Optimal. Leaf size=77 \[ -\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )}{3 e^3}+\frac{(d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{c^2 d^2 (d+e x)^7}{7 e^3} \]
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Rubi [A] time = 0.131867, antiderivative size = 77, 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} \[ -\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )}{3 e^3}+\frac{(d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{c^2 d^2 (d+e x)^7}{7 e^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^2 \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)^4 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2 (d+e x)^4}{e^2}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^5}{e^2}+\frac{c^2 d^2 (d+e x)^6}{e^2}\right ) \, dx\\ &=\frac{\left (c d^2-a e^2\right )^2 (d+e x)^5}{5 e^3}-\frac{c d \left (c d^2-a e^2\right ) (d+e x)^6}{3 e^3}+\frac{c^2 d^2 (d+e x)^7}{7 e^3}\\ \end{align*}
Mathematica [B] time = 0.0315438, size = 160, normalized size = 2.08 \[ a^2 \left (2 d^2 e^4 x^3+2 d^3 e^3 x^2+d^4 e^2 x+d e^5 x^4+\frac{e^6 x^5}{5}\right )+\frac{1}{15} a c d e x^2 \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+\frac{1}{105} c^2 d^2 x^3 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 295, normalized size = 3.8 \begin{align*}{\frac{{e}^{4}{d}^{2}{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{d}^{3}{e}^{3}{c}^{2}+2\,{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) dc \right ){x}^{6}}{6}}+{\frac{ \left ({d}^{4}{e}^{2}{c}^{2}+4\,{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+{e}^{2} \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) ec+2\,de \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\,{e}^{3}ad \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{2} \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +4\,{d}^{2}{e}^{2}a \left ( a{e}^{2}+c{d}^{2} \right ) +{e}^{4}{a}^{2}{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{d}^{3}ae \left ( a{e}^{2}+c{d}^{2} \right ) +2\,{d}^{3}{e}^{3}{a}^{2} \right ){x}^{2}}{2}}+{d}^{4}{a}^{2}{e}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13418, size = 232, normalized size = 3.01 \begin{align*} \frac{1}{7} \, c^{2} d^{2} e^{4} x^{7} + a^{2} d^{4} e^{2} x + \frac{1}{3} \,{\left (2 \, c^{2} d^{3} e^{3} + a c d e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, c^{2} d^{4} e^{2} + 8 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{5} +{\left (c^{2} d^{5} e + 3 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x^{4} + \frac{1}{3} \,{\left (c^{2} d^{6} + 8 \, a c d^{4} e^{2} + 6 \, a^{2} d^{2} e^{4}\right )} x^{3} +{\left (a c d^{5} e + 2 \, a^{2} d^{3} e^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.38587, size = 390, normalized size = 5.06 \begin{align*} \frac{1}{7} x^{7} e^{4} d^{2} c^{2} + \frac{2}{3} x^{6} e^{3} d^{3} c^{2} + \frac{1}{3} x^{6} e^{5} d c a + \frac{6}{5} x^{5} e^{2} d^{4} c^{2} + \frac{8}{5} x^{5} e^{4} d^{2} c a + \frac{1}{5} x^{5} e^{6} a^{2} + x^{4} e d^{5} c^{2} + 3 x^{4} e^{3} d^{3} c a + x^{4} e^{5} d a^{2} + \frac{1}{3} x^{3} d^{6} c^{2} + \frac{8}{3} x^{3} e^{2} d^{4} c a + 2 x^{3} e^{4} d^{2} a^{2} + x^{2} e d^{5} c a + 2 x^{2} e^{3} d^{3} a^{2} + x e^{2} d^{4} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.135621, size = 185, normalized size = 2.4 \begin{align*} a^{2} d^{4} e^{2} x + \frac{c^{2} d^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac{a c d e^{5}}{3} + \frac{2 c^{2} d^{3} e^{3}}{3}\right ) + x^{5} \left (\frac{a^{2} e^{6}}{5} + \frac{8 a c d^{2} e^{4}}{5} + \frac{6 c^{2} d^{4} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{5} + 3 a c d^{3} e^{3} + c^{2} d^{5} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{4} + \frac{8 a c d^{4} e^{2}}{3} + \frac{c^{2} d^{6}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e^{3} + a c d^{5} e\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20042, size = 238, normalized size = 3.09 \begin{align*} \frac{1}{7} \, c^{2} d^{2} x^{7} e^{4} + \frac{2}{3} \, c^{2} d^{3} x^{6} e^{3} + \frac{6}{5} \, c^{2} d^{4} x^{5} e^{2} + c^{2} d^{5} x^{4} e + \frac{1}{3} \, c^{2} d^{6} x^{3} + \frac{1}{3} \, a c d x^{6} e^{5} + \frac{8}{5} \, a c d^{2} x^{5} e^{4} + 3 \, a c d^{3} x^{4} e^{3} + \frac{8}{3} \, a c d^{4} x^{3} e^{2} + a c d^{5} x^{2} e + \frac{1}{5} \, a^{2} x^{5} e^{6} + a^{2} d x^{4} e^{5} + 2 \, a^{2} d^{2} x^{3} e^{4} + 2 \, a^{2} d^{3} x^{2} e^{3} + a^{2} d^{4} x e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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