\(\int \frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx\) [51]

Optimal result

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

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

Mathematica [A] (verified)

Time = 2.51 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.20 \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}} \, dx=\frac {\frac {4 \sqrt {b} c \sqrt {1+(c+d x)^2}}{\sqrt {a+b \text {arcsinh}(c+d x)}}-2 c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+2 c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+2 c \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )-\sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+\sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\frac {2 \sqrt {b} \sinh (2 \text {arcsinh}(c+d x))}{\sqrt {a+b \text {arcsinh}(c+d x)}}}{2 b^{3/2} d^2} \] Input:

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

Output:

((4*Sqrt[b]*c*Sqrt[1 + (c + d*x)^2])/Sqrt[a + b*ArcSinh[c + d*x]] - 2*c*Sq 
rt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] + Sqrt[2*Pi]*C 
osh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + 2*c*Sq 
rt[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*Ar 
cSinh[c + d*x]])/(2*b^(3/2)*d^2)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6274, 25, 27, 6244, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {x}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -\frac {d x}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{d^2}\)

\(\Big \downarrow \) 6244

\(\displaystyle -\frac {\int \left (\frac {c}{(a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {c+d x}{(a+b \text {arcsinh}(c+d x))^{3/2}}\right )d(c+d x)}{d^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {\sqrt {\pi } c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2}}+\frac {\sqrt {\pi } c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2 c \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}}{d^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6244

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_S 
ymbol] :> 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 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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