Optimal. Leaf size=44 \[ \frac{b^2 x}{a^3}-\frac{b^3 \log (a x+b)}{a^4}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0222153, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{b^2 x}{a^3}-\frac{b^3 \log (a x+b)}{a^4}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{a+\frac{b}{x}} \, dx &=\int \frac{x^3}{b+a x} \, dx\\ &=\int \left (\frac{b^2}{a^3}-\frac{b x}{a^2}+\frac{x^2}{a}-\frac{b^3}{a^3 (b+a x)}\right ) \, dx\\ &=\frac{b^2 x}{a^3}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a}-\frac{b^3 \log (b+a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0034744, size = 44, normalized size = 1. \[ \frac{b^2 x}{a^3}-\frac{b^3 \log (a x+b)}{a^4}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 41, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}x}{{a}^{3}}}-{\frac{b{x}^{2}}{2\,{a}^{2}}}+{\frac{{x}^{3}}{3\,a}}-{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01937, size = 57, normalized size = 1.3 \begin{align*} -\frac{b^{3} \log \left (a x + b\right )}{a^{4}} + \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41328, size = 92, normalized size = 2.09 \begin{align*} \frac{2 \, a^{3} x^{3} - 3 \, a^{2} b x^{2} + 6 \, a b^{2} x - 6 \, b^{3} \log \left (a x + b\right )}{6 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.268416, size = 37, normalized size = 0.84 \begin{align*} \frac{x^{3}}{3 a} - \frac{b x^{2}}{2 a^{2}} + \frac{b^{2} x}{a^{3}} - \frac{b^{3} \log{\left (a x + b \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09174, size = 58, normalized size = 1.32 \begin{align*} -\frac{b^{3} \log \left ({\left | a x + b \right |}\right )}{a^{4}} + \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]