Optimal. Leaf size=49 \[ \frac{d x (b c-a d)}{b^2}+\frac{(b c-a d)^2 \log (a+b x)}{b^3}+\frac{(c+d x)^2}{2 b} \]
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Rubi [A] time = 0.0186455, antiderivative size = 49, 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} \[ \frac{d x (b c-a d)}{b^2}+\frac{(b c-a d)^2 \log (a+b x)}{b^3}+\frac{(c+d x)^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{a+b x} \, dx &=\int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx\\ &=\frac{d (b c-a d) x}{b^2}+\frac{(c+d x)^2}{2 b}+\frac{(b c-a d)^2 \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0160432, size = 43, normalized size = 0.88 \[ \frac{b d x (-2 a d+4 b c+b d x)+2 (b c-a d)^2 \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 74, normalized size = 1.5 \begin{align*}{\frac{{d}^{2}{x}^{2}}{2\,b}}-{\frac{a{d}^{2}x}{{b}^{2}}}+2\,{\frac{dxc}{b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ){d}^{2}}{{b}^{3}}}-2\,{\frac{a\ln \left ( bx+a \right ) cd}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{2}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976674, size = 82, normalized size = 1.67 \begin{align*} \frac{b d^{2} x^{2} + 2 \,{\left (2 \, b c d - a d^{2}\right )} x}{2 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06895, size = 135, normalized size = 2.76 \begin{align*} \frac{b^{2} d^{2} x^{2} + 2 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.378542, size = 44, normalized size = 0.9 \begin{align*} \frac{d^{2} x^{2}}{2 b} - \frac{x \left (a d^{2} - 2 b c d\right )}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06733, size = 81, normalized size = 1.65 \begin{align*} \frac{b d^{2} x^{2} + 4 \, b c d x - 2 \, a d^{2} x}{2 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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