Integrand size = 36, antiderivative size = 546 \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\left (5 b^2 d e^2+6 a^2 d f^2+a b f (10 d e+c f)\right ) x^3 \sqrt {c+d x^2}}{5 b^2 d (b c+a d) \sqrt {a-b x^2}}-\frac {f^2 x^5 \sqrt {c+d x^2}}{5 b d \sqrt {a-b x^2}}+\frac {\left (\frac {24 a^2 d f^2}{b}+5 a f (8 d e+c f)+b \left (15 d e^2+10 c e f-\frac {4 c^2 f^2}{d}\right )\right ) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{15 b^2 d (b c+a d)}-\frac {\sqrt {a} \left (48 a^3 d^3 f^2+16 a^2 b d^2 f (5 d e+c f)+3 a b^2 d \left (10 d^2 e^2+10 c d e f-3 c^2 f^2\right )+b^3 c \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 b^{7/2} d^3 (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} c \left (24 a^2 d^2 f^2+a b d f (40 d e-13 c f)+b^2 \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 b^{7/2} d^3 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:
1/5*(5*b^2*d*e^2+6*a^2*d*f^2+a*b*f*(c*f+10*d*e))*x^3*(d*x^2+c)^(1/2)/b^2/d /(a*d+b*c)/(-b*x^2+a)^(1/2)-1/5*f^2*x^5*(d*x^2+c)^(1/2)/b/d/(-b*x^2+a)^(1/ 2)+1/15*(24*a^2*d*f^2/b+5*a*f*(c*f+8*d*e)+b*(15*d*e^2+10*c*e*f-4*c^2*f^2/d ))*x*(-b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d/(a*d+b*c)-1/15*a^(1/2)*(48*a^3 *d^3*f^2+16*a^2*b*d^2*f*(c*f+5*d*e)+3*a*b^2*d*(-3*c^2*f^2+10*c*d*e*f+10*d^ 2*e^2)+b^3*c*(8*c^2*f^2-20*c*d*e*f+15*d^2*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^3/(a*d+b*c )/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+1/15*a^(1/2)*c*(24*a^2*d^2*f^2+a*b*d* f*(-13*c*f+40*d*e)+b^2*(8*c^2*f^2-20*c*d*e*f+15*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^3/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.29 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.84 \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {-\frac {b}{a}} d x \left (c+d x^2\right ) \left (-24 a^3 d^2 f^2+b^3 c f x^2 \left (10 d e-4 c f+3 d f x^2\right )+a^2 b d f \left (-40 d e-5 c f+6 d f x^2\right )+a b^2 \left (4 c^2 f^2+2 c d f \left (-5 e+f x^2\right )+d^2 \left (-15 e^2+10 e f x^2+3 f^2 x^4\right )\right )\right )+i c \left (48 a^3 d^3 f^2+16 a^2 b d^2 f (5 d e+c f)+3 a b^2 d \left (10 d^2 e^2+10 c d e f-3 c^2 f^2\right )+b^3 c \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (b c+a d) \left (24 a^2 d^2 f^2+a b d f (40 d e-13 c f)+b^2 \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{15 b^3 \sqrt {-\frac {b}{a}} d^3 (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(x^4*(e + f*x^2)^2)/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[-(b/a)]*d*x*(c + d*x^2)*(-24*a^3*d^2*f^2 + b^3*c*f*x^2*(10*d*e - 4 *c*f + 3*d*f*x^2) + a^2*b*d*f*(-40*d*e - 5*c*f + 6*d*f*x^2) + a*b^2*(4*c^2 *f^2 + 2*c*d*f*(-5*e + f*x^2) + d^2*(-15*e^2 + 10*e*f*x^2 + 3*f^2*x^4)))) + I*c*(48*a^3*d^3*f^2 + 16*a^2*b*d^2*f*(5*d*e + c*f) + 3*a*b^2*d*(10*d^2*e ^2 + 10*c*d*e*f - 3*c^2*f^2) + b^3*c*(15*d^2*e^2 - 20*c*d*e*f + 8*c^2*f^2) )*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)] *x], -((a*d)/(b*c))] - I*c*(b*c + a*d)*(24*a^2*d^2*f^2 + a*b*d*f*(40*d*e - 13*c*f) + b^2*(15*d^2*e^2 - 20*c*d*e*f + 8*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))])/ (15*b^3*Sqrt[-(b/a)]*d^3*(b*c + a*d)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
Time = 1.86 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.58, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.528, Rules used = {448, 440, 25, 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.
