Optimal. Leaf size=51 \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]
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Rubi [A] time = 0.0385264, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(c+d x)^2} \, dx &=\int \left (\frac{b^2}{d^2}+\frac{(-b c+a d)^2}{d^2 (c+d x)^2}-\frac{2 b (b c-a d)}{d^2 (c+d x)}\right ) \, dx\\ &=\frac{b^2 x}{d^2}-\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0352948, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{c+d x}+2 b (a d-b c) \log (c+d x)+b^2 d x}{d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 86, normalized size = 1.7 \begin{align*}{\frac{{b}^{2}x}{{d}^{2}}}-{\frac{{a}^{2}}{d \left ( dx+c \right ) }}+2\,{\frac{abc}{{d}^{2} \left ( dx+c \right ) }}-{\frac{{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+2\,{\frac{b\ln \left ( dx+c \right ) a}{{d}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( dx+c \right ) c}{{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964595, size = 90, normalized size = 1.76 \begin{align*} \frac{b^{2} x}{d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{4} x + c d^{3}} - \frac{2 \,{\left (b^{2} c - a b d\right )} \log \left (d x + c\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71685, size = 184, normalized size = 3.61 \begin{align*} \frac{b^{2} d^{2} x^{2} + b^{2} c d x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \,{\left (b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{d^{4} x + c d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.503073, size = 60, normalized size = 1.18 \begin{align*} \frac{b^{2} x}{d^{2}} + \frac{2 b \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{3}} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{c d^{3} + d^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0702, size = 132, normalized size = 2.59 \begin{align*} \frac{{\left (d x + c\right )} b^{2}}{d^{3}} + \frac{2 \,{\left (b^{2} c - a b d\right )} \log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d^{3}} - \frac{\frac{b^{2} c^{2} d}{d x + c} - \frac{2 \, a b c d^{2}}{d x + c} + \frac{a^{2} d^{3}}{d x + c}}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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