3.370 \(\int \frac{(1+c^2 x^2)^{3/2}}{x^3 (a+b \sinh ^{-1}(c x))} \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{\left (c^2 x^2+1\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])), x]

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Rubi [A]  time = 0.14518, 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 = {} \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])), x]

Rubi steps

\begin{align*} \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx &=\int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 4.59578, size = 0, normalized size = 0. \[ \int \frac{\left (1+c^2 x^2\right )^{3/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])), x]

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Maple [A]  time = 0.44, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) } \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*x^2+1)^(3/2)/x^3/(a+b*arcsinh(c*x)),x)

[Out]

int((c^2*x^2+1)^(3/2)/x^3/(a+b*arcsinh(c*x)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(3/2)/x^3/(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

integrate((c^2*x^2 + 1)^(3/2)/((b*arcsinh(c*x) + a)*x^3), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{b x^{3} \operatorname{arsinh}\left (c x\right ) + a x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(3/2)/x^3/(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((c^2*x^2 + 1)^(3/2)/(b*x^3*arcsinh(c*x) + a*x^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c^{2} x^{2} + 1\right )^{\frac{3}{2}}}{x^{3} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c**2*x**2+1)**(3/2)/x**3/(a+b*asinh(c*x)),x)

[Out]

Integral((c**2*x**2 + 1)**(3/2)/(x**3*(a + b*asinh(c*x))), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(3/2)/x^3/(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

integrate((c^2*x^2 + 1)^(3/2)/((b*arcsinh(c*x) + a)*x^3), x)