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 = 0.02, 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 = {1584, 43} \[ \frac {a}{b^2 (a+b x)}+\frac {\log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 1584
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 = 0.01, size = 20, normalized size = 0.87 \[ \frac {\frac {a}{a+b x}+\log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 28, normalized size = 1.22 \[ \frac {{\left (b x + a\right )} \log \left (b x + a\right ) + a}{b^{3} x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 24, normalized size = 1.04 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{b^{2}} + \frac {a}{{\left (b x + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 1.04 \[ \frac {a}{\left (b x +a \right ) b^{2}}+\frac {\ln \left (b x +a \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 26, normalized size = 1.13 \[ \frac {a}{b^{3} x + a b^{2}} + \frac {\log \left (b x + a\right )}{b^{2}} \]
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 \[ \frac {\ln \left (a+b\,x\right )}{b^2}+\frac {a}{b^2\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.87 \[ \frac {a}{a b^{2} + b^{3} x} + \frac {\log {\left (a + b x \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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