Optimal. Leaf size=59 \[ \frac{2 b (b c-a d)}{d^3 (c+d x)}-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d^3} \]
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Rubi [A] time = 0.0382741, antiderivative size = 59, 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} \[ \frac{2 b (b c-a d)}{d^3 (c+d x)}-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(c+d x)^3} \, dx &=\int \left (\frac{(-b c+a d)^2}{d^2 (c+d x)^3}-\frac{2 b (b c-a d)}{d^2 (c+d x)^2}+\frac{b^2}{d^2 (c+d x)}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{2 d^3 (c+d x)^2}+\frac{2 b (b c-a d)}{d^3 (c+d x)}+\frac{b^2 \log (c+d x)}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0233252, size = 48, normalized size = 0.81 \[ \frac{\frac{(b c-a d) (a d+3 b c+4 b d x)}{(c+d x)^2}+2 b^2 \log (c+d x)}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 92, normalized size = 1.6 \begin{align*} -2\,{\frac{ab}{{d}^{2} \left ( dx+c \right ) }}+2\,{\frac{{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}+{\frac{{b}^{2}\ln \left ( dx+c \right ) }{{d}^{3}}}-{\frac{{a}^{2}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{abc}{{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( dx+c \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977537, size = 108, normalized size = 1.83 \begin{align*} \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} + \frac{b^{2} \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 = 1.86822, size = 205, normalized size = 3.47 \begin{align*} \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.605171, size = 80, normalized size = 1.36 \begin{align*} \frac{b^{2} \log{\left (c + d x \right )}}{d^{3}} - \frac{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 c^{2} d^{3} + 4 c d^{4} x + 2 d^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05654, size = 93, normalized size = 1.58 \begin{align*} \frac{b^{2} \log \left ({\left | d x + c \right |}\right )}{d^{3}} + \frac{4 \,{\left (b^{2} c - a b d\right )} x + \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{2 \,{\left (d x + c\right )}^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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