Optimal. Leaf size=58 \[ -\frac {b^4}{a^5 (a x+b)}-\frac {4 b^3 \log (a x+b)}{a^5}+\frac {3 b^2 x}{a^4}-\frac {b x^2}{a^3}+\frac {x^3}{3 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 58, 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} \[ -\frac {b^4}{a^5 (a x+b)}+\frac {3 b^2 x}{a^4}-\frac {4 b^3 \log (a x+b)}{a^5}-\frac {b x^2}{a^3}+\frac {x^3}{3 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^4}{(b+a x)^2} \, dx\\ &=\int \left (\frac {3 b^2}{a^4}-\frac {2 b x}{a^3}+\frac {x^2}{a^2}+\frac {b^4}{a^4 (b+a x)^2}-\frac {4 b^3}{a^4 (b+a x)}\right ) \, dx\\ &=\frac {3 b^2 x}{a^4}-\frac {b x^2}{a^3}+\frac {x^3}{3 a^2}-\frac {b^4}{a^5 (b+a x)}-\frac {4 b^3 \log (b+a x)}{a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 54, normalized size = 0.93 \[ \frac {a^3 x^3-3 a^2 b x^2-\frac {3 b^4}{a x+b}-12 b^3 \log (a x+b)+9 a b^2 x}{3 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.89, size = 73, normalized size = 1.26 \[ \frac {a^{4} x^{4} - 2 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a b^{3} x - 3 \, b^{4} - 12 \, {\left (a b^{3} x + b^{4}\right )} \log \left (a x + b\right )}{3 \, {\left (a^{6} x + a^{5} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 62, normalized size = 1.07 \[ -\frac {4 \, b^{3} \log \left ({\left | a x + b \right |}\right )}{a^{5}} - \frac {b^{4}}{{\left (a x + b\right )} a^{5}} + \frac {a^{4} x^{3} - 3 \, a^{3} b x^{2} + 9 \, a^{2} b^{2} x}{3 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 57, normalized size = 0.98 \[ \frac {x^{3}}{3 a^{2}}-\frac {b \,x^{2}}{a^{3}}+\frac {3 b^{2} x}{a^{4}}-\frac {b^{4}}{\left (a x +b \right ) a^{5}}-\frac {4 b^{3} \ln \left (a x +b \right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.05, size = 59, normalized size = 1.02 \[ -\frac {b^{4}}{a^{6} x + a^{5} b} - \frac {4 \, b^{3} \log \left (a x + b\right )}{a^{5}} + \frac {a^{2} x^{3} - 3 \, a b x^{2} + 9 \, b^{2} x}{3 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.05, size = 62, normalized size = 1.07 \[ \frac {x^3}{3\,a^2}-\frac {4\,b^3\,\ln \left (b+a\,x\right )}{a^5}-\frac {b\,x^2}{a^3}+\frac {3\,b^2\,x}{a^4}-\frac {b^4}{a\,\left (x\,a^5+b\,a^4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.22, size = 54, normalized size = 0.93 \[ - \frac {b^{4}}{a^{6} x + a^{5} b} + \frac {x^{3}}{3 a^{2}} - \frac {b x^{2}}{a^{3}} + \frac {3 b^{2} x}{a^{4}} - \frac {4 b^{3} \log {\left (a x + b \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________