Optimal. Leaf size=77 \[ -\frac{2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac{(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3} \]
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Rubi [A] time = 0.0792925, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {626, 43} \[ -\frac{2 c d (d+e x)^5 \left (c d^2-a e^2\right )}{5 e^3}+\frac{(d+e x)^4 \left (c d^2-a e^2\right )^2}{4 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx &=\int (a e+c d x)^2 (d+e x)^3 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2 (d+e x)^3}{e^2}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^4}{e^2}+\frac{c^2 d^2 (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right )^2 (d+e x)^4}{4 e^3}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^5}{5 e^3}+\frac{c^2 d^2 (d+e x)^6}{6 e^3}\\ \end{align*}
Mathematica [A] time = 0.0323462, size = 120, normalized size = 1.56 \[ \frac{1}{60} x \left (15 a^2 e^2 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+6 a c d e x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+c^2 d^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 195, normalized size = 2.5 \begin{align*}{\frac{{d}^{2}{e}^{3}{c}^{2}{x}^{6}}{6}}+{\frac{ \left ({d}^{3}{e}^{2}{c}^{2}+2\,{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) dc \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) ec+e \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\,{e}^{2}ad \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{d}^{2}ae \left ( a{e}^{2}+c{d}^{2} \right ) +{a}^{2}{d}^{2}{e}^{3} \right ){x}^{2}}{2}}+{a}^{2}{d}^{3}{e}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63677, size = 189, normalized size = 2.45 \begin{align*} \frac{1}{6} \, c^{2} d^{2} e^{3} x^{6} + a^{2} d^{3} e^{2} x + \frac{1}{5} \,{\left (3 \, c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{4} + \frac{1}{3} \,{\left (c^{2} d^{5} + 6 \, a c d^{3} e^{2} + 3 \, a^{2} d e^{4}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a c d^{4} e + 3 \, a^{2} d^{2} e^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.42418, size = 312, normalized size = 4.05 \begin{align*} \frac{1}{6} x^{6} e^{3} d^{2} c^{2} + \frac{3}{5} x^{5} e^{2} d^{3} c^{2} + \frac{2}{5} x^{5} e^{4} d c a + \frac{3}{4} x^{4} e d^{4} c^{2} + \frac{3}{2} x^{4} e^{3} d^{2} c a + \frac{1}{4} x^{4} e^{5} a^{2} + \frac{1}{3} x^{3} d^{5} c^{2} + 2 x^{3} e^{2} d^{3} c a + x^{3} e^{4} d a^{2} + x^{2} e d^{4} c a + \frac{3}{2} x^{2} e^{3} d^{2} a^{2} + x e^{2} d^{3} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.107471, size = 150, normalized size = 1.95 \begin{align*} a^{2} d^{3} e^{2} x + \frac{c^{2} d^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac{2 a c d e^{4}}{5} + \frac{3 c^{2} d^{3} e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{5}}{4} + \frac{3 a c d^{2} e^{3}}{2} + \frac{3 c^{2} d^{4} e}{4}\right ) + x^{3} \left (a^{2} d e^{4} + 2 a c d^{3} e^{2} + \frac{c^{2} d^{5}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e^{3}}{2} + a c d^{4} e\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15863, size = 188, normalized size = 2.44 \begin{align*} \frac{1}{6} \, c^{2} d^{2} x^{6} e^{3} + \frac{3}{5} \, c^{2} d^{3} x^{5} e^{2} + \frac{3}{4} \, c^{2} d^{4} x^{4} e + \frac{1}{3} \, c^{2} d^{5} x^{3} + \frac{2}{5} \, a c d x^{5} e^{4} + \frac{3}{2} \, a c d^{2} x^{4} e^{3} + 2 \, a c d^{3} x^{3} e^{2} + a c d^{4} x^{2} e + \frac{1}{4} \, a^{2} x^{4} e^{5} + a^{2} d x^{3} e^{4} + \frac{3}{2} \, a^{2} d^{2} x^{2} e^{3} + a^{2} d^{3} x e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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