3.197 \(\int \frac{x^m \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=102 \[ \frac{x^{m+1} \sinh ^{-1}(a x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-a^2 x^2\right )}{m+1}-\frac{a x^{m+2} \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-a^2 x^2\right )}{m^2+3 m+2} \]

[Out]

(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)*Hyper
geometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(a^2*x^2)])/(2 + 3*m + m^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0715032, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {5762} \[ \frac{x^{m+1} \sinh ^{-1}(a x) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right )}{m+1}-\frac{a x^{m+2} \, _3F_2\left (1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;-a^2 x^2\right )}{m^2+3 m+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)*Hyper
geometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(a^2*x^2)])/(2 + 3*m + m^2)

Rule 5762

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x
)^(m + 1)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/(Sqrt[d]*f*(m + 1)),
x] - Simp[(b*c*(f*x)^(m + 2)*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)])/(Sqrt
[d]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{x^m \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x^{1+m} \sinh ^{-1}(a x) \, _2F_1\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}\\ \end{align*}

Mathematica [A]  time = 0.0312955, size = 97, normalized size = 0.95 \[ \frac{x^{m+1} \left ((m+2) \sinh ^{-1}(a x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},-a^2 x^2\right )-a x \text{HypergeometricPFQ}\left (\left \{1,\frac{m}{2}+1,\frac{m}{2}+1\right \},\left \{\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2\right \},-a^2 x^2\right )\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.227, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\it Arcsinh} \left ( ax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m} \operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \operatorname{asinh}{\left (a x \right )}}{\sqrt{a^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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