Integrand size = 36, antiderivative size = 529 \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\frac {\left (8 (b c-a d) f (7 b d e-3 b c f+3 a d f)-5 b d \left (7 b d e^2+5 a c f^2\right )\right ) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{105 b^3 d^3}-\frac {2 f (7 b d e-3 b c f+3 a d f) x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{35 b^2 d^2}-\frac {f^2 x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}+\frac {2 \sqrt {a} \left (9 a b c d f (7 b d e-3 b c f+3 a d f)+(b c-a d) \left (8 (b c-a d) f (7 b d e-3 b c f+3 a d f)-5 b d \left (7 b d e^2+5 a c f^2\right )\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{105 b^{7/2} d^4 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} c \left (24 a^3 d^3 f^2+a^2 b d^2 f (56 d e-17 c f)+a b^2 d \left (35 d^2 e^2-42 c d e f+16 c^2 f^2\right )-2 b^3 c \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{105 b^{7/2} d^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
1/105*(8*(-a*d+b*c)*f*(3*a*d*f-3*b*c*f+7*b*d*e)-5*b*d*(5*a*c*f^2+7*b*d*e^2 ))*x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^3/d^3-2/35*f*(3*a*d*f-3*b*c*f+7*b* d*e)*x^3*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^2-1/7*f^2*x^5*(-b*x^2+a)^( 1/2)*(d*x^2+c)^(1/2)/b/d+2/105*a^(1/2)*(9*a*b*c*d*f*(3*a*d*f-3*b*c*f+7*b*d *e)+(-a*d+b*c)*(8*(-a*d+b*c)*f*(3*a*d*f-3*b*c*f+7*b*d*e)-5*b*d*(5*a*c*f^2+ 7*b*d*e^2)))*(1-b*x^2/a)^(1/2)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2) ,(-a*d/b/c)^(1/2))/b^(7/2)/d^4/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)-1/105*a^ (1/2)*c*(24*a^3*d^3*f^2+a^2*b*d^2*f*(-17*c*f+56*d*e)+a*b^2*d*(16*c^2*f^2-4 2*c*d*e*f+35*d^2*e^2)-2*b^3*c*(24*c^2*f^2-56*c*d*e*f+35*d^2*e^2))*(1-b*x^2 /a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/ b^(7/2)/d^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.52 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.87 \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {-\frac {b}{a}} d x \left (a-b x^2\right ) \left (c+d x^2\right ) \left (24 a^2 d^2 f^2+a b d f \left (56 d e-23 c f+18 d f x^2\right )+b^2 \left (24 c^2 f^2-2 c d f \left (28 e+9 f x^2\right )+d^2 \left (35 e^2+42 e f x^2+15 f^2 x^4\right )\right )\right )-2 i c \left (24 a^3 d^3 f^2+4 a^2 b d^2 f (14 d e-5 c f)+b^3 c \left (-35 d^2 e^2+56 c d e f-24 c^2 f^2\right )+a b^2 d \left (35 d^2 e^2-49 c d e f+20 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c \left (24 a^3 d^3 f^2+a^2 b d^2 f (56 d e-17 c f)+a b^2 d \left (35 d^2 e^2-42 c d e f+16 c^2 f^2\right )-2 b^3 c \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{105 b^3 \sqrt {-\frac {b}{a}} d^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(x^4*(e + f*x^2)^2)/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[-(b/a)]*d*x*(a - b*x^2)*(c + d*x^2)*(24*a^2*d^2*f^2 + a*b*d*f*(56* d*e - 23*c*f + 18*d*f*x^2) + b^2*(24*c^2*f^2 - 2*c*d*f*(28*e + 9*f*x^2) + d^2*(35*e^2 + 42*e*f*x^2 + 15*f^2*x^4)))) - (2*I)*c*(24*a^3*d^3*f^2 + 4*a^ 2*b*d^2*f*(14*d*e - 5*c*f) + b^3*c*(-35*d^2*e^2 + 56*c*d*e*f - 24*c^2*f^2) + a*b^2*d*(35*d^2*e^2 - 49*c*d*e*f + 20*c^2*f^2))*Sqrt[1 - (b*x^2)/a]*Sqr t[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] + I* c*(24*a^3*d^3*f^2 + a^2*b*d^2*f*(56*d*e - 17*c*f) + a*b^2*d*(35*d^2*e^2 - 42*c*d*e*f + 16*c^2*f^2) - 2*b^3*c*(35*d^2*e^2 - 56*c*d*e*f + 24*c^2*f^2)) *Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]* x], -((a*d)/(b*c))])/(105*b^3*Sqrt[-(b/a)]*d^4*Sqrt[a - b*x^2]*Sqrt[c + d* x^2])
Time = 1.83 (sec) , antiderivative size = 870, normalized size of antiderivative = 1.64, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {448, 444, 444, 399, 323, 323, 321, 331, 330, 327, 444, 399, 323, 323, 321, 331, 330, 327}
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.
