Integrand size = 29, antiderivative size = 546 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 a b x \sqrt {1-c^2 x^2}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^3 (a+b \arcsin (c x))}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \arcsin (c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{3 c^6 d^3}-\frac {22 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \]
1/3*x^4*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)+1/3*b^2/c^6/d^2/(-c ^2*d*x^2+d)^(1/2)+2*b^2*(-c^2*x^2+1)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-4/3*x^2* (a+b*arcsin(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*x^3*(a+b*arcsin(c*x ))/c^3/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+16/3*a*b*x*(-c^2*x^2+1) ^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+16/3*b^2*x*arcsin(c*x)*(-c^2*x^2+1)^(1 /2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-11/3*b*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^( 1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-22/3*I*b*(a+b*arcsin(c*x))*arctan(I*c*x+ (-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)+11/3*I *b^2*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^6/d^2/( -c^2*d*x^2+d)^(1/2)-11/3*I*b^2*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c ^2*x^2+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*(a+b*arcsin(c*x))^2*(-c^2 *d*x^2+d)^(1/2)/c^6/d^3
Time = 1.52 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.09 \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (-64 a^2+22 b^2+96 a^2 c^2 x^2-24 a^2 c^4 x^4-50 a b \arcsin (c x)-25 b^2 \arcsin (c x)^2+28 b^2 \cos (2 \arcsin (c x))-72 a b \arcsin (c x) \cos (2 \arcsin (c x))-36 b^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))+6 b^2 \cos (4 \arcsin (c x))-6 a b \arcsin (c x) \cos (4 \arcsin (c x))-3 b^2 \arcsin (c x)^2 \cos (4 \arcsin (c x))+66 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+22 b^2 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )-66 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-22 b^2 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )-66 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-22 a b \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+66 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+22 a b \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+88 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-88 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+8 a b \sin (2 \arcsin (c x))+8 b^2 \arcsin (c x) \sin (2 \arcsin (c x))+6 a b \sin (4 \arcsin (c x))+6 b^2 \arcsin (c x) \sin (4 \arcsin (c x))\right )}{24 c^6 d^3 \left (-1+c^2 x^2\right )^2} \]
(Sqrt[d - c^2*d*x^2]*(-64*a^2 + 22*b^2 + 96*a^2*c^2*x^2 - 24*a^2*c^4*x^4 - 50*a*b*ArcSin[c*x] - 25*b^2*ArcSin[c*x]^2 + 28*b^2*Cos[2*ArcSin[c*x]] - 7 2*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 36*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c *x]] + 6*b^2*Cos[4*ArcSin[c*x]] - 6*a*b*ArcSin[c*x]*Cos[4*ArcSin[c*x]] - 3 *b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 66*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c* x]*Log[1 - I*E^(I*ArcSin[c*x])] + 22*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Lo g[1 - I*E^(I*ArcSin[c*x])] - 66*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 22*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + I*E^( I*ArcSin[c*x])] - 66*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[Ar cSin[c*x]/2]] - 22*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[Arc Sin[c*x]/2]] + 66*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSi n[c*x]/2]] + 22*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin [c*x]/2]] + (88*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x ])] - (88*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])] + 8*a *b*Sin[2*ArcSin[c*x]] + 8*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]]))/(24*c^6*d^3*(-1 + c ^2*x^2)^2)
Time = 2.30 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5206, 5206, 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.
