d)/(b*c)] - 3*a^2*c*(-(b*c) + a*d)^3*f^4*EllipticPi[(a*f)/(b*e) , I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(3*a^2*Sqrt[b/a]*c^2*(b*c - a*d) ^4*e*(b*e - a*f)^2*(d*e - c*f)^2*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))
Time = 1.61 (sec) , antiderivative size = 1031, normalized size of antiderivative = 1.05, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {421, 25, 402, 25, 402, 27, 402, 400, 313, 320, 421, 25, 402, 25, 400, 313, 320, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^{5/2}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^{5/2}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {6 d f a^2-b (3 d e+5 c f) a+5 b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}dx}{3 a (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {6 d f a^2-3 b d e a-5 b c f a+5 b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}dx}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {3 d \left (b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-2 d f a^2+b d e a+b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {3 d \int \frac {b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-2 d f a^2+b d e a+b^2 c e\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{a (b c-a d)}+\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {3 d \left (\frac {\int \frac {b \left (2 d^2 f a^3-b d (d e-11 c f) a^2-b^2 c (9 d e+5 c f) a+2 b^3 c^2 e\right ) x^2+a \left (4 d^2 f a^3-b d (2 d e+17 c f) a^2+b^2 c (9 d e+5 c f) a+b^3 c^2 e\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\frac {3 d \left (\frac {\frac {\left (-4 a^4 d^3 f+a^3 b d^2 (19 c f+2 d e)-2 a^2 b^2 c d (5 d e-3 c f)-5 a b^3 c^2 (c f+2 d e)+2 b^4 c^3 e\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}-\frac {a b \left (-2 a^3 d^2 f+a^2 b d (28 c f+d e)-2 a b^2 c (5 c f+9 d e)+b^3 c^2 e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\frac {3 d \left (\frac {\frac {\sqrt {a+b x^2} \left (-4 a^4 d^3 f+a^3 b d^2 (19 c f+2 d e)-2 a^2 b^2 c d (5 d e-3 c f)-5 a b^3 c^2 (c f+2 d e)+2 b^4 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a b \left (-2 a^3 d^2 f+a^2 b d (28 c f+d e)-2 a b^2 c (5 c f+9 d e)+b^3 c^2 e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 d \left (\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {\frac {\sqrt {a+b x^2} \left (-4 a^4 d^3 f+a^3 b d^2 (19 c f+2 d e)-2 a^2 b^2 c d (5 d e-3 c f)-5 a b^3 c^2 (c f+2 d e)+2 b^4 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (-2 a^3 d^2 f+a^2 b d (28 c f+d e)-2 a b^2 c (5 c f+9 d e)+b^3 c^2 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 d \left (\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {\frac {\sqrt {a+b x^2} \left (-4 a^4 d^3 f+a^3 b d^2 (19 c f+2 d e)-2 a^2 b^2 c d (5 d e-3 c f)-5 a b^3 c^2 (c f+2 d e)+2 b^4 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (-2 a^3 d^2 f+a^2 b d (28 c f+d e)-2 a b^2 c (5 c f+9 d e)+b^3 c^2 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 d \left (\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {\frac {\sqrt {a+b x^2} \left (-4 a^4 d^3 f+a^3 b d^2 (19 c f+2 d e)-2 a^2 b^2 c d (5 d e-3 c f)-5 a b^3 c^2 (c f+2 d e)+2 b^4 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (-2 a^3 d^2 f+a^2 b d (28 c f+d e)-2 a b^2 c (5 c f+9 d e)+b^3 c^2 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\int -\frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 d \left (\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {\frac {\sqrt {a+b x^2} \left (-4 a^4 d^3 f+a^3 b d^2 (19 c f+2 d e)-2 a^2 b^2 c d (5 d e-3 c f)-5 a b^3 c^2 (c f+2 d e)+2 b^4 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (-2 a^3 d^2 f+a^2 b d (28 c f+d e)-2 a b^2 c (5 c f+9 d e)+b^3 c^2 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (-\frac {\int \frac {b d (d e-c f) x^2+a d (2 d e-5 c f)-3 b c (d e-2 c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {d x \sqrt {a+b x^2} (d e-c f)}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (11 a^2 d f-5 a b c f-8 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 d \left (\frac {x \sqrt {a+b x^2} \left (2 a^3 d^2 f-a^2 b d (d e-11 c f)-a b^2 c (5 c f+9 d e)+2 b^3 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {\frac {\sqrt {a+b x^2} \left (-4 a^4 d^3 f+a^3 b d^2 (19 c f+2 d e)-2 a^2 b^2 c d (5 d e-3 c f)-5 a b^3 c^2 (c f+2 d e)+2 b^4 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (-2 a^3 d^2 f+a^2 b d (28 c f+d e)-2 a b^2 c (5 c f+9 d e)+b^3 c^2 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {d (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {3 d \left (\frac {\left (2 d^2 f a^3-b d (d e-11 c f) a^2-b^2 c (9 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (-4 d^3 f a^4+b d^2 (2 d e+19 c f) a^3-2 b^2 c d (5 d e-3 c f) a^2-5 b^3 c^2 (2 d e+c f) a+2 b^4 c^3 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} \left (-2 d^2 f a^3+b d (d e+28 c f) a^2-2 b^2 c (9 d e+5 c f) a+b^3 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b (a d (d e-4 c f)-3 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {3 d \left (\frac {\left (2 d^2 f a^3-b d (d e-11 c f) a^2-b^2 c (9 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (-4 d^3 f a^4+b d^2 (2 d e+19 c f) a^3-2 b^2 c d (5 d e-3 c f) a^2-5 b^3 c^2 (2 d e+c f) a+2 b^4 c^3 e\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {c} \left (-2 d^2 f a^3+b d (d e+28 c f) a^2-2 b^2 c (9 d e+5 c f) a+b^3 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\left (\frac {\int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {3 d \left (\frac {\left (2 d^2 f a^3-b d (d e-11 c f) a^2-b^2 c (9 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (-4 d^3 f a^4+b d^2 (2 d e+19 c f) a^3-2 b^2 c d (5 d e-3 c f) a^2-5 b^3 c^2 (2 d e+c f) a+2 b^4 c^3 e\right ) \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} \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 {b \sqrt {c} \left (-2 d^2 f a^3+b d (d e+28 c f) a^2-2 b^2 c (9 d e+5 c f) a+b^3 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\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}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\left (\frac {\sqrt {\frac {b x^2}{a}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {d x^2+c} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2 \sqrt {b x^2+a}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {3 d \left (\frac {\left (2 d^2 f a^3-b d (d e-11 c f) a^2-b^2 c (9 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (-4 d^3 f a^4+b d^2 (2 d e+19 c f) a^3-2 b^2 c d (5 d e-3 c f) a^2-5 b^3 c^2 (2 d e+c f) a+2 b^4 c^3 e\right ) \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} \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 {b \sqrt {c} \left (-2 d^2 f a^3+b d (d e+28 c f) a^2-2 b^2 c (9 d e+5 c f) a+b^3 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\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}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\left (\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (f x^2+e\right )}dx f^2}{(d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {3 d \left (\frac {\left (2 d^2 f a^3-b d (d e-11 c f) a^2-b^2 c (9 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (-4 d^3 f a^4+b d^2 (2 d e+19 c f) a^3-2 b^2 c d (5 d e-3 c f) a^2-5 b^3 c^2 (2 d e+c f) a+2 b^4 c^3 e\right ) \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} \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 {b \sqrt {c} \left (-2 d^2 f a^3+b d (d e+28 c f) a^2-2 b^2 c (9 d e+5 c f) a+b^3 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\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}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\left (\frac {\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 ) f^2}{\sqrt {b} e (d e-c f)^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}+\frac {d \left (-\frac {d (d e-c f) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}-\frac {\frac {\sqrt {d} (b c (4 d e-7 c f)-a d (2 d e-5 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}}+\frac {b \sqrt {c} (a d (d e-4 c f)-3 b c (d e-2 c 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}}}{3 c (b c-a d)}\right )}{(d e-c f)^2}\right ) f^2}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) x}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {\frac {b \left (11 d f a^2-8 b d e a-5 b c f a+2 b^2 c e\right ) x}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {3 d \left (\frac {\left (2 d^2 f a^3-b d (d e-11 c f) a^2-b^2 c (9 d e+5 c f) a+2 b^3 c^2 e\right ) \sqrt {b x^2+a} x}{3 c (b c-a d) \left (d x^2+c\right )^{3/2}}+\frac {\frac {\left (-4 d^3 f a^4+b d^2 (2 d e+19 c f) a^3-2 b^2 c d (5 d e-3 c f) a^2-5 b^3 c^2 (2 d e+c f) a+2 b^4 c^3 e\right ) \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} \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 {b \sqrt {c} \left (-2 d^2 f a^3+b d (d e+28 c f) a^2-2 b^2 c (9 d e+5 c f) a+b^3 c^2 e\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\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}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}\right )}{(b e-a f)^2}\) |
Input:
Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
Output:
(b*((b*(b*e - a*f)*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2) ) + ((b*(2*b^2*c*e - 8*a*b*d*e - 5*a*b*c*f + 11*a^2*d*f)*x)/(a*(b*c - a*d) *Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (3*d*(((2*b^3*c^2*e + 2*a^3*d^2*f - a^2*b*d*(d*e - 11*c*f) - a*b^2*c*(9*d*e + 5*c*f))*x*Sqrt[a + b*x^2])/(3*c* (b*c - a*d)*(c + d*x^2)^(3/2)) + (((2*b^4*c^3*e - 4*a^4*d^3*f - 2*a^2*b^2* c*d*(5*d*e - 3*c*f) - 5*a*b^3*c^2*(2*d*e + c*f) + a^3*b*d^2*(2*d*e + 19*c* f))*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d) ])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt [c + d*x^2]) - (b*Sqrt[c]*(b^3*c^2*e - 2*a^3*d^2*f - 2*a*b^2*c*(9*d*e + 5* c*f) + a^2*b*d*(d*e + 28*c*f))*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x )/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a *(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*c*(b*c - a*d))))/(a*(b*c - a*d)))/(3*a *(b*c - a*d))))/(b*e - a*f)^2 + (f^2*((d*(-1/3*(d*(d*e - c*f)*x*Sqrt[a + b *x^2])/(c*(b*c - a*d)*(c + d*x^2)^(3/2)) - ((Sqrt[d]*(b*c*(4*d*e - 7*c*f) - a*d*(2*d*e - 5*c*f))*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c ]], 1 - (b*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d* x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*(a*d*(d*e - 4*c*f) - 3*b*c*(d*e - 2*c *f))*Sqrt[a + b*x^2]*EllipticF[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]))/(3*c*(b*c - a*d))))/(d*e - c*f)^2 + (Sqrt[-a]*f^2*Sqrt[1 + (b*...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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[(-(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[(((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]
Leaf count of result is larger than twice the leaf count of optimal. \(8975\) vs. \(2(947)=1894\).
