Optimal. Leaf size=72 \[ \frac{3 b^2 x^2}{2 a^4}+\frac{b^5}{a^6 (a x+b)}-\frac{4 b^3 x}{a^5}+\frac{5 b^4 \log (a x+b)}{a^6}-\frac{2 b x^3}{3 a^3}+\frac{x^4}{4 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0478724, antiderivative size = 72, 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} \[ \frac{3 b^2 x^2}{2 a^4}+\frac{b^5}{a^6 (a x+b)}-\frac{4 b^3 x}{a^5}+\frac{5 b^4 \log (a x+b)}{a^6}-\frac{2 b x^3}{3 a^3}+\frac{x^4}{4 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^5}{(b+a x)^2} \, dx\\ &=\int \left (-\frac{4 b^3}{a^5}+\frac{3 b^2 x}{a^4}-\frac{2 b x^2}{a^3}+\frac{x^3}{a^2}-\frac{b^5}{a^5 (b+a x)^2}+\frac{5 b^4}{a^5 (b+a x)}\right ) \, dx\\ &=-\frac{4 b^3 x}{a^5}+\frac{3 b^2 x^2}{2 a^4}-\frac{2 b x^3}{3 a^3}+\frac{x^4}{4 a^2}+\frac{b^5}{a^6 (b+a x)}+\frac{5 b^4 \log (b+a x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.0163348, size = 66, normalized size = 0.92 \[ \frac{18 a^2 b^2 x^2-8 a^3 b x^3+3 a^4 x^4+\frac{12 b^5}{a x+b}-48 a b^3 x+60 b^4 \log (a x+b)}{12 a^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 67, normalized size = 0.9 \begin{align*} -4\,{\frac{{b}^{3}x}{{a}^{5}}}+{\frac{3\,{b}^{2}{x}^{2}}{2\,{a}^{4}}}-{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+{\frac{{x}^{4}}{4\,{a}^{2}}}+{\frac{{b}^{5}}{{a}^{6} \left ( ax+b \right ) }}+5\,{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03985, size = 95, normalized size = 1.32 \begin{align*} \frac{b^{5}}{a^{7} x + a^{6} b} + \frac{5 \, b^{4} \log \left (a x + b\right )}{a^{6}} + \frac{3 \, a^{3} x^{4} - 8 \, a^{2} b x^{3} + 18 \, a b^{2} x^{2} - 48 \, b^{3} x}{12 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.42428, size = 186, normalized size = 2.58 \begin{align*} \frac{3 \, a^{5} x^{5} - 5 \, a^{4} b x^{4} + 10 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} - 48 \, a b^{4} x + 12 \, b^{5} + 60 \,{\left (a b^{4} x + b^{5}\right )} \log \left (a x + b\right )}{12 \,{\left (a^{7} x + a^{6} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.356564, size = 71, normalized size = 0.99 \begin{align*} \frac{b^{5}}{a^{7} x + a^{6} b} + \frac{x^{4}}{4 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{3 b^{2} x^{2}}{2 a^{4}} - \frac{4 b^{3} x}{a^{5}} + \frac{5 b^{4} \log{\left (a x + b \right )}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14321, size = 99, normalized size = 1.38 \begin{align*} \frac{5 \, b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{6}} + \frac{b^{5}}{{\left (a x + b\right )} a^{6}} + \frac{3 \, a^{6} x^{4} - 8 \, a^{5} b x^{3} + 18 \, a^{4} b^{2} x^{2} - 48 \, a^{3} b^{3} x}{12 \, a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]