Optimal. Leaf size=35 \[ \frac{(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0141029, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 37} \[ \frac{(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 626
Rule 37
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)^6} \, dx &=\int \frac{(a e+c d x)^2}{(d+e x)^4} \, dx\\ &=\frac{(a e+c d x)^3}{3 \left (c d^2-a e^2\right ) (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.0277713, size = 59, normalized size = 1.69 \[ -\frac{a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 83, normalized size = 2.4 \begin{align*} -{\frac{cd \left ( a{e}^{2}-c{d}^{2} \right ) }{{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0719, size = 127, normalized size = 3.63 \begin{align*} -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66687, size = 186, normalized size = 5.31 \begin{align*} -\frac{3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.54047, size = 99, normalized size = 2.83 \begin{align*} - \frac{a^{2} e^{4} + a c d^{2} e^{2} + c^{2} d^{4} + 3 c^{2} d^{2} e^{2} x^{2} + x \left (3 a c d e^{3} + 3 c^{2} d^{3} e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22456, size = 185, normalized size = 5.29 \begin{align*} -\frac{{\left (3 \, c^{2} d^{2} x^{4} e^{4} + 9 \, c^{2} d^{3} x^{3} e^{3} + 10 \, c^{2} d^{4} x^{2} e^{2} + 5 \, c^{2} d^{5} x e + c^{2} d^{6} + 3 \, a c d x^{3} e^{5} + 7 \, a c d^{2} x^{2} e^{4} + 5 \, a c d^{3} x e^{3} + a c d^{4} e^{2} + a^{2} x^{2} e^{6} + 2 \, a^{2} d x e^{5} + a^{2} d^{2} e^{4}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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