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

Optimal. Leaf size=815 \[ \text{result too large to display} \]

[Out]

(-5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b])/(128*c^2*E^((I*a
)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (5*d^2*E^((I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSi
n[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(128*c^2*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) -
(3^(1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b])/(128*c
^2*E^(((3*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (3^(1 - n)*d^2*E^(((3*I)*a)/b)*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(128*c^2*Sqrt[1 - c^2*x^2]*((
I*(a + b*ArcSin[c*x]))/b)^n) - (d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-5*I)*(a + b*ArcS
in[c*x]))/b])/(128*5^n*c^2*E^(((5*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (d^2*E^(((5*I
)*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((5*I)*(a + b*ArcSin[c*x]))/b])/(128*5^n*c^2*Sq
rt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) - (7^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gam
ma[1 + n, ((-7*I)*(a + b*ArcSin[c*x]))/b])/(128*c^2*E^(((7*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x
]))/b)^n) - (7^(-1 - n)*d^2*E^(((7*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((7*I)*(a +
b*ArcSin[c*x]))/b])/(128*c^2*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n)

________________________________________________________________________________________

Rubi [A]  time = 0.734652, antiderivative size = 815, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {4725, 4723, 4406, 3308, 2181} \[ -\frac{5 d^2 e^{-\frac{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{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}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{1-n} d^2 e^{-\frac{3 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{3 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}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5^{-n} d^2 e^{-\frac{5 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{5 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}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{7^{-n-1} d^2 e^{-\frac{7 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{7 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}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5 d^2 e^{\frac{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{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{1-n} d^2 e^{\frac{3 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{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5^{-n} d^2 e^{\frac{5 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{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{7^{-n-1} d^2 e^{\frac{7 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{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x*(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])^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b])/(128*c^2*E^((I*a
)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (5*d^2*E^((I*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSi
n[c*x])^n*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/(128*c^2*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) -
(3^(1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b])/(128*c
^2*E^(((3*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (3^(1 - n)*d^2*E^(((3*I)*a)/b)*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(128*c^2*Sqrt[1 - c^2*x^2]*((
I*(a + b*ArcSin[c*x]))/b)^n) - (d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((-5*I)*(a + b*ArcS
in[c*x]))/b])/(128*5^n*c^2*E^(((5*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (d^2*E^(((5*I
)*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((5*I)*(a + b*ArcSin[c*x]))/b])/(128*5^n*c^2*Sq
rt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n) - (7^(-1 - n)*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gam
ma[1 + n, ((-7*I)*(a + b*ArcSin[c*x]))/b])/(128*c^2*E^(((7*I)*a)/b)*Sqrt[1 - c^2*x^2]*(((-I)*(a + b*ArcSin[c*x
]))/b)^n) - (7^(-1 - n)*d^2*E^(((7*I)*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n*Gamma[1 + n, ((7*I)*(a +
b*ArcSin[c*x]))/b])/(128*c^2*Sqrt[1 - c^2*x^2]*((I*(a + b*ArcSin[c*x]))/b)^n)

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])

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 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]

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]

Rubi steps

\begin{align*} \int x \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 \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 (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \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}{64} (a+b x)^n \sin (x)+\frac{9}{64} (a+b x)^n \sin (3 x)+\frac{5}{64} (a+b x)^n \sin (5 x)+\frac{1}{64} (a+b x)^n \sin (7 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \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 \sin (7 x) \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \sin (5 x) \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (9 d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \sin (3 x) \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2 \sqrt{1-c^2 x^2}}\\ &=\frac{\left (i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-7 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{7 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (5 i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (5 i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-5 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (5 i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{5 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}+\frac{\left (9 i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{\left (9 i d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2 \sqrt{1-c^2 x^2}}\\ &=-\frac{5 d^2 e^{-\frac{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{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5 d^2 e^{\frac{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{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{1-n} d^2 e^{-\frac{3 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{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{1-n} d^2 e^{\frac{3 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{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5^{-n} d^2 e^{-\frac{5 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{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5^{-n} d^2 e^{\frac{5 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{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{7^{-1-n} d^2 e^{-\frac{7 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{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{7^{-1-n} d^2 e^{\frac{7 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{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 3.9743, size = 603, normalized size = 0.74 \[ -\frac{d^3 5^{-n} 21^{-n-1} e^{-\frac{7 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 )^{-3 n} \left (\left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (9\ 5^n 7^{n+1} e^{\frac{4 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{2 n} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^n \text{Gamma}\left (n+1,-\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+105^{n+1} e^{\frac{8 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+9\ 5^n 7^{n+1} e^{\frac{10 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+3^{n+1} \left (7^{n+1} e^{\frac{12 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+5^n \left (e^{\frac{14 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,\frac{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{3 n} \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{7 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )+7^{n+1} e^{\frac{2 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{3 n} \left (-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )+105^{n+1} e^{\frac{6 i a}{b}} \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \text{Gamma}\left (n+1,-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{128 c^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(21^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(105^(1 + n)*E^(((6*I)*a)/b)*((I*(a + b*ArcSin[c*x])
)/b)^n*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b] + (((-I)*(a + b*ArcSin[c*x
]))/b)^n*(105^(1 + n)*E^(((8*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b
] + 9*5^n*7^(1 + n)*E^(((4*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^(2*n)*((a + b*ArcSin[c*x])^2/b^2)^n*Gamma[1 +
n, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 9*5^n*7^(1 + n)*E^(((10*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[
1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b] + 3^(1 + n)*(7^(1 + n)*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*
((I*(a + b*ArcSin[c*x]))/b)^(3*n)*Gamma[1 + n, ((-5*I)*(a + b*ArcSin[c*x]))/b] + 7^(1 + n)*E^(((12*I)*a)/b)*((
a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((5*I)*(a + b*ArcSin[c*x]))/b] + 5^n*((((-I)*(a + b*ArcSin[c*x]))
/b)^n*((I*(a + b*ArcSin[c*x]))/b)^(3*n)*Gamma[1 + n, ((-7*I)*(a + b*ArcSin[c*x]))/b] + E^(((14*I)*a)/b)*((a +
b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((7*I)*(a + b*ArcSin[c*x]))/b])))))/(128*5^n*c^2*E^(((7*I)*a)/b)*Sqrt
[d - c^2*d*x^2]*((a + b*ArcSin[c*x])^2/b^2)^(3*n))

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Maple [F]  time = 0.174, size = 0, normalized size = 0. \begin{align*} \int x \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x)*sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-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, x)