| \(\displaystyle \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 448 |
| \(\displaystyle \frac {f \int \frac {x^6 \left (f x^2+e\right )}{\left (a-b x^2\right )^{3/2} \sqrt {d x^2+c}}dx}{e^2}+e \int \frac {x^4 \left (f x^2+e\right )}{\left (a-b x^2\right )^{3/2} \sqrt {d x^2+c}}dx\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle \frac {f \left (\frac {\int -\frac {x^4 \left ((5 b d e+b c f+6 a d f) x^2+5 c (b e+a f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{b (a d+b c)}+\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}\right )}{e^2}+e \left (\frac {\int -\frac {x^2 \left ((3 b d e+b c f+4 a d f) x^2+3 c (b e+a f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{b (a d+b c)}+\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\int \frac {x^4 \left ((5 b d e+b c f+6 a d f) x^2+5 c (b e+a f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\int \frac {x^2 \left ((3 b d e+b c f+4 a d f) x^2+3 c (b e+a f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {\left (c (3 d e-2 c f) b^2+3 a d (2 d e+c f) b+8 a^2 d^2 f\right ) x^2+a c (3 b d e+b c f+4 a d f)}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (a d+b c) (4 a d f-2 b c f+3 b d e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )+\frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )+\frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )+\frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\int \frac {\left (-2 c^2 (5 d e-4 c f) b^3+3 a c d (5 d e-3 c f) b^2+8 a^2 d^2 (5 d e+2 c f) b+48 a^3 d^3 f\right ) x^2+a c \left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {f \left (\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\right )}{e^2}+e \left (\frac {x^3 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (8 a^2 d^2 f+3 a b d (c f+2 d e)+b^2 c (3 d e-2 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (4 a d f-2 b c f+3 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {x \sqrt {a-b x^2} \sqrt {c+d x^2} (4 a d f+b c f+3 b d e)}{3 b d}}{b (a d+b c)}\right )\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle e \left (\frac {(b e+a f) x^3 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\sqrt {a} \left (c (3 d e-2 c f) b^2+3 a d (2 d e+c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (3 b d e-2 b c f+4 a d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(3 b d e+b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{b (b c+a d)}\right )+\frac {f \left (\frac {(b e+a f) x^5 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\frac {\left (-2 c^2 (5 d e-4 c f) b^3+3 a c d (5 d e-3 c f) b^2+8 a^2 d^2 (5 d e+2 c f) b+48 a^3 d^3 f\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (b c+a d) \left (-2 c (5 d e-4 c f) b^2+a d (20 d e-13 c f) b+24 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+5 (4 d e+c f) a+\frac {b c (5 d e-4 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(5 b d e+b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{b (b c+a d)}\right )}{e^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle e \left (\frac {(b e+a f) x^3 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\sqrt {a} \left (c (3 d e-2 c f) b^2+3 a d (2 d e+c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (3 b d e-2 b c f+4 a d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(3 b d e+b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{b (b c+a d)}\right )+\frac {f \left (\frac {(b e+a f) x^5 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\frac {\left (-2 c^2 (5 d e-4 c f) b^3+3 a c d (5 d e-3 c f) b^2+8 a^2 d^2 (5 d e+2 c f) b+48 a^3 d^3 f\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c (b c+a d) \left (-2 c (5 d e-4 c f) b^2+a d (20 d e-13 c f) b+24 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+5 (4 d e+c f) a+\frac {b c (5 d e-4 