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 = 0.024534, antiderivative size = 32, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0101296, 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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Maple [A] time = 0.045, size = 39, normalized size = 1.2 \begin{align*}{\frac{d\ln \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ad}{{b}^{2} \left ( bx+a \right ) }}-{\frac{c}{b \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0242, size = 47, normalized size = 1.47 \begin{align*} -\frac{b c - a d}{b^{3} x + a b^{2}} + \frac{d \log \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82422, size = 80, normalized size = 2.5 \begin{align*} -\frac{b c - a d -{\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.510869, size = 27, normalized size = 0.84 \begin{align*} \frac{a d - b c}{a b^{2} + b^{3} x} + \frac{d \log{\left (a + b x \right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23979, size = 45, normalized size = 1.41 \begin{align*} \frac{d \log \left ({\left | b x + a \right |}\right )}{b^{2}} - \frac{b c - a d}{{\left (b x + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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