Integrand size = 29, antiderivative size = 332 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 x}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^2 (a+b \arcsin (c x))}{3 c d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c^3 d^2 \sqrt {d-c^2 d x^2}} \]
1/3*x^3*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)+1/3*b^2*x/c^2/d^2/(-c^2 *d*x^2+d)^(1/2)-1/3*b*x^2*(a+b*arcsin(c*x))/c/d^2/(-c^2*x^2+1)^(1/2)/(-c^2 *d*x^2+d)^(1/2)-1/3*b^2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(-c^2*d*x^2 +d)^(1/2)+1/3*I*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^3/d^2/(-c^2*d*x^2 +d)^(1/2)-2/3*b*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2 *x^2+1)^(1/2)/c^3/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*I*b^2*polylog(2,-(I*c*x+(-c ^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(-c^2*d*x^2+d)^(1/2)
Time = 1.16 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {-b^2 c x-a^2 c^3 x^3+b^2 c^3 x^3+a b \sqrt {1-c^2 x^2}+i b^2 \left (i c^3 x^3-\sqrt {1-c^2 x^2}+c^2 x^2 \sqrt {1-c^2 x^2}\right ) \arcsin (c x)^2+b \arcsin (c x) \left (-2 a c^3 x^3+b \sqrt {1-c^2 x^2}+2 b \left (1-c^2 x^2\right )^{3/2} \log \left (1+e^{2 i \arcsin (c x)}\right )\right )+a b \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )-a b c^2 x^2 \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )-i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{3 c^3 d^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]
(-(b^2*c*x) - a^2*c^3*x^3 + b^2*c^3*x^3 + a*b*Sqrt[1 - c^2*x^2] + I*b^2*(I *c^3*x^3 - Sqrt[1 - c^2*x^2] + c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + b*ArcSin[c*x]*(-2*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2] + 2*b*(1 - c^2*x^2)^(3/2 )*Log[1 + E^((2*I)*ArcSin[c*x])]) + a*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - a*b*c^2*x^2*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - I*b^2*(1 - c^2*x^2)^(3 /2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(3*c^3*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])
Time = 0.92 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.66, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {5186, 5206, 252, 223, 5180, 3042, 4202, 2620, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5186 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \int \frac {x^3 (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5206 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {b \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2}}dx}{2 c}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{c^2}\right )}{2 c}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx}{c^2}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5180 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-c^2 x^2} \left (-\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )}{c^4}+\frac {x^2 (a+b \arcsin (c x))}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\) |
(x^3*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) - (2*b*c*Sqrt[1 - c^2*x^2]*((x^2*(a + b*ArcSin[c*x]))/(2*c^2*(1 - c^2*x^2)) - (b*(x/(c^2*Sqr t[1 - c^2*x^2]) - ArcSin[c*x]/c^3))/(2*c) - (((I/2)*(a + b*ArcSin[c*x])^2) /b - (2*I)*((-1/2*I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/4))/c^4))/(3*d^2*Sqrt[d - c^2*d*x^2 ])
3.3.57.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 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*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[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/(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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3297 vs. \(2 (312 ) = 624\).
Time = 0.27 (sec) , antiderivative size = 3298, normalized size of antiderivative = 9.93
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3298\) |
| parts | \(\text {Expression too large to display}\) | \(3298\) |
2*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2 +1)*c^4*arcsin(c*x)*x^7-a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^ 6+10*c^4*x^4-5*c^2*x^2+1)*c*(-c^2*x^2+1)^(1/2)*x^4-1/3*I*a*b*(-d*(c^2*x^2- 1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^4*x^7-2*a*b*( -d*(c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2 *arcsin(c*x)*x^5+a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^ 4*x^4-5*c^2*x^2+1)/c*(-c^2*x^2+1)^(1/2)*x^2+1/3*I*a*b*(-d*(c^2*x^2-1))^(1/ 2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*(-c^2*x^2+1)*x^3+2/3*I *a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+ 1)*c^2*x^5+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/(c^2*x^2- 1)/c^3*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d ^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*x^3-1/3*I*a*b*(-d*(c^2*x^2 -1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2*(-c^2*x^2+ 1)*x^5-2/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^ 4-5*c^2*x^2+1)/c^3*arcsin(c*x)*(-c^2*x^2+1)^(1/2)-4/3*I*a*b*(-c^2*x^2+1)^( 1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)/c^3*arcsin(c*x)+a^2*(1/2*x/c^2 /d/(-c^2*d*x^2+d)^(3/2)-1/2/c^2*(1/3/d*x/(-c^2*d*x^2+d)^(3/2)+2/3/d^2*x/(- c^2*d*x^2+d)^(1/2)))-1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6 *x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2*(-c^2*x^2+1)*arcsin(c*x)*x^5-2*I*b^2*(-d* (c^2*x^2-1))^(1/2)/d^3/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c*(...
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(-(b^2*x^2*arcsin(c*x)^2 + 2*a*b*x^2*arcsin(c*x) + a^2*x^2)*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)
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*b*c*(1/(c^6*d^(5/2)*x^2 - c^4*d^(5/2)) - log(c*x + 1)/(c^4*d^(5/2)) - log(c*x - 1)/(c^4*d^(5/2))) - 2/3*a*b*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d))*arcsin(c*x) - 1/3*a^2*(x/(sqrt(-c^2*d* x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d)) + b^2*integrate(x^2* 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. \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^2 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]