| \(\displaystyle \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {x^6 \left (f x^2+e\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{e^2}+e \int \frac {x^4 \left (f x^2+e\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f \left (\frac {\int \frac {x^4 \left ((7 b d e-6 b c f+6 a d f) x^2+5 a c f\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\int \frac {x^2 \left ((5 b d e-4 b c f+4 a d f) x^2+3 a c f\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\int \frac {\left (-2 c (5 d e-4 c f) b^2+a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) x^2+a c (5 b d e-4 b c f+4 a d f)}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\frac {\left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\frac {\left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\frac {\left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle e \left (\frac {\frac {\frac {\left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )+\frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle e \left (\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )+\frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle e \left (\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )+\frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {f \left (\frac {\frac {\int \frac {x^2 \left (\left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2+3 a c (7 b d e-6 b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f \left (\frac {\frac {\frac {\int \frac {\left (8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2+8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) x^2+a c \left (-4 c (7 d e-6 c f) b^2+a d (28 d e-23 c f) b+24 a^2 d^2 f\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {f \left (\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f \left (\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {\frac {d x^2}{c}+1} \left (24 a^3 d^3 f+a^2 b d^2 (28 d e-17 c f)-a b^2 c d (21 d e-16 c f)+8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {c+d x^2}}{7 b d}\right )}{e^2}+e \left (\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \left (4 a^2 d^2 f+a b d (5 d e-3 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle e \left (\frac {\frac {\frac {\sqrt {a} \left (-2 c (5 d e-4 c f) b^2+a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \left (-2 c (5 d e-4 c f) b^2+a d (5 d e-3 c f) b+4 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(5 b d e-4 b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}\right )+\frac {f \left (\frac {\frac {\frac {\frac {\left (8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2+8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \left (8 c^2 (7 d e-6 c f) b^3-a c d (21 d e-16 c f) b^2+a^2 d^2 (28 d e-17 c f) b+24 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+(28 d e-23 c f) a-\frac {4 b c (7 d e-6 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(7 b d e-6 b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {d x^2+c}}{7 b d}\right )}{e^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle e \left (\frac {\frac {\frac {\sqrt {a} \left (-2 c (5 d e-4 c f) b^2+a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \left (-2 c (5 d e-4 c f) b^2+a d (5 d e-3 c f) b+4 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(5 b d e-4 b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}\right )+\frac {f \left (\frac {\frac {\frac {\frac {\left (8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2+8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c \left (8 c^2 (7 d e-6 c f) b^3-a c d (21 d e-16 c f) b^2+a^2 d^2 (28 d e-17 c f) b+24 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+(28 d e-23 c f) a-\frac {4 b c (7 d e-6 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(7 b d e-6 b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {d x^2+c}}{7 b d}\right )}{e^2}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle e \left (\frac {\frac {\frac {\sqrt {a} \left (-2 c (5 d e-4 c f) b^2+a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \left (-2 c (5 d e-4 c f) b^2+a d (5 d e-3 c f) b+4 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(5 b d e-4 b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}\right )+\frac {f \left (\frac {\frac {\frac {\frac {\left (8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2+8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c \left (8 c^2 (7 d e-6 c f) b^3-a c d (21 d e-16 c f) b^2+a^2 d^2 (28 d e-17 c f) b+24 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+(28 d e-23 c f) a-\frac {4 b c (7 d e-6 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(7 b d e-6 b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {d x^2+c}}{7 b d}\right )}{e^2}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle e \left (\frac {\frac {\frac {\sqrt {a} \left (-2 c (5 d e-4 c f) b^2+a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \left (-2 c (5 d e-4 c f) b^2+a d (5 d e-3 c f) b+4 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(5 b d e-4 b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}\right )+\frac {f \left (\frac {\frac {\frac {\frac {\left (8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2+8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \left (8 c^2 (7 d e-6 c f) b^3-a c d (21 d e-16 c f) b^2+a^2 d^2 (28 d e-17 c f) b+24 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+(28 d e-23 c f) a-\frac {4 b c (7 d e-6 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(7 b d e-6 b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {d x^2+c}}{7 b d}\right )}{e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle e \left (\frac {\frac {\frac {\sqrt {a} \left (-2 c (5 d e-4 c f) b^2+a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \left (-2 c (5 d e-4 c f) b^2+a d (5 d e-3 c f) b+4 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(5 b d e-4 b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{5 b d}-\frac {f x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}\right )+\frac {f \left (\frac {\frac {\frac {\frac {\sqrt {a} \left (8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2+8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \left (8 c^2 (7 d e-6 c f) b^3-a c d (21 d e-16 c f) b^2+a^2 d^2 (28 d e-17 c f) b+24 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+(28 d e-23 c f) a-\frac {4 b c (7 d e-6 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(7 b d e-6 b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{7 b d}-\frac {f x^5 \sqrt {a-b x^2} \sqrt {d x^2+c}}{7 b d}\right )}{e^2}\) |
Input:
Int[(x^4*(e + f*x^2)^2)/(Sqrt[a - b*x^2]*Sqrt[c + d*x^2]),x]
Output:
e*(-1/5*(f*x^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-1/3*((5*b*d*e - 4*b*c*f + 4*a*d*f)*x*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + ((Sqrt[a]*(8 *a^2*d^2*f + a*b*d*(10*d*e - 7*c*f) - 2*b^2*c*(5*d*e - 4*c*f))*Sqrt[1 - (b *x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b *c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a]*c*(4*a^2 *d^2*f - 2*b^2*c*(5*d*e - 4*c*f) + a*b*d*(5*d*e - 3*c*f))*Sqrt[1 - (b*x^2) /a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b* c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]))/(3*b*d))/(5*b*d)) + (f* (-1/7*(f*x^5*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-1/5*((7*b*d*e - 6* b*c*f + 6*a*d*f)*x^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-1/3*(((24* a^2*d*f)/b + a*(28*d*e - 23*c*f) - (4*b*c*(7*d*e - 6*c*f))/d)*x*Sqrt[a - b *x^2]*Sqrt[c + d*x^2]) + ((Sqrt[a]*(48*a^3*d^3*f - a*b^2*c*d*(49*d*e - 40* c*f) + 8*b^3*c^2*(7*d*e - 6*c*f) + 8*a^2*b*d^2*(7*d*e - 5*c*f))*Sqrt[1 - ( b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/( b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a]*c*(24*a ^3*d^3*f + a^2*b*d^2*(28*d*e - 17*c*f) - a*b^2*c*d*(21*d*e - 16*c*f) + 8*b ^3*c^2*(7*d*e - 6*c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*S qrt[c + d*x^2]))/(3*b*d))/(5*b*d))/(7*b*d)))/e^2
