Optimal. Leaf size=48 \[ \frac{e \log (a e+c d x)}{c^2 d^2}-\frac{c d^2-a e^2}{c^2 d^2 (a e+c d x)} \]
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Rubi [A] time = 0.0363662, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 43} \[ \frac{e \log (a e+c d x)}{c^2 d^2}-\frac{c d^2-a e^2}{c^2 d^2 (a e+c d x)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{d+e x}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac{c d^2-a e^2}{c d (a e+c d x)^2}+\frac{e}{c d (a e+c d x)}\right ) \, dx\\ &=-\frac{c d^2-a e^2}{c^2 d^2 (a e+c d x)}+\frac{e \log (a e+c d x)}{c^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.012727, size = 47, normalized size = 0.98 \[ \frac{a e^2-c d^2}{c^2 d^2 (a e+c d x)}+\frac{e \log (a e+c d x)}{c^2 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 55, normalized size = 1.2 \begin{align*}{\frac{a{e}^{2}}{{c}^{2}{d}^{2} \left ( cdx+ae \right ) }}-{\frac{1}{c \left ( cdx+ae \right ) }}+{\frac{e\ln \left ( cdx+ae \right ) }{{c}^{2}{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02123, size = 70, normalized size = 1.46 \begin{align*} -\frac{c d^{2} - a e^{2}}{c^{3} d^{3} x + a c^{2} d^{2} e} + \frac{e \log \left (c d x + a e\right )}{c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88916, size = 109, normalized size = 2.27 \begin{align*} -\frac{c d^{2} - a e^{2} -{\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3} x + a c^{2} d^{2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.444943, size = 46, normalized size = 0.96 \begin{align*} \frac{a e^{2} - c d^{2}}{a c^{2} d^{2} e + c^{3} d^{3} x} + \frac{e \log{\left (a e + c d x \right )}}{c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20888, size = 435, normalized size = 9.06 \begin{align*} \frac{{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac{e \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{2} d^{2}} - \frac{c^{3} d^{7} - 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} - a^{3} d e^{6} +{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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