Optimal. Leaf size=23 \[ \frac {b}{a^2 (b+a x)}+\frac {\log (b+a x)}{a^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, 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 = {269, 45}
\begin {gather*} \frac {b}{a^2 (a x+b)}+\frac {\log (a x+b)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x} \, dx &=\int \frac {x}{(b+a x)^2} \, dx\\ &=\int \left (-\frac {b}{a (b+a x)^2}+\frac {1}{a (b+a x)}\right ) \, dx\\ &=\frac {b}{a^2 (b+a x)}+\frac {\log (b+a x)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.87 \begin {gather*} \frac {\frac {b}{b+a x}+\log (b+a x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 24, normalized size = 1.04
method | result | size |
default | \(\frac {b}{a^{2} \left (a x +b \right )}+\frac {\ln \left (a x +b \right )}{a^{2}}\) | \(24\) |
norman | \(\frac {b}{a^{2} \left (a x +b \right )}+\frac {\ln \left (a x +b \right )}{a^{2}}\) | \(24\) |
risch | \(\frac {b}{a^{2} \left (a x +b \right )}+\frac {\ln \left (a x +b \right )}{a^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 1.13 \begin {gather*} \frac {b}{a^{3} x + a^{2} b} + \frac {\log \left (a x + b\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 28, normalized size = 1.22 \begin {gather*} \frac {{\left (a x + b\right )} \log \left (a x + b\right ) + b}{a^{3} x + a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 20, normalized size = 0.87 \begin {gather*} \frac {b}{a^{3} x + a^{2} b} + \frac {\log {\left (a x + b \right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 24, normalized size = 1.04 \begin {gather*} \frac {\log \left ({\left | a x + b \right |}\right )}{a^{2}} + \frac {b}{{\left (a x + b\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 23, normalized size = 1.00 \begin {gather*} \frac {\ln \left (b+a\,x\right )}{a^2}+\frac {b}{a^2\,\left (b+a\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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