Optimal. Leaf size=91 \[ \frac{e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac{2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3} \]
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Rubi [A] time = 0.0992741, antiderivative size = 91, 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} \[ \frac{e^2 (a e+c d x)^6}{6 c^3 d^3}+\frac{2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3} \]
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 )^3}{d+e x} \, dx &=\int (a e+c d x)^3 (d+e x)^2 \, dx\\ &=\int \left (\frac{\left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^2 d^2}+\frac{2 e \left (c d^2-a e^2\right ) (a e+c d x)^4}{c^2 d^2}+\frac{e^2 (a e+c d x)^5}{c^2 d^2}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right )^2 (a e+c d x)^4}{4 c^3 d^3}+\frac{2 e \left (c d^2-a e^2\right ) (a e+c d x)^5}{5 c^3 d^3}+\frac{e^2 (a e+c d x)^6}{6 c^3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0391323, size = 123, normalized size = 1.35 \[ \frac{1}{60} x \left (15 a^2 c d e^2 x \left (6 d^2+8 d e x+3 e^2 x^2\right )+20 a^3 e^3 \left (3 d^2+3 d e x+e^2 x^2\right )+6 a c^2 d^2 e x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+c^3 d^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 205, normalized size = 2.3 \begin{align*}{\frac{{c}^{3}{d}^{3}{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( a{e}^{3}{d}^{2}{c}^{2}+2\,{c}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) e \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,a{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) dc+cd \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( ae \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\,c{d}^{2}ae \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{e}^{2}d \left ( a{e}^{2}+c{d}^{2} \right ) +c{d}^{3}{a}^{2}{e}^{2} \right ){x}^{2}}{2}}+{a}^{3}{e}^{3}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0788, size = 203, normalized size = 2.23 \begin{align*} \frac{1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac{1}{5} \,{\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52703, size = 309, normalized size = 3.4 \begin{align*} \frac{1}{6} \, c^{3} d^{3} e^{2} x^{6} + a^{3} d^{2} e^{3} x + \frac{1}{5} \,{\left (2 \, c^{3} d^{4} e + 3 \, a c^{2} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{5} + 6 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a c^{2} d^{4} e + 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{3} e^{2} + 2 \, a^{3} d e^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.152798, size = 160, normalized size = 1.76 \begin{align*} a^{3} d^{2} e^{3} x + \frac{c^{3} d^{3} e^{2} x^{6}}{6} + x^{5} \left (\frac{3 a c^{2} d^{2} e^{3}}{5} + \frac{2 c^{3} d^{4} e}{5}\right ) + x^{4} \left (\frac{3 a^{2} c d e^{4}}{4} + \frac{3 a c^{2} d^{3} e^{2}}{2} + \frac{c^{3} d^{5}}{4}\right ) + x^{3} \left (\frac{a^{3} e^{5}}{3} + 2 a^{2} c d^{2} e^{3} + a c^{2} d^{4} e\right ) + x^{2} \left (a^{3} d e^{4} + \frac{3 a^{2} c d^{3} e^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24299, size = 213, normalized size = 2.34 \begin{align*} \frac{1}{60} \,{\left (10 \, c^{3} d^{3} x^{6} e^{8} + 24 \, c^{3} d^{4} x^{5} e^{7} + 15 \, c^{3} d^{5} x^{4} e^{6} + 36 \, a c^{2} d^{2} x^{5} e^{9} + 90 \, a c^{2} d^{3} x^{4} e^{8} + 60 \, a c^{2} d^{4} x^{3} e^{7} + 45 \, a^{2} c d x^{4} e^{10} + 120 \, a^{2} c d^{2} x^{3} e^{9} + 90 \, a^{2} c d^{3} x^{2} e^{8} + 20 \, a^{3} x^{3} e^{11} + 60 \, a^{3} d x^{2} e^{10} + 60 \, a^{3} d^{2} x e^{9}\right )} e^{\left (-6\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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