Optimal. Leaf size=46 \[ -\frac {2 b x}{a^3}+\frac {x^2}{2 a^2}+\frac {b^3}{a^4 (b+a x)}+\frac {3 b^2 \log (b+a x)}{a^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {269, 45}
\begin {gather*} \frac {b^3}{a^4 (a x+b)}+\frac {3 b^2 \log (a x+b)}{a^4}-\frac {2 b x}{a^3}+\frac {x^2}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{x}\right )^2} \, dx &=\int \frac {x^3}{(b+a x)^2} \, dx\\ &=\int \left (-\frac {2 b}{a^3}+\frac {x}{a^2}-\frac {b^3}{a^3 (b+a x)^2}+\frac {3 b^2}{a^3 (b+a x)}\right ) \, dx\\ &=-\frac {2 b x}{a^3}+\frac {x^2}{2 a^2}+\frac {b^3}{a^4 (b+a x)}+\frac {3 b^2 \log (b+a x)}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.93 \begin {gather*} \frac {-4 a b x+a^2 x^2+\frac {2 b^3}{b+a x}+6 b^2 \log (b+a x)}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 45, normalized size = 0.98
method | result | size |
default | \(\frac {\frac {1}{2} a \,x^{2}-2 b x}{a^{3}}+\frac {b^{3}}{a^{4} \left (a x +b \right )}+\frac {3 b^{2} \ln \left (a x +b \right )}{a^{4}}\) | \(45\) |
risch | \(-\frac {2 b x}{a^{3}}+\frac {x^{2}}{2 a^{2}}+\frac {b^{3}}{a^{4} \left (a x +b \right )}+\frac {3 b^{2} \ln \left (a x +b \right )}{a^{4}}\) | \(45\) |
norman | \(\frac {\frac {3 b^{3}}{a^{4}}+\frac {x^{3}}{2 a}-\frac {3 b \,x^{2}}{2 a^{2}}}{a x +b}+\frac {3 b^{2} \ln \left (a x +b \right )}{a^{4}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 1.02 \begin {gather*} \frac {b^{3}}{a^{5} x + a^{4} b} + \frac {3 \, b^{2} \log \left (a x + b\right )}{a^{4}} + \frac {a x^{2} - 4 \, b x}{2 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 62, normalized size = 1.35 \begin {gather*} \frac {a^{3} x^{3} - 3 \, a^{2} b x^{2} - 4 \, a b^{2} x + 2 \, b^{3} + 6 \, {\left (a b^{2} x + b^{3}\right )} \log \left (a x + b\right )}{2 \, {\left (a^{5} x + a^{4} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 44, normalized size = 0.96 \begin {gather*} \frac {b^{3}}{a^{5} x + a^{4} b} + \frac {x^{2}}{2 a^{2}} - \frac {2 b x}{a^{3}} + \frac {3 b^{2} \log {\left (a x + b \right )}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 48, normalized size = 1.04 \begin {gather*} \frac {3 \, b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{4}} + \frac {b^{3}}{{\left (a x + b\right )} a^{4}} + \frac {a^{2} x^{2} - 4 \, a b x}{2 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 50, normalized size = 1.09 \begin {gather*} \frac {x^2}{2\,a^2}+\frac {3\,b^2\,\ln \left (b+a\,x\right )}{a^4}+\frac {b^3}{a\,\left (x\,a^4+b\,a^3\right )}-\frac {2\,b\,x}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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