Integrand size = 29, antiderivative size = 332 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x (a+b \arcsin (c x))}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \arcsin (c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {10 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \]
1/3*x^2*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)+1/3*b^2/c^4/d^2/(-c ^2*d*x^2+d)^(1/2)-2/3*(a+b*arcsin(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/3 *b*x*(a+b*arcsin(c*x))/c^3/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-10/ 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^4/d^2/(-c^2*d*x^2+d)^(1/2)+5/3*I*b^2*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^ (1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-5/3*I*b^2*polylog( 2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^4/d^2/(-c^2*d*x^2+d)^ (1/2)
Time = 0.83 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.54 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {8 a^2-2 b^2-12 a^2 c^2 x^2+4 a b \arcsin (c x)+2 b^2 \arcsin (c x)^2-2 b^2 \cos (2 \arcsin (c x))+12 a b \arcsin (c x) \cos (2 \arcsin (c x))+6 b^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))-15 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-5 b^2 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )+15 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+5 b^2 \arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )+15 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+5 a b \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-15 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-5 a b \cos (3 \arcsin (c x)) \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-20 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+20 i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+2 a b \sin (2 \arcsin (c x))+2 b^2 \arcsin (c x) \sin (2 \arcsin (c x))}{12 c^4 d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
(8*a^2 - 2*b^2 - 12*a^2*c^2*x^2 + 4*a*b*ArcSin[c*x] + 2*b^2*ArcSin[c*x]^2 - 2*b^2*Cos[2*ArcSin[c*x]] + 12*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 6*b^2 *ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 15*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*L og[1 - I*E^(I*ArcSin[c*x])] - 5*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - I*E^(I*ArcSin[c*x])] + 15*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^( I*ArcSin[c*x])] + 5*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + I*E^(I*ArcS in[c*x])] + 15*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c *x]/2]] + 5*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x ]/2]] - 15*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/ 2]] - 5*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2] ] - (20*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2 0*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])] + 2*a*b*Sin[2 *ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/(12*c^4*d^2*(-1 + c^ 2*x^2)*Sqrt[d - c^2*d*x^2])
Time = 1.86 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5206, 5182, 5164, 3042, 4669, 2715, 2838, 5206, 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.
| \(\displaystyle \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {x^2 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{2 c^2}-\frac {b \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c}+\frac {x (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx}{2 c^2}+\frac {x (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c^3}+\frac {x (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c^3}+\frac {x (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c^3}+\frac {x (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c^3}+\frac {x (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 \left (\frac {(a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c^3}+\frac {x (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b}{2 c^3 \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
(x^2*(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]*(-1/2*b/(c^3*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(2*c^ 2*(1 - c^2*x^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 ])])/(2*c^3)))/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (2*((a + b*ArcSin[c*x])^2/( c^2*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sqrt[1 - c^2*x^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^2*d*Sqrt[d - c^2*d*x^2])))/(3*c^ 2*d)
3.3.56.3.1 Defintions of rubi rules used
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[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]
Time = 0.21 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(a^{2} \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -c^{2} x^{2}-2 \arcsin \left (c x \right )^{2}+1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}+\frac {5 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right ) c^{4}}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-4 \arcsin \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(505\) |
| parts | \(a^{2} \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (3 \arcsin \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -c^{2} x^{2}-2 \arcsin \left (c x \right )^{2}+1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}+\frac {5 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right ) c^{4}}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-4 \arcsin \left (c x \right )\right )}{6 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{4}}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(505\) |
a^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 /3*(-d*(c^2*x^2-1))^(1/2)*(3*arcsin(c*x)^2*x^2*c^2-(-c^2*x^2+1)^(1/2)*arcs in(c*x)*x*c-c^2*x^2-2*arcsin(c*x)^2+1)/(c^2*x^2-1)^2/d^3/c^4+5/3*(-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))))/d^3/(c^2*x^2- 1)/c^4)+2*a*b*(1/6*(-d*(c^2*x^2-1))^(1/2)*(6*c^2*x^2*arcsin(c*x)-c*x*(-c^2 *x^2+1)^(1/2)-4*arcsin(c*x))/(c^2*x^2-1)^2/d^3/c^4-5/6*(-c^2*x^2+1)^(1/2)* (-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)/c^4*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)+ 5/6*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)/c^4*ln(I*c*x +(-c^2*x^2+1)^(1/2)-I))
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(-(b^2*x^3*arcsin(c*x)^2 + 2*a*b*x^3*arcsin(c*x) + a^2*x^3)*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)
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/6*a*b*c*(2*x/(c^6*d^(5/2)*x^2 - c^4*d^(5/2)) + 5*log(c*x + 1)/(c^5*d^(5/ 2)) - 5*log(c*x - 1)/(c^5*d^(5/2))) + 2/3*a*b*(3*x^2/((-c^2*d*x^2 + d)^(3/ 2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d))*arcsin(c*x) + 1/3*a^2*(3*x^2 /((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d)) + b^2* integrate(x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
Exception generated. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]