3.533 \(\int \frac{(a+b \cosh ^{-1}(c x))^2}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}},x\right ) \]

[Out]

Unintegrable[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x]

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Rubi [A]  time = 0.0469609, 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 = {} \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2),x]

[Out]

Defer[Int][(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx &=\int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 7.25814, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2),x]

[Out]

Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x]

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Maple [A]  time = 0.24, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2} \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )} \sqrt{e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)*sqrt(e*x^2 + d)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2
+ d^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**2/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2/(e*x^2 + d)^(5/2), x)