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(5 b d e+b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{b (b c+a d)}\right )}{e^2}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle e \left (\frac {(b e+a f) x^3 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\sqrt {a} \left (c (3 d e-2 c f) b^2+3 a d (2 d e+c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (3 b d e-2 b c f+4 a d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(3 b d e+b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{b (b c+a d)}\right )+\frac {f \left (\frac {(b e+a f) x^5 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\frac {\left (-2 c^2 (5 d e-4 c f) b^3+3 a c d (5 d e-3 c f) b^2+8 a^2 d^2 (5 d e+2 c f) b+48 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c (b c+a d) \left (-2 c (5 d e-4 c f) b^2+a d (20 d e-13 c f) b+24 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+5 (4 d e+c f) a+\frac {b c (5 d e-4 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(5 b d e+b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{b (b c+a d)}\right )}{e^2}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle e \left (\frac {(b e+a f) x^3 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\sqrt {a} \left (c (3 d e-2 c f) b^2+3 a d (2 d e+c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (3 b d e-2 b c f+4 a d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(3 b d e+b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{b (b c+a d)}\right )+\frac {f \left (\frac {(b e+a f) x^5 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\frac {\left (-2 c^2 (5 d e-4 c f) b^3+3 a c d (5 d e-3 c f) b^2+8 a^2 d^2 (5 d e+2 c f) b+48 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) \left (-2 c (5 d e-4 c f) b^2+a d (20 d e-13 c f) b+24 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+5 (4 d e+c f) a+\frac {b c (5 d e-4 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(5 b d e+b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{b (b c+a d)}\right )}{e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle e \left (\frac {(b e+a f) x^3 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\sqrt {a} \left (c (3 d e-2 c f) b^2+3 a d (2 d e+c f) b+8 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) (3 b d e-2 b c f+4 a d f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {(3 b d e+b c f+4 a d f) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{3 b d}}{b (b c+a d)}\right )+\frac {f \left (\frac {(b e+a f) x^5 \sqrt {d x^2+c}}{b (b c+a d) \sqrt {a-b x^2}}-\frac {\frac {\frac {\frac {\sqrt {a} \left (-2 c^2 (5 d e-4 c f) b^3+3 a c d (5 d e-3 c f) b^2+8 a^2 d^2 (5 d e+2 c f) b+48 a^3 d^3 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (b c+a d) \left (-2 c (5 d e-4 c f) b^2+a d (20 d e-13 c f) b+24 a^2 d^2 f\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {d x^2+c}}}{3 b d}-\frac {1}{3} \left (\frac {24 d f a^2}{b}+5 (4 d e+c f) a+\frac {b c (5 d e-4 c f)}{d}\right ) x \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}-\frac {(5 b d e+b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}{5 b d}}{b (b c+a d)}\right )}{e^2}\) |
Input:
Int[(x^4*(e + f*x^2)^2)/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
e*(((b*e + a*f)*x^3*Sqrt[c + d*x^2])/(b*(b*c + a*d)*Sqrt[a - b*x^2]) - (-1 /3*((3*b*d*e + 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 + b^2*c*(3*d*e - 2*c*f) + 3*a*b*d*(2*d*e + c*f))*S qrt[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*(b*c + a*d)*(3*b*d*e - 2*b*c*f + 4*a*d*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))/(b*(b*c + a*d))) + (f*(((b *e + a*f)*x^5*Sqrt[c + d*x^2])/(b*(b*c + a*d)*Sqrt[a - b*x^2]) - (-1/5*((5 *b*d*e + 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 + (b*c*(5*d*e - 4*c*f))/d + 5*a*(4*d*e + c*f))*x*Sqrt[ a - b*x^2]*Sqrt[c + d*x^2]) + ((Sqrt[a]*(48*a^3*d^3*f - 2*b^3*c^2*(5*d*e - 4*c*f) + 3*a*b^2*c*d*(5*d*e - 3*c*f) + 8*a^2*b*d^2*(5*d*e + 2*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* (b*c + a*d)*(24*a^2*d^2*f + a*b*d*(20*d*e - 13*c*f) - 2*b^2*c*(5*d*e - 4*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))/(b*(b*c + a*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[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
