3.492 \(\int x^2 (d-c^2 d x^2)^{5/2} (a+b \sin ^{-1}(c x))^n \, dx\)

Optimal. Leaf size=906 \[ -\frac {i 2^{-n-7} d^2 e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-2 (n+4)} d^2 e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-3 n-11} d^2 e^{-\frac {8 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt {1-c^2 x^2}}+\frac {i 2^{-n-7} d^2 e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-2 (n+4)} d^2 e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-n-7} 3^{-n-1} d^2 e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-3 n-11} d^2 e^{\frac {8 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}} \]

[Out]

5/128*d^2*(a+b*arcsin(c*x))^(1+n)*(-c^2*d*x^2+d)^(1/2)/b/c^3/(1+n)/(-c^2*x^2+1)^(1/2)-I*2^(-7-n)*d^2*(a+b*arcs
in(c*x))^n*GAMMA(1+n,-2*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^3/exp(2*I*a/b)/((-I*(a+b*arcsin(c*x))/b)
^n)/(-c^2*x^2+1)^(1/2)+I*2^(-7-n)*d^2*exp(2*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,2*I*(a+b*arcsin(c*x))/b)*(-c^
2*d*x^2+d)^(1/2)/c^3/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)+I*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,-4*I*(
a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(2^(8+2*n))/c^3/exp(4*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1
)^(1/2)-I*d^2*exp(4*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,4*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/(2^(8+2
*n))/c^3/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)+I*2^(-7-n)*3^(-1-n)*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,
-6*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^3/exp(6*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2
)-I*2^(-7-n)*3^(-1-n)*d^2*exp(6*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,6*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(
1/2)/c^3/((I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)+I*2^(-11-3*n)*d^2*(a+b*arcsin(c*x))^n*GAMMA(1+n,-8*I*(
a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^3/exp(8*I*a/b)/((-I*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)-I*2^
(-11-3*n)*d^2*exp(8*I*a/b)*(a+b*arcsin(c*x))^n*GAMMA(1+n,8*I*(a+b*arcsin(c*x))/b)*(-c^2*d*x^2+d)^(1/2)/c^3/((I
*(a+b*arcsin(c*x))/b)^n)/(-c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.96, antiderivative size = 906, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4725, 4723, 4406, 3307, 2181} \[ -\frac {i 2^{-n-7} d^2 e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-2 (n+4)} d^2 e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-3 n-11} d^2 e^{-\frac {8 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n}}{c^3 \sqrt {1-c^2 x^2}}+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt {1-c^2 x^2}}+\frac {i 2^{-n-7} d^2 e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-2 (n+4)} d^2 e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-n-7} 3^{-n-1} d^2 e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-3 n-11} d^2 e^{\frac {8 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n,x]

[Out]

(5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[1 - c^2*x^2]) - (I*2^(-7 - n)*
d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(c^3*E^(((2*I)*a)/
b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*2^(-7 - n)*d^2*E^(((2*I)*a)/b)*Sqrt[d - c^2*d*x^2]
*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(c^3*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c
*x]))/b)^n) + (I*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b])/(
2^(2*(4 + n))*c^3*E^(((4*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (I*d^2*E^(((4*I)*a)/b)
*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b])/(2^(2*(4 + n))*c^3*Sqr
t[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) + (I*2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin
[c*x])^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b])/(c^3*E^(((6*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*Ar
cSin[c*x]))/b)^n) - (I*2^(-7 - n)*3^(-1 - n)*d^2*E^(((6*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gam
ma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b])/(c^3*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) + (I*2^(-11 -
3*n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-8*I)*(a + b*ArcSin[c*x]))/b])/(c^3*E^(((8*I
)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (I*2^(-11 - 3*n)*d^2*E^(((8*I)*a)/b)*Sqrt[d - c^
2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((8*I)*(a + b*ArcSin[c*x]))/b])/(c^3*Sqrt[1 - c^2*x^2]*((I*(a + b*
ArcSin[c*x]))/b)^n)

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 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 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 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 4725

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(d^IntPar
t[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[x^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x]
/; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] &&  !(Integ
erQ[p] || GtQ[d, 0])

