Integrand size = 27, antiderivative size = 294 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{3 c^2 d^2 \sqrt {d-c^2 d x^2}} \]
1/3*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)+1/3*b^2/c^2/d^2/(-c^2*d *x^2+d)^(1/2)-1/3*b*x*(a+b*arcsin(c*x))/c/d^2/(-c^2*x^2+1)^(1/2)/(-c^2*d*x ^2+d)^(1/2)+2/3*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^2/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*I*b^2*polylog(2,-I*(I*c*x +(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^2/d^2/(-c^2*d*x^2+d)^(1/2)+1/3* I*b^2*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1)^(1/2)/c^2/d^2/( -c^2*d*x^2+d)^(1/2)
Time = 1.80 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.57 \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {a^2 \sqrt {-d \left (-1+c^2 x^2\right )}}{3 c^2 d^3 \left (-1+c^2 x^2\right )^2}+\frac {a b \left (8 \arcsin (c x)+3 \sqrt {1-c^2 x^2} \left (\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )+\cos (3 \arcsin (c x)) \left (\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-\log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )\right )-2 \sin (2 \arcsin (c x))\right )}{12 c^2 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}}+\frac {b^2 \left (2+4 \arcsin (c x)^2+2 \cos (2 \arcsin (c x))-3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-\arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )+3 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+\arcsin (c x) \cos (3 \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )-4 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+4 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-2 \arcsin (c x) \sin (2 \arcsin (c x))\right )}{12 c^2 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}} \]
(a^2*Sqrt[-(d*(-1 + c^2*x^2))])/(3*c^2*d^3*(-1 + c^2*x^2)^2) + (a*b*(8*Arc Sin[c*x] + 3*Sqrt[1 - c^2*x^2]*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2 ]] - Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[3*ArcSin[c*x]]*(L og[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c*x]/2] + Sin [ArcSin[c*x]/2]]) - 2*Sin[2*ArcSin[c*x]]))/(12*c^2*d*(d*(1 - c^2*x^2))^(3/ 2)) + (b^2*(2 + 4*ArcSin[c*x]^2 + 2*Cos[2*ArcSin[c*x]] - 3*Sqrt[1 - c^2*x^ 2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - ArcSin[c*x]*Cos[3*ArcSin[c*x ]]*Log[1 - I*E^(I*ArcSin[c*x])] + 3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + I*E^(I*ArcSi n[c*x])] - (4*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])] - 2*ArcSin[c*x]* Sin[2*ArcSin[c*x]]))/(12*c^2*d*(d*(1 - c^2*x^2))^(3/2))
Time = 0.73 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.67, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5182, 5162, 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.
| \(\displaystyle \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5162 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx-\frac {1}{2} b c \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {1}{2} \int \frac {a+b \arcsin (c x)}{1-c^2 x^2}dx+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5164 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int (a+b \arcsin (c x)) \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {-b \int \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {i b \int e^{-i \arcsin (c x)} \log \left (1-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {(a+b \arcsin (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{2 c}+\frac {x (a+b \arcsin (c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {1-c^2 x^2}}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}\) |
(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*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(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*c*d^2*Sqrt[d - c^2*d*x^2])
3.3.58.3.1 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)^(p_), x_ Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1 ))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b*Ar cSin[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2 *x^2)^p] Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x ]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
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]
Time = 0.32 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.59
| method | result | size |
| default | \(\frac {a^{2}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -c^{2} x^{2}+\arcsin \left (c x \right )^{2}+1\right )}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right ) c^{2}}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-c x \sqrt {-c^{2} x^{2}+1}+2 \arcsin \left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{2} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{2} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(468\) |
| parts | \(\frac {a^{2}}{3 c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -c^{2} x^{2}+\arcsin \left (c x \right )^{2}+1\right )}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{3 d^{3} \left (c^{2} x^{2}-1\right ) c^{2}}\right )+2 a b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-c x \sqrt {-c^{2} x^{2}+1}+2 \arcsin \left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{2} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{2} d^{3} \left (c^{2} x^{2}-1\right )}\right )\) | \(468\) |
1/3*a^2/c^2/d/(-c^2*d*x^2+d)^(3/2)+b^2*(1/3*(-d*(c^2*x^2-1))^(1/2)*(-(-c^2 *x^2+1)^(1/2)*arcsin(c*x)*x*c-c^2*x^2+arcsin(c*x)^2+1)/d^3/(c^4*x^4-2*c^2* x^2+1)/c^2-1/3*(-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^2)+2*a*b*(1/6*(-d*(c^2*x^2-1))^(1/2)*(-c*x*(- c^2*x^2+1)^(1/2)+2*arcsin(c*x))/d^3/(c^4*x^4-2*c^2*x^2+1)/c^2+1/6*(-c^2*x^ 2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)/c^2*ln(I*c*x+(-c^2*x^2+1 )^(1/2)+I)-1/6*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)/c ^2*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I))
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*x*arcsin(c*x)^2 + 2*a*b*x*arcsin(c*x) + a^2*x)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-sqrt(d)*integrate((b^2*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2 *a*b*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x) + 1/3*a^2/(( -c^2*d*x^2 + d)^(3/2)*c^2*d)
Exception generated. \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]