Optimal. Leaf size=25 \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]
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Rubi [A] time = 0.0207689, antiderivative size = 25, 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} \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]
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)^2} \, dx &=\frac{\int \frac{b^2 c+b^2 d x}{a+b x} \, dx}{b^2}\\ &=\frac{\int \left (b d+\frac{b (b c-a d)}{a+b x}\right ) \, dx}{b^2}\\ &=\frac{d x}{b}+\frac{(b c-a d) \log (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0071398, size = 25, normalized size = 1. \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 32, normalized size = 1.3 \begin{align*}{\frac{dx}{b}}-{\frac{\ln \left ( bx+a \right ) ad}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) c}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0755, size = 34, normalized size = 1.36 \begin{align*} \frac{d x}{b} + \frac{{\left (b c - a d\right )} \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 = 1.71667, size = 54, normalized size = 2.16 \begin{align*} \frac{b d x +{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.578064, size = 20, normalized size = 0.8 \begin{align*} \frac{d x}{b} - \frac{\left (a d - b c\right ) \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 [B] time = 1.1685, size = 158, normalized size = 6.32 \begin{align*} b d{\left (\frac{2 \, a \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{b x + a}{b^{3}} - \frac{a^{2}}{{\left (b x + a\right )} b^{3}}\right )} - \frac{{\left (b c + a d\right )}{\left (\frac{\log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}\right )}}{b} - \frac{a c}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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