3.652 \(\int \sqrt{d+e x^2} (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\sqrt{d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2,x\right ) \]

[Out]

Unintegrable[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2, x]

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Rubi [A]  time = 0.0420453, 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 = {} \[ \int \sqrt{d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2,x]

[Out]

Defer[Int][Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2, x]

Rubi steps

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

Mathematica [A]  time = 13.7045, size = 0, normalized size = 0. \[ \int \sqrt{d+e x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2,x]

[Out]

Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2, x]

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Maple [A]  time = 0.242, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}\sqrt{e{x}^{2}+d}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*sqrt(e*x^2 + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asinh(c*x))**2*sqrt(d + e*x**2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x^{2} + d}{\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((a+b*arcsinh(c*x))^2*(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arcsinh(c*x) + a)^2, x)