3.101 \(\int x (a+b \sinh ^{-1}(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=326 \[ -\frac {3 \sqrt {\pi } b^{3/2} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 \sqrt {\pi } b^{3/2} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}}{16 d^2}+\frac {3 b c \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac {\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]

[Out]

-c*(d*x+c)*(a+b*arcsinh(d*x+c))^(3/2)/d^2+1/4*(a+b*arcsinh(d*x+c))^(3/2)*cosh(2*arcsinh(d*x+c))/d^2-3/128*b^(3
/2)*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d^2+3/128*b^(3/2)*erfi(2^(1/2)
*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d^2/exp(2*a/b)-3/8*b^(3/2)*c*exp(a/b)*erf((a+b*arcsinh(d
*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d^2-3/8*b^(3/2)*c*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d^2/exp(a/b
)-3/16*b*sinh(2*arcsinh(d*x+c))*(a+b*arcsinh(d*x+c))^(1/2)/d^2+3/2*b*c*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c)
)^(1/2)/d^2

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Rubi [A]  time = 0.96, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5865, 5805, 6741, 6742, 5325, 5324, 5299, 2205, 2204, 5298} \[ -\frac {3 \sqrt {\pi } b^{3/2} c e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 \sqrt {\pi } b^{3/2} c e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 d^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d^2}-\frac {3 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}}{16 d^2}+\frac {3 b c \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d^2}+\frac {\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSinh[c + d*x])^(3/2),x]

[Out]

(3*b*c*Sqrt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/(2*d^2) - (c*(c + d*x)*(a + b*ArcSinh[c + d*x])^(3/
2))/d^2 + ((a + b*ArcSinh[c + d*x])^(3/2)*Cosh[2*ArcSinh[c + d*x]])/(4*d^2) - (3*b^(3/2)*c*E^(a/b)*Sqrt[Pi]*Er
f[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*d^2) - (3*b^(3/2)*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*A
rcSinh[c + d*x]])/Sqrt[b]])/(64*d^2) - (3*b^(3/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*d^
2*E^(a/b)) + (3*b^(3/2)*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(64*d^2*E^((2*a)/b))
- (3*b*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[2*ArcSinh[c + d*x]])/(16*d^2)

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]

Rule 5298

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5299

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5324

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

Rule 5325

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

Rule 5805

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 5865

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 6741

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

Rule 6742

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

Rubi steps

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

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Mathematica [A]  time = 5.19, size = 582, normalized size = 1.79 \[ \frac {-16 \sqrt {b} c \left (\sqrt {\pi } (3 b-2 a) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {\pi } (2 a+3 b) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+4 \sqrt {b} \left (2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt {(c+d x)^2+1}\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}\right )+4 a \left (-\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}\right )+\sqrt {b} \left (\sqrt {2 \pi } (4 a-3 b) \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } (4 a+3 b) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {b} \left (4 \sinh ^{-1}(c+d x) \cosh \left (2 \sinh ^{-1}(c+d x)\right )-3 \sinh \left (2 \sinh ^{-1}(c+d x)\right )\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}\right )-64 a c e^{-\frac {a}{b}} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}}}-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)}}\right )}{128 d^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcSinh[c + d*x])^(3/2),x]

[Out]

((-64*a*c*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[
c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) - 16*Sqrt
[b]*c*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2*(c + d*x)*ArcSinh[c + d*x]) + (2*a
+ 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Er
f[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + 4*a*(8*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*
ArcSinh[c + d*x]] + Sqrt[b]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(-Cosh[(2*a)/b] +
Sinh[(2*a)/b]) - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[
(2*a)/b])) + Sqrt[b]*((4*a + 3*b)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/
b] - Sinh[(2*a)/b]) + (4*a - 3*b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b
] + Sinh[(2*a)/b]) + 8*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(4*ArcSinh[c + d*x]*Cosh[2*ArcSinh[c + d*x]] - 3*S
inh[2*ArcSinh[c + d*x]])))/(128*d^2)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsinh(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsinh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)^(3/2)*x, x)

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maple [F]  time = 0.50, size = 0, normalized size = 0.00 \[ \int x \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsinh(d*x+c))^(3/2),x)

[Out]

int(x*(a+b*arcsinh(d*x+c))^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsinh(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arcsinh(d*x + c) + a)^(3/2)*x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asinh(c + d*x))^(3/2),x)

[Out]

int(x*(a + b*asinh(c + d*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asinh(d*x+c))**(3/2),x)

[Out]

Integral(x*(a + b*asinh(c + d*x))**(3/2), x)

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