3.517 \(\int x^2 (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x))^n \, dx\)

Optimal. Leaf size=616 \[ \frac{d 2^{-n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}+\frac{d 2^{-2 n-7} e^{-\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-n-7} e^{-\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}+\frac{d 2^{-n-7} e^{\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-2 n-7} e^{\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-n-7} 3^{-n-1} e^{\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{16 b c^3 (n+1) \sqrt{c^2 x^2+1}} \]

[Out]

-(d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(16*b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-7 - n)*3^(-1
- n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((6*a)/b)
*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - 2*n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n
*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n)
- (2^(-7 - n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c^3*E^(
(2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*d*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a +
b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n
) - (2^(-7 - 2*n)*d*E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]
))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d*E^((6*a)/b)*Sqrt[d + c^2*
d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh
[c*x])/b)^n)

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Rubi [A]  time = 0.832672, antiderivative size = 616, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {5782, 5779, 5448, 3307, 2181} \[ \frac{d 2^{-n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}+\frac{d 2^{-2 n-7} e^{-\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-n-7} e^{-\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}+\frac{d 2^{-n-7} e^{\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-2 n-7} e^{\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-n-7} 3^{-n-1} e^{\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{c^2 x^2+1}}-\frac{d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{16 b c^3 (n+1) \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n,x]

[Out]

-(d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(16*b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-7 - n)*3^(-1
- n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((6*a)/b)
*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - 2*n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n
*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n)
- (2^(-7 - n)*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(c^3*E^(
(2*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*d*E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(a +
b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n
) - (2^(-7 - 2*n)*d*E^((4*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]
))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d*E^((6*a)/b)*Sqrt[d + c^2*
d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])/(c^3*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh
[c*x])/b)^n)

Rule 5782

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

Rule 5779

Int[((a_.) + ArcSinh[(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*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 5448

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

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]

Rubi steps

\begin{align*} \int x^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh ^4(x) \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt{1+c^2 x^2}}\\ &=\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{16} (a+b x)^n-\frac{1}{32} (a+b x)^n \cosh (2 x)+\frac{1}{16} (a+b x)^n \cosh (4 x)+\frac{1}{32} (a+b x)^n \cosh (6 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt{1+c^2 x^2}}\\ &=-\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{16 b c^3 (1+n) \sqrt{1+c^2 x^2}}-\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (6 x) \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^n \cosh (4 x) \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^3 \sqrt{1+c^2 x^2}}\\ &=-\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{16 b c^3 (1+n) \sqrt{1+c^2 x^2}}+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-6 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt{1+c^2 x^2}}-\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt{1+c^2 x^2}}-\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{6 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt{1+c^2 x^2}}+\frac{\left (d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt{1+c^2 x^2}}\\ &=-\frac{d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{16 b c^3 (1+n) \sqrt{1+c^2 x^2}}+\frac{2^{-7-n} 3^{-1-n} d e^{-\frac{6 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1+c^2 x^2}}+\frac{2^{-7-2 n} d e^{-\frac{4 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1+c^2 x^2}}-\frac{2^{-7-n} d e^{-\frac{2 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1+c^2 x^2}}+\frac{2^{-7-n} d e^{\frac{2 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1+c^2 x^2}}-\frac{2^{-7-2 n} d e^{\frac{4 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1+c^2 x^2}}-\frac{2^{-7-n} 3^{-1-n} d e^{\frac{6 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 3.08736, size = 429, normalized size = 0.7 \[ -\frac{d^2 2^{-2 n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (2^n e^{\frac{6 a}{b}} \left (b (n+1) e^{\frac{6 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2^{n+3} 3^{n+1} \left (a+b \sinh ^{-1}(c x)\right ) \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n\right )+b \left (-2^n\right ) (n+1) \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b 3^{n+1} (n+1) e^{\frac{2 a}{b}} \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 2^n 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b 2^n 3^{n+1} (n+1) e^{\frac{8 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} (n+1) e^{\frac{10 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{b c^3 (n+1) \sqrt{c^2 d x^2+d}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n,x]

[Out]

-((2^(-7 - 2*n)*3^(-1 - n)*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(-(2^n*b*(1 + n)*(a/b + ArcSinh[c*x])^
n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b]) - 3^(1 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1
+ n, (-4*(a + b*ArcSinh[c*x]))/b] + 2^n*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-
2*(a + b*ArcSinh[c*x]))/b] - 2^n*3^(1 + n)*b*E^((8*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (2
*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((10*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a
+ b*ArcSinh[c*x]))/b] + 2^n*E^((6*a)/b)*(2^(3 + n)*3^(1 + n)*(a + b*ArcSinh[c*x])*(-((a + b*ArcSinh[c*x])^2/b^
2))^n + b*E^((6*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])))/(b*c^3
*E^((6*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])^2/b^2))^n))

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Maple [F]  time = 0.188, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^n,x)

[Out]

int(x^2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(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{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^n,x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 + d)^(3/2)*(b*arcsinh(c*x) + a)^n*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^n,x, algorithm="fricas")

[Out]

integral((c^2*d*x^4 + d*x^2)*sqrt(c^2*d*x^2 + d)*(b*arcsinh(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**2*(c**2*d*x**2+d)**(3/2)*(a+b*asinh(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{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^n,x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 + d)^(3/2)*(b*arcsinh(c*x) + a)^n*x^2, x)