Rubi steps

\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos ^6(x) \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {5}{128} (a+b x)^n+\frac {1}{32} (a+b x)^n \cos (2 x)-\frac {1}{32} (a+b x)^n \cos (4 x)-\frac {1}{32} (a+b x)^n \cos (6 x)-\frac {1}{128} (a+b x)^n \cos (8 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos (8 x) \, dx,x,\sin ^{-1}(c x)\right )}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos (4 x) \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos (6 x) \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-8 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{256 c^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{8 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{256 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c^3 \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-4 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{4 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-6 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c^3 \sqrt {1-c^2 x^2}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{6 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-n} d^2 e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} d^2 e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 4^{-4-n} d^2 e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 4^{-4-n} d^2 e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}+\frac {i 2^{-11-3 n} d^2 e^{-\frac {8 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}-\frac {i 2^{-11-3 n} d^2 e^{\frac {8 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 4.41, size = 989, normalized size = 1.09 \[ \frac {2^{-3 n-11} 3^{-n-1} d^3 e^{-\frac {8 i a}{b}} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (i 3^{n+1} 4^{n+2} b e^{\frac {10 i a}{b}} (n+1) \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n-i 2^{n+3} 3^{n+1} b e^{\frac {12 i a}{b}} \Gamma \left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n-i 2^{n+3} 3^{n+1} b e^{\frac {12 i a}{b}} n \Gamma \left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n-i 4^{n+2} b e^{\frac {14 i a}{b}} \Gamma \left (n+1,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n-i 4^{n+2} b e^{\frac {14 i a}{b}} n \Gamma \left (n+1,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n-i 3^{n+1} b e^{\frac {16 i a}{b}} \Gamma \left (n+1,\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n-i 3^{n+1} b e^{\frac {16 i a}{b}} n \Gamma \left (n+1,\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right ) \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n+5\ 2^{3 n+4} 3^{n+1} a e^{\frac {8 i a}{b}} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^n+5\ 2^{3 n+4} 3^{n+1} b e^{\frac {8 i a}{b}} \sin ^{-1}(c x) \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^n-i 3^{n+1} 4^{n+2} b e^{\frac {6 i a}{b}} (n+1) \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 2^{n+3} 3^{n+1} b e^{\frac {4 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 2^{n+3} 3^{n+1} b e^{\frac {4 i a}{b}} n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 4^{n+2} b e^{\frac {2 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 4^{n+2} b e^{\frac {2 i a}{b}} n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 3^{n+1} b \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 3^{n+1} b n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{b c^3 (n+1) \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n,x]

[Out]

(2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(5*2^(4 + 3*n)*3^(1 + n)*a*E^(((8*I)*a)/
b)*((a + b*ArcSin[c*x])^2/b^2)^n + 5*2^(4 + 3*n)*3^(1 + n)*b*E^(((8*I)*a)/b)*ArcSin[c*x]*((a + b*ArcSin[c*x])^
2/b^2)^n - I*3^(1 + n)*4^(2 + n)*b*E^(((6*I)*a)/b)*(1 + n)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-2*I)*
(a + b*ArcSin[c*x]))/b] + I*3^(1 + n)*4^(2 + n)*b*E^(((10*I)*a)/b)*(1 + n)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Ga
mma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b] + I*2^(3 + n)*3^(1 + n)*b*E^(((4*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b
)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b] + I*2^(3 + n)*3^(1 + n)*b*E^(((4*I)*a)/b)*n*((I*(a + b*ArcSin
[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b] - I*2^(3 + n)*3^(1 + n)*b*E^(((12*I)*a)/b)*(((-I)*(a
+ b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b] - I*2^(3 + n)*3^(1 + n)*b*E^(((12*I)*a)/b)
*n*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b] + I*4^(2 + n)*b*E^(((2*I)*a)/b
)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b] + I*4^(2 + n)*b*E^(((2*I)*a)/b)*n
*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b] - I*4^(2 + n)*b*E^(((14*I)*a)/b)*(
((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b] - I*4^(2 + n)*b*E^(((14*I)*a)/b)*n
*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b] + I*3^(1 + n)*b*((I*(a + b*ArcSi
n[c*x]))/b)^n*Gamma[1 + n, ((-8*I)*(a + b*ArcSin[c*x]))/b] + I*3^(1 + n)*b*n*((I*(a + b*ArcSin[c*x]))/b)^n*Gam
ma[1 + n, ((-8*I)*(a + b*ArcSin[c*x]))/b] - I*3^(1 + n)*b*E^(((16*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Ga
mma[1 + n, ((8*I)*(a + b*ArcSin[c*x]))/b] - I*3^(1 + n)*b*E^(((16*I)*a)/b)*n*(((-I)*(a + b*ArcSin[c*x]))/b)^n*
Gamma[1 + n, ((8*I)*(a + b*ArcSin[c*x]))/b]))/(b*c^3*E^(((8*I)*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*((a + b*ArcSi
n[c*x])^2/b^2)^n)

________________________________________________________________________________________

fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^6 - 2*c^2*d^2*x^4 + d^2*x^2)*sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)^n*x^2, x)

________________________________________________________________________________________

maple [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x)

[Out]

int(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))^n,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arcsin(c*x) + a)^n*x^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*asin(c*x))^n*(d - c^2*d*x^2)^(5/2),x)

[Out]

int(x^2*(a + b*asin(c*x))^n*(d - c^2*d*x^2)^(5/2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x))**n,x)

[Out]

Timed out

________________________________________________________________________________________