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 = 21.08 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.19
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) a x \left (a^{2} f^{2}+2 a b f e +b^{2} e^{2}\right )}{b^{4} \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {f^{2} x^{3} \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{5 b^{2} d}-\frac {\left (-\frac {f \left (a f +2 b e \right )}{b^{2}}-\frac {f^{2} \left (4 a d -4 b c \right )}{5 b^{2} d}\right ) x \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 b d}+\frac {\left (-\frac {c a \left (a^{2} f^{2}+2 a b f e +b^{2} e^{2}\right )}{b^{3} \left (a d +b c \right )}+\frac {\left (-\frac {f \left (a f +2 b e \right )}{b^{2}}-\frac {f^{2} \left (4 a d -4 b c \right )}{5 b^{2} d}\right ) a c}{3 b d}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (-\frac {a^{2} f^{2}+2 a b f e +b^{2} e^{2}}{b^{3}}-\frac {d a \left (a^{2} f^{2}+2 a b f e +b^{2} e^{2}\right )}{b^{3} \left (a d +b c \right )}-\frac {3 f^{2} a c}{5 b^{2} d}+\frac {\left (-\frac {f \left (a f +2 b e \right )}{b^{2}}-\frac {f^{2} \left (4 a d -4 b c \right )}{5 b^{2} d}\right ) \left (2 a d -2 b c \right )}{3 b d}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(652\) |
| risch | \(\frac {f x \left (3 b d f \,x^{2}+9 a d f -4 b c f +10 b d e \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{3} d^{2}}-\frac {\left (-\frac {\left (33 a^{2} d^{2} f^{2}-17 a b c d \,f^{2}+50 a b \,d^{2} e f +8 b^{2} c^{2} f^{2}-20 b^{2} c d e f +15 b^{2} d^{2} e^{2}\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}+\frac {a \left (15 a^{2} d^{2} f^{2}+9 a b c d \,f^{2}+30 a b \,d^{2} e f -4 b^{2} c^{2} f^{2}+10 b^{2} c d e f +15 b^{2} d^{2} e^{2}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{b \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {15 a^{2} d^{2} \left (a^{2} f^{2}+2 a b f e +b^{2} e^{2}\right ) \left (\frac {\left (-b d \,x^{2}-b c \right ) x}{a \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\left (-\frac {1}{a}+\frac {b c}{a \left (a d +b c \right )}\right ) \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 )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b 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 )}{a \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{b}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b^{3} d^{2} \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(774\) |
| default | \(\text {Expression too large to display}\) | \(1621\) |
Input:
int(x^4*(f*x^2+e)^2/(-b*x^2+a)^(3/2)/(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)*(-(-b*d*x^2- b*c)/b^4*a/(a*d+b*c)*x*(a^2*f^2+2*a*b*e*f+b^2*e^2)/((x^2-a/b)*(-b*d*x^2-b* c))^(1/2)+1/5*f^2/b^2/d*x^3*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)-1/3*(-1/b ^2*f*(a*f+2*b*e)-1/5*f^2/b^2/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/b^3*c*a/(a*d+b*c)*(a^2*f^2+2*a*b*e*f+b^2*e^2)+1/3*(-1/b ^2*f*(a*f+2*b*e)-1/5*f^2/b^2/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)*Elliptic F(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-(-1/b^3*(a^2*f^2+2*a*b*e*f+b^2*e ^2)-1/b^3*d*a*(a^2*f^2+2*a*b*e*f+b^2*e^2)/(a*d+b*c)-3/5*f^2/b^2/d*a*c+1/3* (-1/b^2*f*(a*f+2*b*e)-1/5*f^2/b^2/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))))
Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (494) = 988\).
