\(\int (a+b \text {arcsinh}(c x))^{3/2} \, dx\) [141]

Optimal result

Integrand size = 12, antiderivative size = 135 \[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c} \]

[Out]

x*(a+b*arcsinh(c*x))^(3/2)+3/8*b^(3/2)*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c+3/8*b^(3/2)*e
rfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c/exp(a/b)-3/2*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^(1/2)/c

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5772, 5798, 5774, 3388, 2211, 2236, 2235} \[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {3 \sqrt {\pi } b^{3/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2} \]

[In]

Int[(a + b*ArcSinh[c*x])^(3/2),x]

[Out]

(-3*b*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(2*c) + x*(a + b*ArcSinh[c*x])^(3/2) + (3*b^(3/2)*E^(a/b)*Sq
rt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(8*c) + (3*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b
]])/(8*c*E^(a/b))

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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 3388

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 5772

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 5774

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[-a/b + x/b], x], x
, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^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]

Rubi steps \begin{align*} \text {integral}& = x (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} (3 b c) \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2}+\frac {1}{4} \left (3 b^2\right ) \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx \\ & = -\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2}+\frac {(3 b) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 c} \\ & = -\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2}+\frac {(3 b) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 c}+\frac {(3 b) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 c} \\ & = -\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2}+\frac {(3 b) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 c}+\frac {(3 b) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 c} \\ & = -\frac {3 b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{2 c}+x (a+b \text {arcsinh}(c x))^{3/2}+\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.86 \[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {a e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}}\right )}{2 c}+\frac {\sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \text {arcsinh}(c x)} \left (-3 \sqrt {1+c^2 x^2}+2 c x \text {arcsinh}(c x)\right )+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(-2 a+3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{8 c} \]

[In]

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

[Out]

(a*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[
3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (Sqrt[b]*(4*Sqrt[b]*Sqrt[a +
b*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*
x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b]
+ Sinh[a/b])))/(8*c)

Maple [F]

\[\int \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2} \,d x \]

[In]

int((a + b*asinh(c*x))^(3/2),x)

[Out]

int((a + b*asinh(c*x))^(3/2), x)