3.1.94 \(\int x^2 (a+b \sinh ^{-1}(c+d x))^n \, dx\) [94]

Optimal. Leaf size=545 \[ \frac {3^{-1-n} e^{-\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}-\frac {2^{-2-n} c e^{-\frac {2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac {e^{-\frac {a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}+\frac {c^2 e^{-\frac {a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}+\frac {e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}-\frac {c^2 e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac {2^{-2-n} c e^{\frac {2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac {3^{-1-n} e^{\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3} \]

[Out]

1/8*3^(-1-n)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,-3*(a+b*arcsinh(d*x+c))/b)/d^3/exp(3*a/b)/(((-a-b*arcsinh(d*x+c)
)/b)^n)-2^(-2-n)*c*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,-2*(a+b*arcsinh(d*x+c))/b)/d^3/exp(2*a/b)/(((-a-b*arcsinh(
d*x+c))/b)^n)-1/8*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,(-a-b*arcsinh(d*x+c))/b)/d^3/exp(a/b)/(((-a-b*arcsinh(d*x+c
))/b)^n)+1/2*c^2*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,(-a-b*arcsinh(d*x+c))/b)/d^3/exp(a/b)/(((-a-b*arcsinh(d*x+c)
)/b)^n)+1/8*exp(a/b)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,(a+b*arcsinh(d*x+c))/b)/d^3/(((a+b*arcsinh(d*x+c))/b)^n)
-1/2*c^2*exp(a/b)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,(a+b*arcsinh(d*x+c))/b)/d^3/(((a+b*arcsinh(d*x+c))/b)^n)-2^
(-2-n)*c*exp(2*a/b)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,2*(a+b*arcsinh(d*x+c))/b)/d^3/(((a+b*arcsinh(d*x+c))/b)^n
)-1/8*3^(-1-n)*exp(3*a/b)*(a+b*arcsinh(d*x+c))^n*GAMMA(1+n,3*(a+b*arcsinh(d*x+c))/b)/d^3/(((a+b*arcsinh(d*x+c)
)/b)^n)

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Rubi [A]
time = 0.84, antiderivative size = 545, normalized size of antiderivative = 1.00, 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 = {5859, 5830, 6873, 12, 6874, 3388, 2212, 5556, 3389} \begin {gather*} \frac {c^2 e^{-\frac {a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac {c^2 e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}+\frac {3^{-n-1} e^{-\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}-\frac {c 2^{-n-2} e^{-\frac {2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac {e^{-\frac {a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}+\frac {e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}-\frac {c 2^{-n-2} e^{\frac {2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac {3^{-n-1} e^{\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcSinh[c + d*x])^n,x]

[Out]

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

Rule 12

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

Rule 2212

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

Rule 3388

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 3389

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 5556

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 5830

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

Rule 5859

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

Rule 6873

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

Rule 6874

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

Rubi steps

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

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Mathematica [F]
time = 1.01, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*asinh(c + d*x))^n,x)

[Out]

int(x^2*(a + b*asinh(c + d*x))^n, x)

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