Time = 22.34 (sec) , antiderivative size = 8976, normalized size of antiderivative = 9.13
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(8976\) |
| default | \(\text {Expression too large to display}\) | \(11693\) |
Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
result too large to display
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(5/2)/(f*x**2+e),x)
Output:
Integral(1/((a + b*x**2)**(5/2)*(c + d*x**2)**(5/2)*(e + f*x**2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="maxima ")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x)
Output:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)*(e + f*x^2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} d^{3} f \,x^{14}+3 a \,b^{2} d^{3} f \,x^{12}+3 b^{3} c \,d^{2} f \,x^{12}+b^{3} d^{3} e \,x^{12}+3 a^{2} b \,d^{3} f \,x^{10}+9 a \,b^{2} c \,d^{2} f \,x^{10}+3 a \,b^{2} d^{3} e \,x^{10}+3 b^{3} c^{2} d f \,x^{10}+3 b^{3} c \,d^{2} e \,x^{10}+a^{3} d^{3} f \,x^{8}+9 a^{2} b c \,d^{2} f \,x^{8}+3 a^{2} b \,d^{3} e \,x^{8}+9 a \,b^{2} c^{2} d f \,x^{8}+9 a \,b^{2} c \,d^{2} e \,x^{8}+b^{3} c^{3} f \,x^{8}+3 b^{3} c^{2} d e \,x^{8}+3 a^{3} c \,d^{2} f \,x^{6}+a^{3} d^{3} e \,x^{6}+9 a^{2} b \,c^{2} d f \,x^{6}+9 a^{2} b c \,d^{2} e \,x^{6}+3 a \,b^{2} c^{3} f \,x^{6}+9 a \,b^{2} c^{2} d e \,x^{6}+b^{3} c^{3} e \,x^{6}+3 a^{3} c^{2} d f \,x^{4}+3 a^{3} c \,d^{2} e \,x^{4}+3 a^{2} b \,c^{3} f \,x^{4}+9 a^{2} b \,c^{2} d e \,x^{4}+3 a \,b^{2} c^{3} e \,x^{4}+a^{3} c^{3} f \,x^{2}+3 a^{3} c^{2} d e \,x^{2}+3 a^{2} b \,c^{3} e \,x^{2}+a^{3} c^{3} e}d x \] Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**3*c**3*e + a**3*c**3*f*x**2 + 3*a**3*c**2*d*e*x**2 + 3*a**3*c**2*d*f*x**4 + 3*a**3*c*d**2*e*x**4 + 3*a** 3*c*d**2*f*x**6 + a**3*d**3*e*x**6 + a**3*d**3*f*x**8 + 3*a**2*b*c**3*e*x* *2 + 3*a**2*b*c**3*f*x**4 + 9*a**2*b*c**2*d*e*x**4 + 9*a**2*b*c**2*d*f*x** 6 + 9*a**2*b*c*d**2*e*x**6 + 9*a**2*b*c*d**2*f*x**8 + 3*a**2*b*d**3*e*x**8 + 3*a**2*b*d**3*f*x**10 + 3*a*b**2*c**3*e*x**4 + 3*a*b**2*c**3*f*x**6 + 9 *a*b**2*c**2*d*e*x**6 + 9*a*b**2*c**2*d*f*x**8 + 9*a*b**2*c*d**2*e*x**8 + 9*a*b**2*c*d**2*f*x**10 + 3*a*b**2*d**3*e*x**10 + 3*a*b**2*d**3*f*x**12 + b**3*c**3*e*x**6 + b**3*c**3*f*x**8 + 3*b**3*c**2*d*e*x**8 + 3*b**3*c**2*d *f*x**10 + 3*b**3*c*d**2*e*x**10 + 3*b**3*c*d**2*f*x**12 + b**3*d**3*e*x** 12 + b**3*d**3*f*x**14),x)