3.173 \(\int x^2 (a+b \sin ^{-1}(c+d x))^n \, dx\)

Optimal. Leaf size=611 \[ -\frac{i c^2 e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{i c^2 e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}-\frac{i e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}+\frac{c 2^{-n-2} e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^3}+\frac{i 3^{-n-1} e^{-\frac{3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}+\frac{i e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}+\frac{c 2^{-n-2} e^{\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{i 3^{-n-1} e^{\frac{3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3} \]

[Out]

((-I/8)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*E^((I*a)/b)*(((-I)*(a +
b*ArcSin[c + d*x]))/b)^n) - ((I/2)*c^2*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/
b])/(d^3*E^((I*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/8)*E^((I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamm
a[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/2)*c^2*E^((I*a)/b)*(a +
b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n) + (
2^(-2 - n)*c*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*E^(((2*I)*a)/b)*
(((-I)*(a + b*ArcSin[c + d*x]))/b)^n) + (2^(-2 - n)*c*E^(((2*I)*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (
(2*I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/8)*3^(-1 - n)*(a + b*ArcSin[c
+ d*x])^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*E^(((3*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x
]))/b)^n) - ((I/8)*3^(-1 - n)*E^(((3*I)*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c +
d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n)

________________________________________________________________________________________

Rubi [A]  time = 1.11638, antiderivative size = 611, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {4805, 4747, 6741, 12, 6742, 3307, 2181, 4406, 3308} \[ -\frac{i c^2 e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{i c^2 e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}-\frac{i e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}+\frac{c 2^{-n-2} e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^3}+\frac{i 3^{-n-1} e^{-\frac{3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}+\frac{i e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}+\frac{c 2^{-n-2} e^{\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{i 3^{-n-1} e^{\frac{3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcSin[c + d*x])^n,x]

[Out]

((-I/8)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*E^((I*a)/b)*(((-I)*(a +
b*ArcSin[c + d*x]))/b)^n) - ((I/2)*c^2*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/
b])/(d^3*E^((I*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/8)*E^((I*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamm
a[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/2)*c^2*E^((I*a)/b)*(a +
b*ArcSin[c + d*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n) + (
2^(-2 - n)*c*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*E^(((2*I)*a)/b)*
(((-I)*(a + b*ArcSin[c + d*x]))/b)^n) + (2^(-2 - n)*c*E^(((2*I)*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, (
(2*I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n) + ((I/8)*3^(-1 - n)*(a + b*ArcSin[c
+ d*x])^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c + d*x]))/b])/(d^3*E^(((3*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x
]))/b)^n) - ((I/8)*3^(-1 - n)*E^(((3*I)*a)/b)*(a + b*ArcSin[c + d*x])^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c +
d*x]))/b])/(d^3*((I*(a + b*ArcSin[c + d*x]))/b)^n)

Rule 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[I
nt[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0
]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rubi steps

\begin{align*} \int x^2 \left (a+b \sin ^{-1}(c+d x)\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right )^2 \left (a+b \sin ^{-1}(x)\right )^n \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (x) \left (-\frac{c}{d}+\frac{\sin (x)}{d}\right )^2 \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^n \cos (x) (c-\sin (x))^2}{d^2} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (x) (c-\sin (x))^2 \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (c^2 (a+b x)^n \cos (x)-2 c (a+b x)^n \cos (x) \sin (x)+(a+b x)^n \cos (x) \sin ^2(x)\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (x) \sin ^2(x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}-\frac{(2 c) \operatorname{Subst}\left (\int (a+b x)^n \cos (x) \sin (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}+\frac{c^2 \operatorname{Subst}\left (\int (a+b x)^n \cos (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} (a+b x)^n \cos (x)-\frac{1}{4} (a+b x)^n \cos (3 x)\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{2} (a+b x)^n \sin (2 x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}+\frac{c^2 \operatorname{Subst}\left (\int e^{-i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d^3}+\frac{c^2 \operatorname{Subst}\left (\int e^{i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d^3}\\ &=-\frac{i c^2 e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{i c^2 e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (x) \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d^3}-\frac{\operatorname{Subst}\left (\int (a+b x)^n \cos (3 x) \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d^3}-\frac{c \operatorname{Subst}\left (\int (a+b x)^n \sin (2 x) \, dx,x,\sin ^{-1}(c+d x)\right )}{d^3}\\ &=-\frac{i c^2 e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{i c^2 e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{\operatorname{Subst}\left (\int e^{-i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d^3}+\frac{\operatorname{Subst}\left (\int e^{i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d^3}-\frac{\operatorname{Subst}\left (\int e^{-3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d^3}-\frac{\operatorname{Subst}\left (\int e^{3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d^3}-\frac{(i c) \operatorname{Subst}\left (\int e^{-2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d^3}+\frac{(i c) \operatorname{Subst}\left (\int e^{2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d^3}\\ &=-\frac{i e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}-\frac{i c^2 e^{-\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{i e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}+\frac{i c^2 e^{\frac{i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 d^3}+\frac{2^{-2-n} c e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^3}+\frac{2^{-2-n} c e^{\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{d^3}+\frac{i 3^{-1-n} e^{-\frac{3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}-\frac{i 3^{-1-n} e^{\frac{3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}\\ \end{align*}

Mathematica [A]  time = 0.503547, size = 419, normalized size = 0.69 \[ \frac{2^{-n-3} 3^{-n-1} e^{-\frac{3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}\right )^{-n} \left (i \left (4 c^2+1\right ) 2^n 3^{n+1} e^{\frac{4 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-i \left (4 c^2+1\right ) 2^n 3^{n+1} e^{\frac{2 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 c 3^{n+1} e^{\frac{5 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-i 2^n e^{\frac{6 i a}{b}} \left (-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 c 3^{n+1} e^{\frac{i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+i 2^n \left (\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*ArcSin[c + d*x])^n,x]

[Out]

(2^(-3 - n)*3^(-1 - n)*(a + b*ArcSin[c + d*x])^n*((-I)*2^n*3^(1 + n)*(1 + 4*c^2)*E^(((2*I)*a)/b)*((I*(a + b*Ar
cSin[c + d*x]))/b)^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b] + I*2^n*3^(1 + n)*(1 + 4*c^2)*E^(((4*I)*a)
/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b] + 2*3^(1 + n)*c*E^((I*a)/
b)*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + 2*3^(1 + n)*c*E^(((5*I
)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + I*2^n*((I*(a +
b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] - I*2^n*E^(((6*I)*a)/b)*(((-I)*(a +
b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c + d*x]))/b]))/(d^3*E^(((3*I)*a)/b)*((a + b*ArcSin
[c + d*x])^2/b^2)^n)

________________________________________________________________________________________

Maple [F]  time = 0.405, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arcsin(d*x+c))^n,x)

[Out]

int(x^2*(a+b*arcsin(d*x+c))^n,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsin(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*arcsin(d*x + c) + a)^n*x^2, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsin(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*arcsin(d*x + c) + a)^n*x^2, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{asin}{\left (c + d x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*asin(d*x+c))**n,x)

[Out]

Integral(x**2*(a + b*asin(c + d*x))**n, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{n} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arcsin(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x + c) + a)^n*x^2, x)