Integrand size = 29, antiderivative size = 440 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {32 b^2 \sqrt {d-c^2 d x^2}}{9 c^6 d^2}+\frac {2 b^2 \left (d-c^2 d x^2\right )^{3/2}}{27 c^6 d^3}-\frac {10 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^5 d \sqrt {d-c^2 d x^2}}-\frac {2 b x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^6 d \sqrt {d-c^2 d x^2}} \] Output:
-32/9*b^2*(-c^2*d*x^2+d)^(1/2)/c^6/d^2+2/27*b^2*(-c^2*d*x^2+d)^(3/2)/c^6/d ^3-10/3*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^5/d/(-c^2*d*x^2+d)^(1/2 )-2/9*b*x^3*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/c^3/d/(-c^2*d*x^2+d)^(1/2 )+x^4*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+8/3*(-c^2*d*x^2+d)^(1 /2)*(a+b*arcsin(c*x))^2/c^6/d^2+4/3*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c *x))^2/c^4/d^2+4*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*arctan(I*c*x+(-c ^2*x^2+1)^(1/2))/c^6/d/(-c^2*d*x^2+d)^(1/2)-2*I*b^2*(-c^2*x^2+1)^(1/2)*pol ylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^6/d/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*( -c^2*x^2+1)^(1/2)*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^6/d/(-c^2*d*x^ 2+d)^(1/2)
Time = 0.66 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.03 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {576 a^2-378 b^2-288 a^2 c^2 x^2-72 a^2 c^4 x^4+810 a b \arcsin (c x)+405 b^2 \arcsin (c x)^2-376 b^2 \cos (2 \arcsin (c x))+360 a b \arcsin (c x) \cos (2 \arcsin (c x))+180 b^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))+2 b^2 \cos (4 \arcsin (c x))-18 a b \arcsin (c x) \cos (4 \arcsin (c x))-9 b^2 \arcsin (c x)^2 \cos (4 \arcsin (c x))-432 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+432 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+432 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-432 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-432 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+432 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-372 a b \sin (2 \arcsin (c x))-372 b^2 \arcsin (c x) \sin (2 \arcsin (c x))+6 a b \sin (4 \arcsin (c x))+6 b^2 \arcsin (c x) \sin (4 \arcsin (c x))}{216 c^6 d \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(x^5*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]
Output:
(576*a^2 - 378*b^2 - 288*a^2*c^2*x^2 - 72*a^2*c^4*x^4 + 810*a*b*ArcSin[c*x ] + 405*b^2*ArcSin[c*x]^2 - 376*b^2*Cos[2*ArcSin[c*x]] + 360*a*b*ArcSin[c* x]*Cos[2*ArcSin[c*x]] + 180*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + 2*b^2*C os[4*ArcSin[c*x]] - 18*a*b*ArcSin[c*x]*Cos[4*ArcSin[c*x]] - 9*b^2*ArcSin[c *x]^2*Cos[4*ArcSin[c*x]] - 432*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I *E^(I*ArcSin[c*x])] + 432*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I *ArcSin[c*x])] + 432*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[Ar cSin[c*x]/2]] - 432*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[Arc Sin[c*x]/2]] - (432*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c *x])] + (432*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])] - 37 2*a*b*Sin[2*ArcSin[c*x]] - 372*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] + 6*a*b* Sin[4*ArcSin[c*x]] + 6*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]])/(216*c^6*d*Sqrt [d - c^2*d*x^2])
Time = 3.05 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.15, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {5206, 5210, 243, 53, 2009, 5138, 243, 53, 2009, 5182, 2009, 5210, 241, 5164, 3042, 4669, 2715, 2838}
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.
| \(\displaystyle \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^4 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int x^2 (a+b \arcsin (c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx}{3 c}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int x^2 (a+b \arcsin (c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2}{6 c}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int x^2 (a+b \arcsin (c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {b \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2}{6 c}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {1-c^2 x^2} \int x^2 (a+b \arcsin (c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle -\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {1-c^2 x^2}}dx\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx^2\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \int \left (\frac {1}{c^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c^2}\right )dx^2\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {4 \left (\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))}{c^2}}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}}{c^2}-\frac {x^3 (a+b \arcsin (c x))}{3 c^2}+\frac {b \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )}{6 c}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{3 c^2}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{3} x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {2 \left (1-c^2 x^2\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {1-c^2 x^2}}{c^4}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\) |
Input:
Int[(x^5*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]
Output:
(x^4*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (4*(-1/3*(x^2*Sq rt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*Sqrt[1 - c^2*x^2]* (-1/6*(b*c*((-2*Sqrt[1 - c^2*x^2])/c^4 + (2*(1 - c^2*x^2)^(3/2))/(3*c^4))) + (x^3*(a + b*ArcSin[c*x]))/3))/(3*c*Sqrt[d - c^2*d*x^2]) + (2*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d)) + (2*b*Sqrt[1 - c^2*x^2]*(a* x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/(c*Sqrt[d - c^2*d*x^2])))/ (3*c^2)))/(c^2*d) - (2*b*Sqrt[1 - c^2*x^2]*((b*((-2*Sqrt[1 - c^2*x^2])/c^4 + (2*(1 - c^2*x^2)^(3/2))/(3*c^4)))/(6*c) - (x^3*(a + b*ArcSin[c*x]))/(3* c^2) + (-((b*Sqrt[1 - c^2*x^2])/c^3) - (x*(a + b*ArcSin[c*x]))/c^2 + ((-2* I)*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, (-I)*E^( I*ArcSin[c*x])] - I*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/c^3)/c^2))/(c*d*Sqr t[d - c^2*d*x^2])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp [b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG tQ[m, 1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1086 vs. \(2 (417 ) = 834\).
