Optimal. Leaf size=114 \[ -\frac{a e^2+c d^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac{2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac{2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
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Rubi [A] time = 0.0331131, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {614, 616, 31} \[ -\frac{a e^2+c d^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )}-\frac{2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac{2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 614
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=-\frac{c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac{(2 c d e) \int \frac{1}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=-\frac{c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}+\frac{\left (2 c^2 d^2 e^2\right ) \int \frac{1}{c d^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^3}-\frac{\left (2 c^2 d^2 e^2\right ) \int \frac{1}{a e^2+c d e x} \, dx}{\left (c d^2-a e^2\right )^3}\\ &=-\frac{c d^2+a e^2+2 c d e x}{\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )}-\frac{2 c d e \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}+\frac{2 c d e \log (d+e x)}{\left (c d^2-a e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.094808, size = 86, normalized size = 0.75 \[ \frac{\frac{\left (c d^2-a e^2\right ) \left (a e^2+c d (d+2 e x)\right )}{(d+e x) (a e+c d x)}+2 c d e \log (a e+c d x)-2 c d e \log (d+e x)}{\left (a e^2-c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 107, normalized size = 0.9 \begin{align*} -{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}-2\,{\frac{dec\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}}-{\frac{cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdx+ae \right ) }}+2\,{\frac{dec\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06506, size = 319, normalized size = 2.8 \begin{align*} -\frac{2 \, c d e \log \left (c d x + a e\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} + \frac{2 \, c d e \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} - \frac{2 \, c d e x + c d^{2} + a e^{2}}{a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5} +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{2} +{\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92847, size = 545, normalized size = 4.78 \begin{align*} -\frac{c^{2} d^{4} - a^{2} e^{4} + 2 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (c d x + a e\right ) - 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 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 )}{a c^{3} d^{7} e - 3 \, a^{2} c^{2} d^{5} e^{3} + 3 \, a^{3} c d^{3} e^{5} - a^{4} d e^{7} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x^{2} +{\left (c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} + 2 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.53199, size = 484, normalized size = 4.25 \begin{align*} - \frac{2 c d e \log{\left (x + \frac{- \frac{2 a^{4} c d e^{9}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{8 a^{3} c^{2} d^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{12 a^{2} c^{3} d^{5} e^{5}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{8 a c^{4} d^{7} e^{3}}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 a c d e^{3} - \frac{2 c^{5} d^{9} e}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 c^{2} d^{3} e}{4 c^{2} d^{2} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{2 c d e \log{\left (x + \frac{\frac{2 a^{4} c d e^{9}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{8 a^{3} c^{2} d^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{12 a^{2} c^{3} d^{5} e^{5}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{8 a c^{4} d^{7} e^{3}}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 a c d e^{3} + \frac{2 c^{5} d^{9} e}{\left (a e^{2} - c d^{2}\right )^{3}} + 2 c^{2} d^{3} e}{4 c^{2} d^{2} e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{a e^{2} + c d^{2} + 2 c d e x}{a^{3} d e^{5} - 2 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e + x^{2} \left (a^{2} c d e^{5} - 2 a c^{2} d^{3} e^{3} + c^{3} d^{5} e\right ) + x \left (a^{3} e^{6} - a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} + c^{3} d^{6}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31586, size = 238, normalized size = 2.09 \begin{align*} -\frac{4 \, c d \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 ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac{2 \, c d x e + c d^{2} + a e^{2}}{{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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