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

Optimal. Leaf size=816 \[ \frac {2^{-3 n-11} d^2 e^{-\frac {8 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-2 (n+4)} d^2 e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-n-7} d^2 e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {5 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} d^2 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} \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 {2^{-2 (n+4)} d^2 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} \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 {2^{-n-7} 3^{-n-1} d^2 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} \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 {2^{-3 n-11} d^2 e^{\frac {8 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} \Gamma \left (n+1,\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \]

[Out]

-5/128*d^2*(a+b*arcsinh(c*x))^(1+n)*(c^2*d*x^2+d)^(1/2)/b/c^3/(1+n)/(c^2*x^2+1)^(1/2)+2^(-11-3*n)*d^2*(a+b*arc
sinh(c*x))^n*GAMMA(1+n,-8*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/exp(8*a/b)/(((-a-b*arcsinh(c*x))/b)^n)
/(c^2*x^2+1)^(1/2)+2^(-7-n)*3^(-1-n)*d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-6*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)
^(1/2)/c^3/exp(6*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-4*(a+b
*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/(2^(8+2*n))/c^3/exp(4*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)
-2^(-7-n)*d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-2*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/exp(2*a/b)/(((-a
-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+2^(-7-n)*d^2*exp(2*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,2*(a+b*arcsinh
(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)-d^2*exp(4*a/b)*(a+b*arcsinh(c*x
))^n*GAMMA(1+n,4*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/(2^(8+2*n))/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2
+1)^(1/2)-2^(-7-n)*3^(-1-n)*d^2*exp(6*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,6*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d
)^(1/2)/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)-2^(-11-3*n)*d^2*exp(8*a/b)*(a+b*arcsinh(c*x))^n*GAMMA
(1+n,8*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^3/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.97, antiderivative size = 816, normalized size of antiderivative = 1.00, number of steps used = 16, 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 {2^{-3 n-11} d^2 e^{-\frac {8 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 a}{b}} \sqrt {c^2 d x^2+d} \left (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 ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-2 (n+4)} d^2 e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} \left (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 ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-n-7} d^2 e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (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 ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {5 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} d^2 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 {2^{-2 (n+4)} d^2 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 {2^{-n-7} 3^{-n-1} d^2 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 {2^{-3 n-11} d^2 e^{\frac {8 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 {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(128*b*c^3*(1 + n)*Sqrt[1 + c^2*x^2]) + (2^(-11 - 3*
n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSinh[c*x]))/b])/(c^3*E^((8*a)/b)*
Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-7 - n)*3^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[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) + (d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(
4 + n))*c^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (2^(-7 - n)*d^2*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^2*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) - (d^2*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])/(2^(2*(4 + n))*c^3*Sqrt[1 + c^2*x^2]*
((a + b*ArcSinh[c*x])/b)^n) - (2^(-7 - n)*3^(-1 - n)*d^2*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) - (2^(-11 - 3*n
)*d^2*E^((8*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b])/(c^3*Sq
rt[1 + c^2*x^2]*((a + b*ArcSinh[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 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 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 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])

Rubi steps

\begin {align*} \int x^2 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-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 \sinh ^{-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 \cosh ^6(x) \sinh ^2(x) \, dx,x,\sinh ^{-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 \cosh (2 x)+\frac {1}{32} (a+b x)^n \cosh (4 x)+\frac {1}{32} (a+b x)^n \cosh (6 x)+\frac {1}{128} (a+b x)^n \cosh (8 x)\right ) \, dx,x,\sinh ^{-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 \sinh ^{-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 \cosh (8 x) \, dx,x,\sinh ^{-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 \cosh (2 x) \, dx,x,\sinh ^{-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 \cosh (4 x) \, dx,x,\sinh ^{-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 \cosh (6 x) \, dx,x,\sinh ^{-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 \sinh ^{-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 x} (a+b x)^n \, dx,x,\sinh ^{-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 x} (a+b x)^n \, dx,x,\sinh ^{-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^{-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^2 \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 )}{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 x} (a+b x)^n \, dx,x,\sinh ^{-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 x} (a+b x)^n \, dx,x,\sinh ^{-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 x} (a+b x)^n \, dx,x,\sinh ^{-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 x} (a+b x)^n \, dx,x,\sinh ^{-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 \sinh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 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 {8 \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^2 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 {4^{-4-n} d^2 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^2 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^2 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 {4^{-4-n} d^2 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^2 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^{-11-3 n} d^2 e^{\frac {8 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 {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 6.94, size = 667, normalized size = 0.82 \[ -\frac {d^3 2^{-3 n-11} 3^{-n-1} e^{-\frac {8 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 (e^{\frac {8 a}{b}} \left (5 a 2^{3 n+4} 3^{n+1} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n+5 b 2^{3 n+4} 3^{n+1} \sinh ^{-1}(c x) \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n+b \left (-3^{n+1}\right ) 4^{n+2} (n+1) e^{\frac {2 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 2^{n+3} 3^{n+1} (n+1) e^{\frac {4 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 4^{n+2} e^{\frac {6 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 4^{n+2} n e^{\frac {6 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} n e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )+b \left (-3^{n+1}\right ) (n+1) \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b 4^{n+2} (n+1) e^{\frac {2 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b 2^{n+3} 3^{n+1} (n+1) e^{\frac {4 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} 4^{n+2} (n+1) e^{\frac {6 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {2 \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)^(5/2)*(a + b*ArcSinh[c*x])^n,x]

[Out]

-((2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(-(3^(1 + n)*b*(1 + n)*(a/b + ArcSinh
[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSinh[c*x]))/b]) - 4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*G
amma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b] - 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Ga
mma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*4^(2 + n)*b*E^((6*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gam
ma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b] + E^((8*a)/b)*(5*2^(4 + 3*n)*3^(1 + n)*a*(-((a + b*ArcSinh[c*x])^2/b^2)
)^n + 5*2^(4 + 3*n)*3^(1 + n)*b*ArcSinh[c*x]*(-((a + b*ArcSinh[c*x])^2/b^2))^n - 3^(1 + n)*4^(2 + n)*b*E^((2*a
)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b] + 2^(3 + n)*3^(1 + n)*b*E^
((4*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b] + 4^(2 + n)*b*E^((6*a
)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b] + 4^(2 + n)*b*E^((6*a)/b)*n*(-((a
+ b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*(-((a + b*ArcSinh[c
*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*n*(-((a + b*ArcSinh[c*x])/b))^n*
Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b])))/(b*c^3*E^((8*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c
*x])^2/b^2))^n))

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*asinh(c*x))**n,x)

[Out]

Timed out

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