Optimal. Leaf size=50 \[ -\frac{b x (b c-a d)}{d^2}+\frac{(b c-a d)^2 \log (c+d x)}{d^3}+\frac{(a+b x)^2}{2 d} \]
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Rubi [A] time = 0.0208861, antiderivative size = 50, 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} \[ -\frac{b x (b c-a d)}{d^2}+\frac{(b c-a d)^2 \log (c+d x)}{d^3}+\frac{(a+b x)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{c+d x} \, dx &=\int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx\\ &=-\frac{b (b c-a d) x}{d^2}+\frac{(a+b x)^2}{2 d}+\frac{(b c-a d)^2 \log (c+d x)}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0168373, size = 43, normalized size = 0.86 \[ \frac{b d x (4 a d-2 b c+b d x)+2 (b c-a d)^2 \log (c+d x)}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 74, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,d}}+2\,{\frac{abx}{d}}-{\frac{{b}^{2}xc}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){a}^{2}}{d}}-2\,{\frac{\ln \left ( dx+c \right ) abc}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){b}^{2}{c}^{2}}{{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976171, size = 81, normalized size = 1.62 \begin{align*} \frac{b^{2} d x^{2} - 2 \,{\left (b^{2} c - 2 \, a b d\right )} x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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 = 1.77896, size = 135, normalized size = 2.7 \begin{align*} \frac{b^{2} d^{2} x^{2} - 2 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} x + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.36138, size = 44, normalized size = 0.88 \begin{align*} \frac{b^{2} x^{2}}{2 d} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05664, size = 81, normalized size = 1.62 \begin{align*} \frac{b^{2} d x^{2} - 2 \, b^{2} c x + 4 \, a b d x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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