Optimal. Leaf size=62 \[ -\frac{(a d+b c)^2 \log (a-b x)}{2 a b^3}+\frac{(b c-a d)^2 \log (a+b x)}{2 a b^3}-\frac{d^2 x}{b^2} \]
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Rubi [A] time = 0.0455141, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {72} \[ -\frac{(a d+b c)^2 \log (a-b x)}{2 a b^3}+\frac{(b c-a d)^2 \log (a+b x)}{2 a b^3}-\frac{d^2 x}{b^2} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(a-b x) (a+b x)} \, dx &=\int \left (-\frac{d^2}{b^2}+\frac{(b c+a d)^2}{2 a b^2 (a-b x)}+\frac{(-b c+a d)^2}{2 a b^2 (a+b x)}\right ) \, dx\\ &=-\frac{d^2 x}{b^2}-\frac{(b c+a d)^2 \log (a-b x)}{2 a b^3}+\frac{(b c-a d)^2 \log (a+b x)}{2 a b^3}\\ \end{align*}
Mathematica [A] time = 0.025909, size = 54, normalized size = 0.87 \[ \frac{-(a d+b c)^2 \log (a-b x)+(b c-a d)^2 \log (a+b x)-2 a b d^2 x}{2 a b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 107, normalized size = 1.7 \begin{align*} -{\frac{{d}^{2}x}{{b}^{2}}}+{\frac{a\ln \left ( bx+a \right ){d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( bx+a \right ) cd}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{2}}{2\,ab}}-{\frac{a\ln \left ( bx-a \right ){d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( bx-a \right ) cd}{{b}^{2}}}-{\frac{\ln \left ( bx-a \right ){c}^{2}}{2\,ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77663, size = 111, normalized size = 1.79 \begin{align*} -\frac{d^{2} x}{b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, a b^{3}} - \frac{{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x - a\right )}{2 \, a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19701, size = 165, normalized size = 2.66 \begin{align*} -\frac{2 \, a b d^{2} x -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x - a\right )}{2 \, a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.691094, size = 112, normalized size = 1.81 \begin{align*} - \frac{d^{2} x}{b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (x + \frac{2 a^{2} c d + \frac{a \left (a d - b c\right )^{2}}{b}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 a b^{3}} - \frac{\left (a d + b c\right )^{2} \log{\left (x + \frac{2 a^{2} c d - \frac{a \left (a d + b c\right )^{2}}{b}}{a^{2} d^{2} + b^{2} c^{2}} \right )}}{2 a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.76339, size = 113, normalized size = 1.82 \begin{align*} -\frac{d^{2} x}{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 )}{2 \, a b^{3}} - \frac{{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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