Optimal. Leaf size=32 \[ \frac {d \log (a+b x)}{b^2}-\frac {b c-a d}{b^2 (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {24, 43} \[ \frac {d \log (a+b x)}{b^2}-\frac {b c-a d}{b^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 24
Rule 43
Rubi steps
\begin {align*} \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^3} \, dx &=\frac {\int \frac {b^2 c+b^2 d x}{(a+b x)^2} \, dx}{b^2}\\ &=\frac {\int \left (\frac {b (b c-a d)}{(a+b x)^2}+\frac {b d}{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {b c-a d}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.97 \[ \frac {a d-b c}{b^2 (a+b x)}+\frac {d \log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 39, normalized size = 1.22 \[ -\frac {b c - a d - {\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 33, normalized size = 1.03 \[ \frac {d \log \left ({\left | b x + a \right |}\right )}{b^{2}} - \frac {b c - a d}{{\left (b x + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 39, normalized size = 1.22 \[ \frac {a d}{\left (b x +a \right ) b^{2}}-\frac {c}{\left (b x +a \right ) b}+\frac {d \ln \left (b x +a \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 35, normalized size = 1.09 \[ -\frac {b c - a d}{b^{3} x + a b^{2}} + \frac {d \log \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 31, normalized size = 0.97 \[ \frac {a\,d-b\,c}{b^2\,\left (a+b\,x\right )}+\frac {d\,\ln \left (a+b\,x\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 27, normalized size = 0.84 \[ \frac {a d - b c}{a b^{2} + b^{3} x} + \frac {d \log {\left (a + b x \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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