Time = 0.86 (sec) , antiderivative size = 1087, normalized size of antiderivative = 2.47
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1087\) |
| parts | \(\text {Expression too large to display}\) | \(1087\) |
Input:
int(x^5*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
a^2*(-1/3*x^4/c^2/d/(-c^2*d*x^2+d)^(1/2)+4/3/c^2*(-x^2/c^2/d/(-c^2*d*x^2+d )^(1/2)+2/d/c^4/(-c^2*d*x^2+d)^(1/2)))-94/27*b^2*(-d*(c^2*x^2-1))^(1/2)/d^ 2/c^4/(c^2*x^2-1)*x^2-2*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/ c^6/(c^2*x^2-1)*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*b^2*(-c^2 *x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c*x)*ln(1- I*(I*c*x+(-c^2*x^2+1)^(1/2)))-2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^ (1/2)/d^2/c^6/(c^2*x^2-1)*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*I*b^2*(- c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*dilog(1+I*(I*c *x+(-c^2*x^2+1)^(1/2)))+31/9*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1 )*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x-65/24*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^ 6/(c^2*x^2-1)*arcsin(c*x)^2-1/108*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2* x^2-1)*cos(4*arcsin(c*x))+1/24*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2 -1)*cos(4*arcsin(c*x))*arcsin(c*x)^2+377/108*b^2*(-d*(c^2*x^2-1))^(1/2)/d^ 2/c^6/(c^2*x^2-1)+5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c^2*x^2-1)*arcsi n(c*x)^2*x^2-1/36*b^2*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c* x)*sin(4*arcsin(c*x))-1/36*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)* sin(4*arcsin(c*x))+10/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^4/(c^2*x^2-1)*arc sin(c*x)*x^2+31/9*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*(-c^2*x^2 +1)^(1/2)*x-65/12*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c* x)+1/12*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^6/(c^2*x^2-1)*arcsin(c*x)*cos(...
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="frica s")
Output:
integral((b^2*x^5*arcsin(c*x)^2 + 2*a*b*x^5*arcsin(c*x) + a^2*x^5)*sqrt(-c ^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x**5*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(3/2),x)
Output:
Integral(x**5*(a + b*asin(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxim a")
Output:
-1/3*a^2*(x^4/(sqrt(-c^2*d*x^2 + d)*c^2*d) + 4*x^2/(sqrt(-c^2*d*x^2 + d)*c ^4*d) - 8/(sqrt(-c^2*d*x^2 + d)*c^6*d)) + 1/3*((b^2*c^4*x^4 + 4*b^2*c^2*x^ 2 - 8*b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*arctan2(c*x, sqrt(c*x + 1) *sqrt(-c*x + 1))^2 + 3*(c^8*d^2*x^2 - c^6*d^2)*integrate(2/3*(3*sqrt(c*x + 1)*sqrt(-c*x + 1)*a*b*c^5*sqrt(d)*x^5*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c* x + 1)) - (b^2*c^6*x^6 + 3*b^2*c^4*x^4 - 12*b^2*c^2*x^2 + 8*b^2)*sqrt(d)*a rctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(c^9*d^2*x^4 - 2*c^7*d^2*x^2 + c^5*d^2), x))/(c^8*d^2*x^2 - c^6*d^2)
Exception generated. \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^5*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(3/2),x)
Output:
int((x^5*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(3/2), x)
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right ) x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{6}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2} x^{5}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{6}-a^{2} c^{4} x^{4}-4 a^{2} c^{2} x^{2}+8 a^{2}}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{6} d} \] Input:
int(x^5*(a+b*asin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x)
Output:
( - 6*sqrt( - c**2*x**2 + 1)*int((asin(c*x)*x**5)/(sqrt( - c**2*x**2 + 1)* c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*a*b*c**6 - 3*sqrt( - c**2*x**2 + 1) *int((asin(c*x)**2*x**5)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2* x**2 + 1)),x)*b**2*c**6 - a**2*c**4*x**4 - 4*a**2*c**2*x**2 + 8*a**2)/(3*s qrt(d)*sqrt( - c**2*x**2 + 1)*c**6*d)