Optimal. Leaf size=39 \[ -\frac{3 a^2 b}{2 x^2}+a^3 \log (x)-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0208135, 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} \[ -\frac{3 a^2 b}{2 x^2}+a^3 \log (x)-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^3}{x} \, dx &=\int \frac{\left (b+a x^2\right )^3}{x^7} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^3}{x^4}+\frac{3 a b^2}{x^3}+\frac{3 a^2 b}{x^2}+\frac{a^3}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^3}{6 x^6}-\frac{3 a b^2}{4 x^4}-\frac{3 a^2 b}{2 x^2}+a^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0041217, size = 39, normalized size = 1. \[ -\frac{3 a^2 b}{2 x^2}+a^3 \log (x)-\frac{3 a b^2}{4 x^4}-\frac{b^3}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{6\,{x}^{6}}}-{\frac{3\,{b}^{2}a}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+{a}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.980571, size = 53, normalized size = 1.36 \begin{align*} \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) - \frac{18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.47874, size = 90, normalized size = 2.31 \begin{align*} \frac{12 \, a^{3} x^{6} \log \left (x\right ) - 18 \, a^{2} b x^{4} - 9 \, a b^{2} x^{2} - 2 \, b^{3}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.380377, size = 36, normalized size = 0.92 \begin{align*} a^{3} \log{\left (x \right )} - \frac{18 a^{2} b x^{4} + 9 a b^{2} x^{2} + 2 b^{3}}{12 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15831, size = 63, normalized size = 1.62 \begin{align*} \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) - \frac{11 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]