Optimal. Leaf size=63 \[ -\frac{\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac{2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac{c^2 d^2 x}{e^2} \]
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Rubi [A] time = 0.0520952, antiderivative size = 63, 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} \[ -\frac{\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac{2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac{c^2 d^2 x}{e^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^2} \, dx\\ &=\int \left (\frac{c^2 d^2}{e^2}+\frac{\left (-c d^2+a e^2\right )^2}{e^2 (d+e x)^2}-\frac{2 c d \left (c d^2-a e^2\right )}{e^2 (d+e x)}\right ) \, dx\\ &=\frac{c^2 d^2 x}{e^2}-\frac{\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac{2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0409001, size = 59, normalized size = 0.94 \[ \frac{-\frac{\left (c d^2-a e^2\right )^2}{d+e x}+2 c d \left (a e^2-c d^2\right ) \log (d+e x)+c^2 d^2 e x}{e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 92, normalized size = 1.5 \begin{align*}{\frac{{c}^{2}{d}^{2}x}{{e}^{2}}}+2\,{\frac{cd\ln \left ( ex+d \right ) a}{e}}-2\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{3}}}-{\frac{{a}^{2}e}{ex+d}}+2\,{\frac{ac{d}^{2}}{e \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01256, size = 107, normalized size = 1.7 \begin{align*} \frac{c^{2} d^{2} x}{e^{2}} - \frac{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{4} x + d e^{3}} - \frac{2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50685, size = 208, normalized size = 3.3 \begin{align*} \frac{c^{2} d^{2} e^{2} x^{2} + c^{2} d^{3} e x - c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} - 2 \,{\left (c^{2} d^{4} - a c d^{2} e^{2} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.647083, size = 71, normalized size = 1.13 \begin{align*} \frac{c^{2} d^{2} x}{e^{2}} + \frac{2 c d \left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}}{d e^{3} + e^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2321, size = 181, normalized size = 2.87 \begin{align*} c^{2} d^{2} x e^{\left (-2\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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