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[e Int[(g*x)^m*(a + b*x ^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] + Simp[f/e^2 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^(r - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q}, x] && IGtQ[r, 0]
Time = 11.12 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.19
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {f^{2} x^{5} \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{7 b d}-\frac {\left (2 e f +\frac {f^{2} \left (6 a d -6 b c \right )}{7 b d}\right ) x^{3} \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{5 b d}-\frac {\left (e^{2}+\frac {5 a c \,f^{2}}{7 b d}+\frac {\left (2 e f +\frac {f^{2} \left (6 a d -6 b c \right )}{7 b d}\right ) \left (4 a d -4 b c \right )}{5 b d}\right ) x \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 b d}+\frac {\left (e^{2}+\frac {5 a c \,f^{2}}{7 b d}+\frac {\left (2 e f +\frac {f^{2} \left (6 a d -6 b c \right )}{7 b d}\right ) \left (4 a d -4 b c \right )}{5 b d}\right ) a c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{3 b d \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (\frac {3 \left (2 e f +\frac {f^{2} \left (6 a d -6 b c \right )}{7 b d}\right ) a c}{5 b d}+\frac {\left (e^{2}+\frac {5 a c \,f^{2}}{7 b d}+\frac {\left (2 e f +\frac {f^{2} \left (6 a d -6 b c \right )}{7 b d}\right ) \left (4 a d -4 b c \right )}{5 b d}\right ) \left (2 a d -2 b c \right )}{3 b d}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(632\) |
| risch | \(\text {Expression too large to display}\) | \(1026\) |
| default | \(\text {Expression too large to display}\) | \(1719\) |
Input:
int(x^4*(f*x^2+e)^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBO SE)
Output:
((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/7*f^2/b/ d*x^5*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)-1/5*(2*e*f+1/7*f^2/b/d*(6*a*d-6 *b*c))/b/d*x^3*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)-1/3*(e^2+5/7*a/b*c/d*f ^2+1/5*(2*e*f+1/7*f^2/b/d*(6*a*d-6*b*c))/b/d*(4*a*d-4*b*c))/b/d*x*(-b*d*x^ 4+a*d*x^2-b*c*x^2+a*c)^(1/2)+1/3*(e^2+5/7*a/b*c/d*f^2+1/5*(2*e*f+1/7*f^2/b /d*(6*a*d-6*b*c))/b/d*(4*a*d-4*b*c))/b/d*a*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)*EllipticF(x*(b/a)^ (1/2),(-1-(a*d-b*c)/c/b)^(1/2))-(3/5*(2*e*f+1/7*f^2/b/d*(6*a*d-6*b*c))/b/d *a*c+1/3*(e^2+5/7*a/b*c/d*f^2+1/5*(2*e*f+1/7*f^2/b/d*(6*a*d-6*b*c))/b/d*(4 *a*d-4*b*c))/b/d*(2*a*d-2*b*c))*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*(EllipticF(x*(b/a)^(1/2),(- 1-(a*d-b*c)/c/b)^(1/2))-EllipticE(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))) )
Time = 0.10 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.15 \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\frac {2 \, {\left (35 \, {\left (a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} e^{2} - 7 \, {\left (8 \, a b^{3} c^{2} d - 7 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e f + 4 \, {\left (6 \, a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 5 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3}\right )} f^{2}\right )} \sqrt {-b d} x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + {\left (35 \, {\left (2 \, a^{2} b^{2} d^{3} - {\left (2 \, a b^{3} - b^{4}\right )} c d^{2}\right )} e^{2} + 14 \, {\left (8 \, a^{3} b d^{3} + 4 \, {\left (2 \, a b^{3} - b^{4}\right )} c^{2} d - {\left (7 \, a^{2} b^{2} - 4 \, a b^{3}\right )} c d^{2}\right )} e f + {\left (48 \, a^{4} d^{3} - 24 \, {\left (2 \, a b^{3} - b^{4}\right )} c^{3} + {\left (40 \, a^{2} b^{2} - 23 \, a b^{3}\right )} c^{2} d - 8 \, {\left (5 \, a^{3} b - 3 \, a^{2} b^{2}\right )} c d^{2}\right )} f^{2}\right )} \sqrt {-b d} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - {\left (15 \, b^{4} d^{3} f^{2} x^{6} + 6 \, {\left (7 \, b^{4} d^{3} e f - 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} f^{2}\right )} x^{4} - 70 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} e^{2} + 14 \, {\left (8 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 8 \, a^{2} b^{2} d^{3}\right )} e f - 8 \, {\left (6 \, b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 5 \, a^{2} b^{2} c d^{2} - 6 \, a^{3} b d^{3}\right )} f^{2} + {\left (35 \, b^{4} d^{3} e^{2} - 56 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} e f + {\left (24 \, b^{4} c^{2} d - 23 \, a b^{3} c d^{2} + 24 \, a^{2} b^{2} d^{3}\right )} f^{2}\right )} x^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{5} d^{4} x} \] Input:
