\(\int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx\) [209]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {x^{1+m} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-a^2 x^2\right )}{1+m}-\frac {a x^{2+m} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-a^2 x^2\right )}{2+3 m+m^2} \] Output:

x^(1+m)*arcsinh(a*x)*hypergeom([1/2, 1/2+1/2*m],[3/2+1/2*m],-a^2*x^2)/(1+m 
)-a*x^(2+m)*hypergeom([1, 1+1/2*m, 1+1/2*m],[2+1/2*m, 3/2+1/2*m],-a^2*x^2) 
/(m^2+3*m+2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\frac {x^{1+m} \left ((2+m) \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},-a^2 x^2\right )-a x \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-a^2 x^2\right )\right )}{(1+m) (2+m)} \] Input:

Integrate[(x^m*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]
 

Output:

(x^(1 + m)*((2 + m)*ArcSinh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m) 
/2, -(a^2*x^2)] - a*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 
 2 + m/2}, -(a^2*x^2)]))/((1 + m)*(2 + m))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6232}

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.

\(\displaystyle \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}} \, dx\)

\(\Big \downarrow \) 6232

\(\displaystyle \frac {x^{m+1} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},-a^2 x^2\right )}{m+1}-\frac {a x^{m+2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;-a^2 x^2\right )}{m^2+3 m+2}\)

Input:

Int[(x^m*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]
 

Output:

(x^(1 + m)*ArcSinh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(a^2 
*x^2)])/(1 + m) - (a*x^(2 + m)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3 
/2 + m/2, 2 + m/2}, -(a^2*x^2)])/(2 + 3*m + m^2)
 

Defintions of rubi rules used

rule 6232

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_ 
.)*(x_)^2], x_Symbol] :> Simp[((f*x)^(m + 1)/(f*(m + 1)))*Simp[Sqrt[1 + c^2 
*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/ 
2, (3 + m)/2, (-c^2)*x^2], x] - Simp[b*c*((f*x)^(m + 2)/(f^2*(m + 1)*(m + 2 
)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*HypergeometricPFQ[{1, 1 + m/2, 
1 + m/2}, {3/2 + m/2, 2 + m/2}, (-c^2)*x^2], x] /; FreeQ[{a, b, c, d, e, f, 
 m}, x] && EqQ[e, c^2*d] &&  !IntegerQ[m]
 

Maple [F]

\[\int \frac {x^{m} \operatorname {arcsinh}\left (x a \right )}{\sqrt {a^{2} x^{2}+1}}d x\]

Input:

int(x^m*arcsinh(x*a)/(a^2*x^2+1)^(1/2),x)
 

Output:

int(x^m*arcsinh(x*a)/(a^2*x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^m*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(x^m*arcsinh(a*x)/sqrt(a^2*x^2 + 1), x)
 

Sympy [F]

\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^{m} \operatorname {asinh}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \] Input:

integrate(x**m*asinh(a*x)/(a**2*x**2+1)**(1/2),x)
 

Output:

Integral(x**m*asinh(a*x)/sqrt(a**2*x**2 + 1), x)
 

Maxima [F]

\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^m*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^m*arcsinh(a*x)/sqrt(a^2*x^2 + 1), x)
 

Giac [F]

\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:

integrate(x^m*arcsinh(a*x)/(a^2*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^m*arcsinh(a*x)/sqrt(a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^m\,\mathrm {asinh}\left (a\,x\right )}{\sqrt {a^2\,x^2+1}} \,d x \] Input:

int((x^m*asinh(a*x))/(a^2*x^2 + 1)^(1/2),x)
 

Output:

int((x^m*asinh(a*x))/(a^2*x^2 + 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^m \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^{m} \mathit {asinh} \left (a x \right )}{\sqrt {a^{2} x^{2}+1}}d x \] Input:

int(x^m*asinh(a*x)/(a^2*x^2+1)^(1/2),x)
 

Output:

int((x**m*asinh(a*x))/sqrt(a**2*x**2 + 1),x)