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

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{\sqrt{c^2 x^2+1} x^m}{\left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.113262, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^m \sqrt{1+c^2 x^2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.118895, size = 0, normalized size = 0. \[ \int \frac{x^m \sqrt{1+c^2 x^2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.591, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}\sqrt{{c}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (c^{2} x^{2} + 1\right )}^{2} x^{m} +{\left (c^{3} x^{3} + c x\right )} \sqrt{c^{2} x^{2} + 1} x^{m}}{a b c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} a b c^{2} x + a b c +{\left (b^{2} c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{{\left (c^{3}{\left (m + 2\right )} x^{3} + c{\left (m - 1\right )} x\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{m} +{\left (2 \, c^{4}{\left (m + 2\right )} x^{4} + c^{2}{\left (3 \, m + 2\right )} x^{2} + m\right )}{\left (c^{2} x^{2} + 1\right )} x^{m} +{\left (c^{5}{\left (m + 2\right )} x^{5} + c^{3}{\left (2 \, m + 3\right )} x^{3} + c{\left (m + 1\right )} x\right )} \sqrt{c^{2} x^{2} + 1} x^{m}}{a b c^{5} x^{5} +{\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{3} + 2 \, a b c^{3} x^{3} + a b c x +{\left (b^{2} c^{5} x^{5} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{3} + 2 \, b^{2} c^{3} x^{3} + b^{2} c x + 2 \,{\left (b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{4} x^{4} + a b c^{2} x^{2}\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^2*x^2 + 1)^2*x^m + (c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1)*x^m)/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*
b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((c^3*(m +
2)*x^3 + c*(m - 1)*x)*(c^2*x^2 + 1)^(3/2)*x^m + (2*c^4*(m + 2)*x^4 + c^2*(3*m + 2)*x^2 + m)*(c^2*x^2 + 1)*x^m
+ (c^5*(m + 2)*x^5 + c^3*(2*m + 3)*x^3 + c*(m + 1)*x)*sqrt(c^2*x^2 + 1)*x^m)/(a*b*c^5*x^5 + (c^2*x^2 + 1)*a*b
*c^3*x^3 + 2*a*b*c^3*x^3 + a*b*c*x + (b^2*c^5*x^5 + (c^2*x^2 + 1)*b^2*c^3*x^3 + 2*b^2*c^3*x^3 + b^2*c*x + 2*(b
^2*c^4*x^4 + b^2*c^2*x^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^4 + a*b*c^2*x^2)*sqrt
(c^2*x^2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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