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

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

[Out]

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

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Rubi [A]  time = 0.113754, 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 = {} \[ \int \frac{x^m \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.114576, size = 0, normalized size = 0. \[ \int \frac{x^m \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.605, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{a+b{\it Arcsinh} \left ( cx \right ) }\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)),x)

[Out]

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

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

[Out]

integrate(sqrt(c^2*x^2 + 1)*x^m/(b*arcsinh(c*x) + a), 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 \operatorname{arsinh}\left (c x\right ) + a}, 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)),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)*x^m/(b*arcsinh(c*x) + a), 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}}{a + b \operatorname{asinh}{\left (c x \right )}}\, 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)),x)

[Out]

Integral(x**m*sqrt(c**2*x**2 + 1)/(a + b*asinh(c*x)), 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}}{b \operatorname{arsinh}\left (c x\right ) + a}\,{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)),x, algorithm="giac")

[Out]

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