\(\int \frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx\) [108]

Optimal result

Integrand size = 16, antiderivative size = 269 \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2} \]

[Out]

1/2*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)/b^(3/2)/d^2+1/2*erfi(2^(1/2)*(
a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/d^2/exp(2*a/b)+c*exp(a/b)*erf((a+b*arcsinh(d*x+c))
^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/d^2-c*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/d^2/exp(a/b)+
2*c*(1+(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsinh(d*x+c))^(1/2)-2*(d*x+c)*(1+(d*x+c)^2)^(1/2)/b/d^2/(a+b*arcsinh(d*x+
c))^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5859, 5829, 5773, 5819, 3389, 2211, 2236, 2235, 5778, 3388} \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {\sqrt {\pi } c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {\sqrt {\pi } c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {2 c \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \text {arcsinh}(c+d x)}} \]

[In]

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

[Out]

(2*c*Sqrt[1 + (c + d*x)^2])/(b*d^2*Sqrt[a + b*ArcSinh[c + d*x]]) - (2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(b*d^2*
Sqrt[a + b*ArcSinh[c + d*x]]) + (c*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(b^(3/2)*d^2) +
(E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(b^(3/2)*d^2) - (c*Sqrt[Pi]*Erfi
[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(b^(3/2)*d^2*E^(a/b)) + (Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c
+ d*x]])/Sqrt[b]])/(b^(3/2)*d^2*E^((2*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 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 5773

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

Rule 5778

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

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]

Rule 5829

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]

Rule 5859

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
]

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

Mathematica [A] (verified)

Time = 2.61 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.20 \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {\frac {4 \sqrt {b} c \sqrt {1+(c+d x)^2}}{\sqrt {a+b \text {arcsinh}(c+d x)}}-2 c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+2 c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+2 c \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )-\sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+\sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\frac {2 \sqrt {b} \sinh (2 \text {arcsinh}(c+d x))}{\sqrt {a+b \text {arcsinh}(c+d x)}}}{2 b^{3/2} d^2} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x/(a + b*asinh(c + d*x))^(3/2),x)

[Out]

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