Time = 0.12 (sec) , antiderivative size = 1025, normalized size of antiderivative = 1.88 \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
integrate(x^4*(f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="f ricas")
Output:
1/15*(((15*(a*b^4*c*d^2 + 2*a^2*b^3*d^3)*e^2 - 10*(2*a*b^4*c^2*d - 3*a^2*b ^3*c*d^2 - 8*a^3*b^2*d^3)*e*f + (8*a*b^4*c^3 - 9*a^2*b^3*c^2*d + 16*a^3*b^ 2*c*d^2 + 48*a^4*b*d^3)*f^2)*x^3 - (15*(a^2*b^3*c*d^2 + 2*a^3*b^2*d^3)*e^2 - 10*(2*a^2*b^3*c^2*d - 3*a^3*b^2*c*d^2 - 8*a^4*b*d^3)*e*f + (8*a^2*b^3*c ^3 - 9*a^3*b^2*c^2*d + 16*a^4*b*c*d^2 + 48*a^5*d^3)*f^2)*x)*sqrt(-b*d)*sqr t(a/b)*elliptic_e(arcsin(sqrt(a/b)/x), -b*c/(a*d)) - ((15*(2*a^2*b^3*d^3 + (a*b^4 + b^5)*c*d^2)*e^2 + 10*(8*a^3*b^2*d^3 - (2*a*b^4 - b^5)*c^2*d + (3 *a^2*b^3 + 4*a*b^4)*c*d^2)*e*f + (48*a^4*b*d^3 + 4*(2*a*b^4 - b^5)*c^3 - ( 9*a^2*b^3 - 5*a*b^4)*c^2*d + 8*(2*a^3*b^2 + 3*a^2*b^3)*c*d^2)*f^2)*x^3 - ( 15*(2*a^3*b^2*d^3 + (a^2*b^3 + a*b^4)*c*d^2)*e^2 + 10*(8*a^4*b*d^3 - (2*a^ 2*b^3 - a*b^4)*c^2*d + (3*a^3*b^2 + 4*a^2*b^3)*c*d^2)*e*f + (48*a^5*d^3 + 4*(2*a^2*b^3 - a*b^4)*c^3 - (9*a^3*b^2 - 5*a^2*b^3)*c^2*d + 8*(2*a^4*b + 3 *a^3*b^2)*c*d^2)*f^2)*x)*sqrt(-b*d)*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/ x), -b*c/(a*d)) + (3*(b^5*c*d^2 + a*b^4*d^3)*f^2*x^6 + 2*(5*(b^5*c*d^2 + a *b^4*d^3)*e*f - (2*b^5*c^2*d - a*b^4*c*d^2 - 3*a^2*b^3*d^3)*f^2)*x^4 - 15* (a*b^4*c*d^2 + 2*a^2*b^3*d^3)*e^2 + 10*(2*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 - 8*a^3*b^2*d^3)*e*f - (8*a*b^4*c^3 - 9*a^2*b^3*c^2*d + 16*a^3*b^2*c*d^2 + 4 8*a^4*b*d^3)*f^2 + (15*(b^5*c*d^2 + a*b^4*d^3)*e^2 - 20*(b^5*c^2*d - a*b^4 *c*d^2 - 2*a^2*b^3*d^3)*e*f + (8*b^5*c^3 - 5*a*b^4*c^2*d + 11*a^2*b^3*c*d^ 2 + 24*a^3*b^2*d^3)*f^2)*x^2)*sqrt(-b*x^2 + a)*sqrt(d*x^2 + c))/((b^7*c...
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{4} \left (e + f x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**4*(f*x**2+e)**2/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**4*(e + f*x**2)**2/((a - b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2} x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="m axima")
Output:
integrate((f*x^2 + e)^2*x^4/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2} x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="g iac")
Output:
integrate((f*x^2 + e)^2*x^4/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^4\,{\left (f\,x^2+e\right )}^2}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((x^4*(e + f*x^2)^2)/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
Output:
int((x^4*(e + f*x^2)^2)/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )^2}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\text {too large to display} \] Input:
int(x^4*(f*x^2+e)^2/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
(9*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*c*d*f**2*x - 6*sqrt(c + d*x**2)*sqr t(a - b*x**2)*a*d**2*f**2*x**3 - 6*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*c** 2*f**2*x + 15*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*c*d*e*f*x + 4*sqrt(c + d *x**2)*sqrt(a - b*x**2)*b*c*d*f**2*x**3 - 10*sqrt(c + d*x**2)*sqrt(a - b*x **2)*b*d**2*e*f*x**3 - 3*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*d**2*f**2*x** 5 + 24*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**3*d**3*f* *2 - 4*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c*d** 2*f**2 + 40*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d* x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b* d**3*e*f - 24*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2* d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2* b*d**3*f**2*x**2 + 2*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x )*a*b**2*c**2*d*f**2 - 5*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a** 2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x** 6),x)*a*b**2*c*d**2*e*f + 4*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/( a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d* x**6),x)*a*b**2*c*d**2*f**2*x**2 + 15*int((sqrt(c + d*x**2)*sqrt(a - b*...