Optimal. Leaf size=39 \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^6}{6}+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0199808, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^6}{6}+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x^2}\right )^3 x^5 \, dx &=\int \frac{\left (b+a x^2\right )^3}{x} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (3 a b^2+\frac{b^3}{x}+3 a^2 b x+a^3 x^2\right ) \, dx,x,x^2\right )\\ &=\frac{3}{2} a b^2 x^2+\frac{3}{4} a^2 b x^4+\frac{a^3 x^6}{6}+b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0040285, size = 39, normalized size = 1. \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^6}{6}+\frac{3}{2} a b^2 x^2+b^3 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 34, normalized size = 0.9 \begin{align*}{\frac{3\,a{b}^{2}{x}^{2}}{2}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{{a}^{3}{x}^{6}}{6}}+{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.966368, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, a^{3} x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{1}{2} \, b^{3} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.4256, size = 78, normalized size = 2. \begin{align*} \frac{1}{6} \, a^{3} x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{3}{2} \, a b^{2} x^{2} + b^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.25561, size = 37, normalized size = 0.95 \begin{align*} \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{4}}{4} + \frac{3 a b^{2} x^{2}}{2} + b^{3} \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17138, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, a^{3} x^{6} + \frac{3}{4} \, a^{2} b x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{1}{2} \, b^{3} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]