integrate(x^4*(f*x^2+e)^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="f ricas")
Output:
1/105*(2*(35*(a*b^3*c*d^2 - a^2*b^2*d^3)*e^2 - 7*(8*a*b^3*c^2*d - 7*a^2*b^ 2*c*d^2 + 8*a^3*b*d^3)*e*f + 4*(6*a*b^3*c^3 - 5*a^2*b^2*c^2*d + 5*a^3*b*c* d^2 - 6*a^4*d^3)*f^2)*sqrt(-b*d)*x*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/b)/x ), -b*c/(a*d)) + (35*(2*a^2*b^2*d^3 - (2*a*b^3 - b^4)*c*d^2)*e^2 + 14*(8*a ^3*b*d^3 + 4*(2*a*b^3 - b^4)*c^2*d - (7*a^2*b^2 - 4*a*b^3)*c*d^2)*e*f + (4 8*a^4*d^3 - 24*(2*a*b^3 - b^4)*c^3 + (40*a^2*b^2 - 23*a*b^3)*c^2*d - 8*(5* a^3*b - 3*a^2*b^2)*c*d^2)*f^2)*sqrt(-b*d)*x*sqrt(a/b)*elliptic_f(arcsin(sq rt(a/b)/x), -b*c/(a*d)) - (15*b^4*d^3*f^2*x^6 + 6*(7*b^4*d^3*e*f - 3*(b^4* c*d^2 - a*b^3*d^3)*f^2)*x^4 - 70*(b^4*c*d^2 - a*b^3*d^3)*e^2 + 14*(8*b^4*c ^2*d - 7*a*b^3*c*d^2 + 8*a^2*b^2*d^3)*e*f - 8*(6*b^4*c^3 - 5*a*b^3*c^2*d + 5*a^2*b^2*c*d^2 - 6*a^3*b*d^3)*f^2 + (35*b^4*d^3*e^2 - 56*(b^4*c*d^2 - a* b^3*d^3)*e*f + (24*b^4*c^2*d - 23*a*b^3*c*d^2 + 24*a^2*b^2*d^3)*f^2)*x^2)* sqrt(-b*x^2 + a)*sqrt(d*x^2 + c))/(b^5*d^4*x)
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{4} \left (e + f x^{2}\right )^{2}}{\sqrt {a - b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**4*(f*x**2+e)**2/(-b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**4*(e + f*x**2)**2/(sqrt(a - b*x**2)*sqrt(c + d*x**2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2} x^{4}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="m axima")
Output:
integrate((f*x^2 + e)^2*x^4/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2} x^{4}}{\sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="g iac")
Output:
integrate((f*x^2 + e)^2*x^4/(sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^4\,{\left (f\,x^2+e\right )}^2}{\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((x^4*(e + f*x^2)^2)/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
Output:
int((x^4*(e + f*x^2)^2)/((a - b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\sqrt {a-b x^2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
int(x^4*(f*x^2+e)^2/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
( - 24*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a**2*d**2*f**2*x + 23*sqrt(c + d* x**2)*sqrt(a - b*x**2)*a*b*c*d*f**2*x - 56*sqrt(c + d*x**2)*sqrt(a - b*x** 2)*a*b*d**2*e*f*x - 18*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*b*d**2*f**2*x** 3 - 24*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b**2*c**2*f**2*x + 56*sqrt(c + d* x**2)*sqrt(a - b*x**2)*b**2*c*d*e*f*x + 18*sqrt(c + d*x**2)*sqrt(a - b*x** 2)*b**2*c*d*f**2*x**3 - 35*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b**2*d**2*e** 2*x - 42*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b**2*d**2*e*f*x**3 - 15*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b**2*d**2*f**2*x**5 + 48*int((sqrt(c + d*x**2)* sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a**3*d**3 *f**2 - 40*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a**2*b*c*d**2*f**2 + 112*int((sqrt(c + d*x**2)*sqr t(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a**2*b*d**3* e*f + 40*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b* c*x**2 - b*d*x**4),x)*a*b**2*c**2*d*f**2 - 98*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*a*b**2*c*d**2*e *f + 70*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c *x**2 - b*d*x**4),x)*a*b**2*d**3*e**2 - 48*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*b**3*c**3*f**2 + 1 12*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x**2 - b*c*x**2 - b*d*x**4),x)*b**3*c**2*d*e*f - 70*int((sqrt(c + d*x**2)*sqrt(a - b*x...