Optimal. Leaf size=57 \[ \frac{b^3}{2 a^4 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^2}{a^3}+\frac{x^4}{4 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0420627, antiderivative size = 57, 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} \[ \frac{b^3}{2 a^4 \left (a x^2+b\right )}+\frac{3 b^2 \log \left (a x^2+b\right )}{2 a^4}-\frac{b x^2}{a^3}+\frac{x^4}{4 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+\frac{b}{x^2}\right )^2} \, dx &=\int \frac{x^7}{\left (b+a x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{2 b}{a^3}+\frac{x}{a^2}-\frac{b^3}{a^3 (b+a x)^2}+\frac{3 b^2}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b x^2}{a^3}+\frac{x^4}{4 a^2}+\frac{b^3}{2 a^4 \left (b+a x^2\right )}+\frac{3 b^2 \log \left (b+a x^2\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0154264, size = 49, normalized size = 0.86 \[ \frac{a^2 x^4+\frac{2 b^3}{a x^2+b}+6 b^2 \log \left (a x^2+b\right )-4 a b x^2}{4 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 52, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{2}}{{a}^{3}}}+{\frac{{x}^{4}}{4\,{a}^{2}}}+{\frac{{b}^{3}}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}+{\frac{3\,{b}^{2}\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05457, size = 73, normalized size = 1.28 \begin{align*} \frac{b^{3}}{2 \,{\left (a^{5} x^{2} + a^{4} b\right )}} + \frac{3 \, b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac{a x^{4} - 4 \, b x^{2}}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.39442, size = 143, normalized size = 2.51 \begin{align*} \frac{a^{3} x^{6} - 3 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} + 2 \, b^{3} + 6 \,{\left (a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{5} x^{2} + a^{4} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.532448, size = 53, normalized size = 0.93 \begin{align*} \frac{b^{3}}{2 a^{5} x^{2} + 2 a^{4} b} + \frac{x^{4}}{4 a^{2}} - \frac{b x^{2}}{a^{3}} + \frac{3 b^{2} \log{\left (a x^{2} + b \right )}}{2 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18324, size = 74, normalized size = 1.3 \begin{align*} \frac{3 \, b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac{b^{3}}{2 \,{\left (a x^{2} + b\right )} a^{4}} + \frac{a^{2} x^{4} - 4 \, a b x^{2}}{4 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]