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

Optimal. Leaf size=155 \[ \frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}+\frac{15}{4} b^2 x \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]

[Out]

(15*b^2*x*Sqrt[a + b*ArcSinh[c*x]])/4 - (5*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(3/2))/(2*c) + x*(a + b*Ar
cSinh[c*x])^(5/2) + (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c) - (15*b^(5/2)*S
qrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c*E^(a/b))

________________________________________________________________________________________

Rubi [A]  time = 0.406392, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5653, 5717, 5779, 3308, 2180, 2204, 2205} \[ \frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 c}+\frac{15}{4} b^2 x \sqrt{a+b \sinh ^{-1}(c x)}-\frac{5 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{3/2}}{2 c}+x \left (a+b \sinh ^{-1}(c x)\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])^(5/2),x]

[Out]

(15*b^2*x*Sqrt[a + b*ArcSinh[c*x]])/4 - (5*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(3/2))/(2*c) + x*(a + b*Ar
cSinh[c*x])^(5/2) + (15*b^(5/2)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c) - (15*b^(5/2)*S
qrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*c*E^(a/b))

Rule 5653

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

Rule 5717

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[p]), Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -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 3308

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

Mathematica [A]  time = 3.10819, size = 282, normalized size = 1.82 \[ \frac{\sqrt{b} e^{-\frac{a}{b}} \left (\frac{4 \sqrt{b} \left (-2 a^2 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )-2 a^2 \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )+e^{a/b} \left (a+b \sinh ^{-1}(c x)\right ) \left (5 \left (3 b c x-2 a \sqrt{c^2 x^2+1}\right )+2 \sinh ^{-1}(c x) \left (4 a c x-5 b \sqrt{c^2 x^2+1}\right )+4 b c x \sinh ^{-1}(c x)^2\right )\right )}{\sqrt{a+b \sinh ^{-1}(c x)}}+\sqrt{\pi } \left (-\left (4 a^2-15 b^2\right )\right ) e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )+\sqrt{\pi } \left (4 a^2-15 b^2\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )\right )}{16 c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[b]*(-((4*a^2 - 15*b^2)*E^((2*a)/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]) + (4*a^2 - 15*b^2)*Sq
rt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]] + (4*Sqrt[b]*(E^(a/b)*(a + b*ArcSinh[c*x])*(5*(3*b*c*x - 2*a*Sqr
t[1 + c^2*x^2]) + 2*(4*a*c*x - 5*b*Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 4*b*c*x*ArcSinh[c*x]^2) - 2*a^2*E^((2*a)/
b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, a/b + ArcSinh[c*x]] - 2*a^2*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2,
-((a + b*ArcSinh[c*x])/b)]))/Sqrt[a + b*ArcSinh[c*x]]))/(16*c*E^(a/b))

________________________________________________________________________________________

Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \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((a+b*arcsinh(c*x))^(5/2),x)

[Out]

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

________________________________________________________________________________________

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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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