3.6.23 \(\int x (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^n \, dx\) [523]

Optimal. Leaf size=745 \[ \frac {7^{-1-n} d^2 e^{-\frac {7 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-n} d^2 e^{-\frac {5 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{1-n} d^2 e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5 d^2 e^{a/b} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{1-n} d^2 e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-n} d^2 e^{\frac {5 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {7^{-1-n} d^2 e^{\frac {7 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}} \]

[Out]

1/128*7^(-1-n)*d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-7*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/exp(7*a/b)/
(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+1/128*d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-5*(a+b*arcsinh(c*x))/b
)*(c^2*d*x^2+d)^(1/2)/(5^n)/c^2/exp(5*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+1/128*3^(1-n)*d^2*(a+
b*arcsinh(c*x))^n*GAMMA(1+n,-3*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/exp(3*a/b)/(((-a-b*arcsinh(c*x))/
b)^n)/(c^2*x^2+1)^(1/2)+5/128*d^2*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(-a-b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^
2/exp(a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+5/128*d^2*exp(a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(a+
b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+1/128*3^(1-n)*d^2*exp(
3*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,3*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arcsinh(c*x))/b)^
n)/(c^2*x^2+1)^(1/2)+1/128*d^2*exp(5*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,5*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)
^(1/2)/(5^n)/c^2/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)+1/128*7^(-1-n)*d^2*exp(7*a/b)*(a+b*arcsinh(c*x))
^n*GAMMA(1+n,7*(a+b*arcsinh(c*x))/b)*(c^2*d*x^2+d)^(1/2)/c^2/(((a+b*arcsinh(c*x))/b)^n)/(c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.45, antiderivative size = 745, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5819, 5556, 3389, 2212} \begin {gather*} \frac {d^2 7^{-n-1} e^{-\frac {7 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 {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}}+\frac {d^2 5^{-n} e^{-\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}}+\frac {d^2 3^{1-n} e^{-\frac {3 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 {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}}+\frac {5 d^2 e^{-\frac {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 {a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}}+\frac {5 d^2 e^{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 {a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}}+\frac {d^2 3^{1-n} e^{\frac {3 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 {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}}+\frac {d^2 5^{-n} e^{\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}}+\frac {d^2 7^{-n-1} e^{\frac {7 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 {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7^(-1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-7*(a + b*ArcSinh[c*x]))/b])/(128*c^2
*E^((7*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, (-5*(a + b*ArcSinh[c*x]))/b])/(128*5^n*c^2*E^((5*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x]
)/b))^n) + (3^(1 - n)*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b]
)/(128*c^2*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (5*d^2*Sqrt[d + c^2*d*x^2]*(a + b*Ar
cSinh[c*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(128*c^2*E^(a/b)*Sqrt[1 + c^2*x^2]*(-((a + b*ArcSinh[c*
x])/b))^n) + (5*d^2*E^(a/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(
128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (3^(1 - n)*d^2*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*
ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^
n) + (d^2*E^((5*a)/b)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b])/(12
8*5^n*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (7^(-1 - n)*d^2*E^((7*a)/b)*Sqrt[d + c^2*d*x^2]*(a +
b*ArcSinh[c*x])^n*Gamma[1 + n, (7*(a + b*ArcSinh[c*x]))/b])/(128*c^2*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])/
b)^n)

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 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 5819

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

Rubi steps

\begin {align*} \int x \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 \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 ) \text {Subst}\left (\int (a+b x)^n \cosh ^6(x) \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {5}{64} (a+b x)^n \sinh (x)+\frac {9}{64} (a+b x)^n \sinh (3 x)+\frac {5}{64} (a+b x)^n \sinh (5 x)+\frac {1}{64} (a+b x)^n \sinh (7 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \sinh (7 x) \, dx,x,\sinh ^{-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 ) \text {Subst}\left (\int (a+b x)^n \sinh (x) \, dx,x,\sinh ^{-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 ) \text {Subst}\left (\int (a+b x)^n \sinh (5 x) \, dx,x,\sinh ^{-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 ) \text {Subst}\left (\int (a+b x)^n \sinh (3 x) \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-7 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{7 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (5 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (5 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (5 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (5 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (9 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (9 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {7^{-1-n} d^2 e^{-\frac {7 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 {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-n} d^2 e^{-\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{1-n} d^2 e^{-\frac {3 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 {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5 d^2 e^{-\frac {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 {a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5 d^2 e^{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 {a+b \sinh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{1-n} d^2 e^{\frac {3 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 {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-n} d^2 e^{\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}+\frac {7^{-1-n} d^2 e^{\frac {7 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 {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 2.00, size = 685, normalized size = 0.92 \begin {gather*} \frac {105^{-1-n} d^3 e^{-\frac {7 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-2 n} \left (5^{2+n} 21^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (1+n,\frac {a}{b}+\sinh ^{-1}(c x)\right )+15^{1+n} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac {2 a}{b}} \left (5\ 21^{1+n} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+9\ 35^{1+n} e^{\frac {2 a}{b}} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+5^{2+n} 21^{1+n} e^{\frac {4 a}{b}} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )+35^{1+n} e^{\frac {8 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+8\ 35^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-3^{2+n} 7^{1+n} e^{\frac {10 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+8\ 21^{1+n} e^{\frac {10 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (1+n,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+15^{1+n} e^{\frac {12 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{3 n} \Gamma \left (1+n,\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{128 c^2 \sqrt {d+c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(105^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(5^(2 + n)*21^(1 + n)*E^((8*a)/b)*(-((a + b*ArcSinh
[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, a/b + ArcSinh[c*x]] + 15^(1 + n)*(-((a + b*Arc
Sinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-7*(a + b*ArcSinh[c*x]))/b] + E^((2*a)/b)*(5*21^(1 + n)*(-((a + b*ArcSi
nh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b] + 9*35^(1 + n)*E^((2*a)/b)*(-((a + b*ArcSinh[
c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b] + 5^(2 + n)*21^(1 + n)*E^((4*a)/b)*(-((a + b*Arc
Sinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)] + 35^(1 + n)*E^((8*a)/b)*(a/b + ArcSinh[c*x])
^n*(-((a + b*ArcSinh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] + 8*35^(1 + n)*E^((8*a)/b)*(-((a
+ b*ArcSinh[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] - 3^(2
+ n)*7^(1 + n)*E^((10*a)/b)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])/b))^(3*n)*Gamma[1 + n, (5*(a + b*A
rcSinh[c*x]))/b] + 8*21^(1 + n)*E^((10*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))
^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b] + 15^(1 + n)*E^((12*a)/b)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh
[c*x])/b))^(3*n)*Gamma[1 + n, (7*(a + b*ArcSinh[c*x]))/b])))/(128*c^2*E^((7*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a +
b*ArcSinh[c*x])/b))^n*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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