Optimal. Leaf size=101 \[ -\frac{8 a \sqrt{a^2 x^2+1} \text{Unintegrable}\left (\frac{x}{\left (a^2 x^2+1\right )^3 \sinh ^{-1}(a x)^{3/2}},x\right )}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{3 a \left (a^2 c x^2+c\right )^{5/2} \sinh ^{-1}(a x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0905415, 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 = {} \[ \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^{3/2}}-\frac{\left (8 a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^{3/2}} \, dx}{3 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.4787, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.228, size = 0, normalized size = 0. \begin{align*} \int{ \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]