Optimal. Leaf size=23 \[ \frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45}
\begin {gather*} \frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a x^2+b x^3\right )^2} \, dx &=\int \frac {x}{(a+b x)^2} \, dx\\ &=\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx\\ &=\frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 0.87 \begin {gather*} \frac {\frac {a}{a+b x}+\log (a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 24, normalized size = 1.04
method | result | size |
default | \(\frac {a}{b^{2} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{2}}\) | \(24\) |
norman | \(\frac {a}{b^{2} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{2}}\) | \(24\) |
risch | \(\frac {a}{b^{2} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 1.13 \begin {gather*} \frac {a}{b^{3} x + a b^{2}} + \frac {\log \left (b x + a\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 28, normalized size = 1.22 \begin {gather*} \frac {{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \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 {a}{a b^{2} + b^{3} x} + \frac {\log {\left (a + b x \right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 24, normalized size = 1.04 \begin {gather*} \frac {\log \left ({\left | b x + a \right |}\right )}{b^{2}} + \frac {a}{{\left (b x + a\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 23, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,x\right )}{b^2}+\frac {a}{b^2\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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