3.559 \(\int \frac{(a+b \cosh ^{-1}(c x))^{3/2}}{d+e x^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{d+e x^2},x\right ) \]

[Out]

Unintegrable[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2), x]

________________________________________________________________________________________

Rubi [A]  time = 0.0665648, 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 = {} \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{d+e x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2),x]

[Out]

Defer[Int][(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2), x]

Rubi steps

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

Mathematica [A]  time = 2.06754, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{d+e x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2),x]

[Out]

Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2), x]

________________________________________________________________________________________

Maple [A]  time = 0.245, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{e{x}^{2}+d} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x)

[Out]

int((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x)

________________________________________________________________________________________

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*arccosh(c*x))^(3/2)/(e*x^2+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**(3/2)/(e*x**2+d),x)

[Out]

Integral((a + b*acosh(c*x))**(3/2)/(d + e*x**2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^(3/2)/(e*x^2+d),x, algorithm="giac")

[Out]

Exception raised: AttributeError