| \(\displaystyle \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^4 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{2 c^2}-\frac {b \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{2 c^2}-\frac {b \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2}}dx^2}{4 c}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{2 c^2}-\frac {b \int \left (\frac {1}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{c^2 \sqrt {1-c^2 x^2}}\right )dx^2}{4 c}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))}{c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))}{c^2}\right )}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {2 b \sqrt {1-c^2 x^2} \left (a x+b x \arcsin (c x)+\frac {b \sqrt {1-c^2 x^2}}{c}\right )}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^2 d}\right )}{c^2 d}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^3}-\frac {x (a+b \arcsin (c x))}{c^2}-\frac {b \sqrt {1-c^2 x^2}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {2 \sqrt {1-c^2 x^2}}{c^4}+\frac {2}{c^4 \sqrt {1-c^2 x^2}}\right )}{4 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
(x^4*(a + b*ArcSin[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (2*b*Sqrt[1 - c^2*x^2]*(-1/4*(b*(2/(c^4*Sqrt[1 - c^2*x^2]) + (2*Sqrt[1 - c^2*x^2])/c^4 ))/c + (x^3*(a + b*ArcSin[c*x]))/(2*c^2*(1 - c^2*x^2)) - (3*(-((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))/(2*c^2)))/(3*c*d^2*Sqrt[d - c^2*d* x^2]) - (4*((x^2*(a + b*ArcSin[c*x])^2)/(c^2*d*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])))/(c^2*d) - (2*b*Sqrt[1 - c^2*x^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*ArcS in[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*d*Sqrt[d - c^2*d*x^2])))/(3*c^2*d)
3.3.54.3.1 Defintions of rubi rules used
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_) + (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]
Time = 0.58 (sec) , antiderivative size = 815, normalized size of antiderivative = 1.49
| method | result | size |
| default | \(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -c^{2} x^{2}-5 \arcsin \left (c x \right )^{2}+1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} c^{6} d^{3}}+\frac {11 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-10 \arcsin \left (c x \right )\right )}{6 c^{6} \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(815\) |
| parts | \(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \arcsin \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -c^{2} x^{2}-5 \arcsin \left (c x \right )^{2}+1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} c^{6} d^{3}}+\frac {11 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )+i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right )}{2 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (12 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-10 \arcsin \left (c x \right )\right )}{6 c^{6} \left (c^{2} x^{2}-1\right )^{2} d^{3}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(815\) |
a^2*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2) -2/3/d/c^4/(-c^2*d*x^2+d)^(3/2)))+b^2*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^ 2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x)^2-2+2*I*arcsin(c*x))/c^6/d^3/(c ^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)* (arcsin(c*x)^2-2-2*I*arcsin(c*x))/c^6/d^3/(c^2*x^2-1)+1/3*(-d*(c^2*x^2-1)) ^(1/2)*(6*arcsin(c*x)^2*x^2*c^2-(-c^2*x^2+1)^(1/2)*arcsin(c*x)*x*c-c^2*x^2 -5*arcsin(c*x)^2+1)/(c^2*x^2-1)^2/c^6/d^3+11/3*(-c^2*x^2+1)^(1/2)*(-d*(c^2 *x^2-1))^(1/2)*(arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-arcsin(c*x) *ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))-I*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)) )+I*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))))/c^6/d^3/(c^2*x^2-1))+2*a*b*(-1/ 2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(c*x) +I)/c^6/d^3/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x *c+c^2*x^2-1)*(arcsin(c*x)-I)/c^6/d^3/(c^2*x^2-1)+1/6*(-d*(c^2*x^2-1))^(1/ 2)*(12*c^2*x^2*arcsin(c*x)-c*x*(-c^2*x^2+1)^(1/2)-10*arcsin(c*x))/c^6/(c^2 *x^2-1)^2/d^3+11/6*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2* x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)-11/6*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2 -1))^(1/2)/c^6/d^3/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I))
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*a^2*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^ (3/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(3/2)*c^6*d)) - 1/3*((3*b^2*c^4*x^4 - 1 2*b^2*c^2*x^2 + 8*b^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*arctan2(c*x, s qrt(c*x + 1)*sqrt(-c*x + 1))^2 + 3*(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3 )*integrate(2/3*(3*sqrt(c*x + 1)*sqrt(-c*x + 1)*a*b*c^5*sqrt(d)*x^5*arctan 2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) - (3*b^2*c^6*x^6 - 15*b^2*c^4*x^4 + 2 0*b^2*c^2*x^2 - 8*b^2)*sqrt(d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))) /(c^11*d^3*x^6 - 3*c^9*d^3*x^4 + 3*c^7*d^3*x^2 - c^5*d^3), x))/(c^10*d^3*x ^4 - 2*c^8*d^3*x^2 + c^6*d^3)
Exception generated. \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]