\(\int x \text {arcsinh}(a x)^{5/2} \, dx\) [121]

Optimal result

Integrand size = 10, antiderivative size = 152 \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\frac {15 \sqrt {\text {arcsinh}(a x)}}{64 a^2}+\frac {15}{32} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {5 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{3/2}}{8 a}+\frac {\text {arcsinh}(a x)^{5/2}}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2}-\frac {15 \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a^2} \] Output:

15/64*arcsinh(a*x)^(1/2)/a^2+15/32*x^2*arcsinh(a*x)^(1/2)-5/8*x*(a^2*x^2+1 
)^(1/2)*arcsinh(a*x)^(3/2)/a+1/4*arcsinh(a*x)^(5/2)/a^2+1/2*x^2*arcsinh(a* 
x)^(5/2)-15/512*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*arcsinh(a*x)^(1/2))/a^2-15/51 
2*2^(1/2)*Pi^(1/2)*erfi(2^(1/2)*arcsinh(a*x)^(1/2))/a^2
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.34 \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\frac {\frac {\sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {-\text {arcsinh}(a x)}}+\Gamma \left (\frac {7}{2},2 \text {arcsinh}(a x)\right )}{32 \sqrt {2} a^2} \] Input:

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

Output:

((Sqrt[ArcSinh[a*x]]*Gamma[7/2, -2*ArcSinh[a*x]])/Sqrt[-ArcSinh[a*x]] + Ga 
mma[7/2, 2*ArcSinh[a*x]])/(32*Sqrt[2]*a^2)
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6192, 6227, 6192, 6198, 6234, 3042, 25, 3793, 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.

\(\displaystyle \int x \text {arcsinh}(a x)^{5/2} \, dx\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \int \frac {x^2 \text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {\int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {3 \int x \sqrt {\text {arcsinh}(a x)}dx}{4 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 a}-\frac {\int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a^2 x^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int -\frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}+\frac {\int \frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {3 \left (\frac {\int \left (\frac {1}{2 \sqrt {\text {arcsinh}(a x)}}-\frac {\cosh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}\right )}{4 a}-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^{5/2}-\frac {5}{4} a \left (-\frac {\text {arcsinh}(a x)^{5/2}}{5 a^3}-\frac {3 \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\sqrt {\text {arcsinh}(a x)}}{4 a^2}\right )}{4 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}{2 a^2}\right )\)

Input:

Int[x*ArcSinh[a*x]^(5/2),x]
 

Output:

(x^2*ArcSinh[a*x]^(5/2))/2 - (5*a*((x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2) 
)/(2*a^2) - ArcSinh[a*x]^(5/2)/(5*a^3) - (3*((x^2*Sqrt[ArcSinh[a*x]])/2 - 
(-Sqrt[ArcSinh[a*x]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4 + (S 
qrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4)/(4*a^2)))/(4*a)))/4
 

Defintions of rubi rules used

rule 25

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

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6192

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

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6227

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

rule 6234

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> Simp[(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]
 

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {\sqrt {2}\, \left (-128 \operatorname {arcsinh}\left (x a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+160 \operatorname {arcsinh}\left (x a \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -120 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\, \sqrt {\pi }\, a^{2} x^{2}-64 \operatorname {arcsinh}\left (x a \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }-60 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\, \sqrt {\pi }+15 \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\right )+15 \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\right )\right )}{512 \sqrt {\pi }\, a^{2}}\)	\(136\)

Input:

int(x*arcsinh(x*a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/512*2^(1/2)*(-128*arcsinh(x*a)^(5/2)*2^(1/2)*Pi^(1/2)*a^2*x^2+160*arcsi 
nh(x*a)^(3/2)*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*a*x-120*2^(1/2)*arcsinh(x 
*a)^(1/2)*Pi^(1/2)*a^2*x^2-64*arcsinh(x*a)^(5/2)*2^(1/2)*Pi^(1/2)-60*2^(1/ 
2)*arcsinh(x*a)^(1/2)*Pi^(1/2)+15*Pi*erf(2^(1/2)*arcsinh(x*a)^(1/2))+15*Pi 
*erfi(2^(1/2)*arcsinh(x*a)^(1/2)))/Pi^(1/2)/a^2
 

Fricas [F(-2)]

Exception generated. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*arcsinh(a*x)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int x \operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}\, dx \] Input:

integrate(x*asinh(a*x)**(5/2),x)
 

Output:

Integral(x*asinh(a*x)**(5/2), x)
 

Maxima [F]

\[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^{5/2} \,d x \] Input:

int(x*asinh(a*x)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int(x*asinh(a*x)^(5/2), x)
 

Reduce [F]

\[ \int x \text {arcsinh}(a x)^{5/2} \, dx=\int \sqrt {\mathit {asinh} \left (a x \right )}\, \mathit {asinh} \left (a x \right )^{2} x d x \] Input:

int(x*asinh(a*x)^(5/2),x)
 

Output:

int(sqrt(asinh(a*x))*asinh(a*x)**2*x,x)