Optimal. Leaf size=28 \[ -\frac{a \log \left (a-b x^2\right )}{2 b^2}-\frac{x^2}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0220877, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 43} \[ -\frac{a \log \left (a-b x^2\right )}{2 b^2}-\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{a-b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a-b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{b}-\frac{a}{b (-a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{2 b}-\frac{a \log \left (a-b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0058522, size = 28, normalized size = 1. \[ -\frac{a \log \left (a-b x^2\right )}{2 b^2}-\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 26, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{2\,b}}-{\frac{a\ln \left ( b{x}^{2}-a \right ) }{2\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52083, size = 34, normalized size = 1.21 \begin{align*} -\frac{x^{2}}{2 \, b} - \frac{a \log \left (b x^{2} - a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.21094, size = 50, normalized size = 1.79 \begin{align*} -\frac{b x^{2} + a \log \left (b x^{2} - a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.296177, size = 22, normalized size = 0.79 \begin{align*} - \frac{a \log{\left (- a + b x^{2} \right )}}{2 b^{2}} - \frac{x^{2}}{2 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.58651, size = 35, normalized size = 1.25 \begin{align*} -\frac{x^{2}}{2 \, b} - \frac{a \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]