Optimal. Leaf size=44 \[ -\frac {b^3 \log (a x+b)}{a^4}+\frac {b^2 x}{a^3}-\frac {b x^2}{2 a^2}+\frac {x^3}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 43
Rule 263
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 = 0.00, size = 44, normalized size = 1.00 \[ -\frac {b^3 \log (a x+b)}{a^4}+\frac {b^2 x}{a^3}-\frac {b x^2}{2 a^2}+\frac {x^3}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 41, normalized size = 0.93 \[ \frac {2 \, a^{3} x^{3} - 3 \, a^{2} b x^{2} + 6 \, a b^{2} x - 6 \, b^{3} \log \left (a x + b\right )}{6 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 43, normalized size = 0.98 \[ -\frac {b^{3} \log \left ({\left | a x + b \right |}\right )}{a^{4}} + \frac {2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 41, normalized size = 0.93 \[ \frac {x^{3}}{3 a}-\frac {b \,x^{2}}{2 a^{2}}+\frac {b^{2} x}{a^{3}}-\frac {b^{3} \ln \left (a x +b \right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.00, size = 42, normalized size = 0.95 \[ -\frac {b^{3} \log \left (a x + b\right )}{a^{4}} + \frac {2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 40, normalized size = 0.91 \[ \frac {x^3}{3\,a}-\frac {b^3\,\ln \left (b+a\,x\right )}{a^4}-\frac {b\,x^2}{2\,a^2}+\frac {b^2\,x}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.14, size = 37, normalized size = 0.84 \[ \frac {x^{3}}{3 a} - \frac {b x^{2}}{2 a^{2}} + \frac {b^{2} x}{a^{3}} - \frac {b^{3} \log {\left (a x + b \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________