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

Optimal. Leaf size=223 \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}+\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]

[Out]

(15*b^2*Sqrt[a + b*ArcSinh[c*x]])/(64*c^2) + (15*b^2*x^2*Sqrt[a + b*ArcSinh[c*x]])/32 - (5*b*x*Sqrt[1 + c^2*x^
2]*(a + b*ArcSinh[c*x])^(3/2))/(8*c) + (a + b*ArcSinh[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcSinh[c*x])^(5/2))/2
- (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(256*c^2) - (15*b^(5/2)
*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(256*c^2*E^((2*a)/b))

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Rubi [A]  time = 0.750858, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5663, 5758, 5675, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}+\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSinh[c*x])^(5/2),x]

[Out]

(15*b^2*Sqrt[a + b*ArcSinh[c*x]])/(64*c^2) + (15*b^2*x^2*Sqrt[a + b*ArcSinh[c*x]])/32 - (5*b*x*Sqrt[1 + c^2*x^
2]*(a + b*ArcSinh[c*x])^(3/2))/(8*c) + (a + b*ArcSinh[c*x])^(5/2)/(4*c^2) + (x^2*(a + b*ArcSinh[c*x])^(5/2))/2
- (15*b^(5/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(256*c^2) - (15*b^(5/2)
*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(256*c^2*E^((2*a)/b))

Rule 5663

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcSinh[c*x])^n)/
(m + 1), x] - Dist[(b*c*n)/(m + 1), Int[(x^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /;
FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]
), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5675

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSinh[c*x]
)^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && NeQ[n, -1
]

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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int x \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \, dx &=\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{1}{4} (5 b c) \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac{1}{16} \left (15 b^2\right ) \int x \sqrt{a+b \sinh ^{-1}(c x)} \, dx+\frac{(5 b) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{\sqrt{1+c^2 x^2}} \, dx}{8 c}\\ &=\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{1}{64} \left (15 b^3 c\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}} \, dx\\ &=\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2}\\ &=\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}+\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}-\frac{\cosh (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{256 c^2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{256 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{128 c^2}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{128 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sinh ^{-1}(c x)}}{64 c^2}+\frac{15}{32} b^2 x^2 \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{\left (a+b \sinh ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sinh ^{-1}(c x)\right )^{5/2}-\frac{15 b^{5/2} e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}-\frac{15 b^{5/2} e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{256 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0720903, size = 115, normalized size = 0.52 \[ \frac{e^{-\frac{2 a}{b}} \left (b^3 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{7}{2},\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-b^3 \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{32 \sqrt{2} c^2 \sqrt{a+b \sinh ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(b^3*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[7/2, (-2*(a + b*ArcSinh[c*x]))/b]) + b^3*E^((4*a)/b)*Sqrt[a/b + A
rcSinh[c*x]]*Gamma[7/2, (2*(a + b*ArcSinh[c*x]))/b])/(32*Sqrt[2]*c^2*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c*x]])

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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