Optimal. Leaf size=73 \[ -\frac{(c d-b e)^2}{b^2 c (b+c x)}-\frac{2 d \log (x) (c d-b e)}{b^3}+\frac{2 d (c d-b e) \log (b+c x)}{b^3}-\frac{d^2}{b^2 x} \]
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Rubi [A] time = 0.0581193, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{(c d-b e)^2}{b^2 c (b+c x)}-\frac{2 d \log (x) (c d-b e)}{b^3}+\frac{2 d (c d-b e) \log (b+c x)}{b^3}-\frac{d^2}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac{d^2}{b^2 x^2}+\frac{2 d (-c d+b e)}{b^3 x}+\frac{(-c d+b e)^2}{b^2 (b+c x)^2}-\frac{2 c d (-c d+b e)}{b^3 (b+c x)}\right ) \, dx\\ &=-\frac{d^2}{b^2 x}-\frac{(c d-b e)^2}{b^2 c (b+c x)}-\frac{2 d (c d-b e) \log (x)}{b^3}+\frac{2 d (c d-b e) \log (b+c x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0769541, size = 67, normalized size = 0.92 \[ \frac{-\frac{b (c d-b e)^2}{c (b+c x)}+2 d \log (x) (b e-c d)+2 d (c d-b e) \log (b+c x)-\frac{b d^2}{x}}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 106, normalized size = 1.5 \begin{align*} -{\frac{{d}^{2}}{{b}^{2}x}}+2\,{\frac{d\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{{d}^{2}\ln \left ( x \right ) c}{{b}^{3}}}-{\frac{{e}^{2}}{c \left ( cx+b \right ) }}+2\,{\frac{de}{b \left ( cx+b \right ) }}-{\frac{c{d}^{2}}{{b}^{2} \left ( cx+b \right ) }}-2\,{\frac{d\ln \left ( cx+b \right ) e}{{b}^{2}}}+2\,{\frac{{d}^{2}\ln \left ( cx+b \right ) c}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14467, size = 126, normalized size = 1.73 \begin{align*} -\frac{b c d^{2} +{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x}{b^{2} c^{2} x^{2} + b^{3} c x} + \frac{2 \,{\left (c d^{2} - b d e\right )} \log \left (c x + b\right )}{b^{3}} - \frac{2 \,{\left (c d^{2} - b d e\right )} \log \left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74305, size = 297, normalized size = 4.07 \begin{align*} -\frac{b^{2} c d^{2} +{\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x - 2 \,{\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} +{\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \left (c x + b\right ) + 2 \,{\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} +{\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \left (x\right )}{b^{3} c^{2} x^{2} + b^{4} c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.60136, size = 173, normalized size = 2.37 \begin{align*} - \frac{b c d^{2} + x \left (b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{b^{3} c x + b^{2} c^{2} x^{2}} + \frac{2 d \left (b e - c d\right ) \log{\left (x + \frac{2 b^{2} d e - 2 b c d^{2} - 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} - \frac{2 d \left (b e - c d\right ) \log{\left (x + \frac{2 b^{2} d e - 2 b c d^{2} + 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23792, size = 136, normalized size = 1.86 \begin{align*} -\frac{2 \,{\left (c d^{2} - b d e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \,{\left (c^{2} d^{2} - b c d e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac{2 \, c^{2} d^{2} x - 2 \, b c d x e + b c d^{2} + b^{2} x e^{2}}{{\left (c x^{2} + b x\right )} b^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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