Optimal. Leaf size=111 \[ -\frac{3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac{(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4} \]
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Rubi [A] time = 0.160326, antiderivative size = 111, 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} \[ -\frac{3 c^2 d^2 (d+e x)^7 \left (c d^2-a e^2\right )}{7 e^4}+\frac{c d (d+e x)^6 \left (c d^2-a e^2\right )^2}{2 e^4}-\frac{(d+e x)^5 \left (c d^2-a e^2\right )^3}{5 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4} \]
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 )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^4 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 (d+e x)^4}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^5}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^6}{e^3}+\frac{c^3 d^3 (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^5}{5 e^4}+\frac{c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{2 e^4}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{7 e^4}+\frac{c^3 d^3 (d+e x)^8}{8 e^4}\\ \end{align*}
Mathematica [A] time = 0.0633384, size = 211, normalized size = 1.9 \[ \frac{1}{280} x \left (28 a^2 c d e^2 x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+56 a^3 e^3 \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+8 a c^2 d^2 e x^2 \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+c^3 d^3 x^3 \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 531, normalized size = 4.8 \begin{align*}{\frac{{d}^{3}{e}^{4}{c}^{3}{x}^{8}}{8}}+{\frac{ \left ({d}^{4}{e}^{3}{c}^{3}+3\,{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ){d}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{d}^{3} \left ( a{e}^{2}+c{d}^{2} \right ){e}^{2}{c}^{2}+e \left ( a{d}^{3}{e}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( a{d}^{3}{e}^{3}{c}^{2}+2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}dec+dec \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) +e \left ( 4\,a{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,a{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) c+ \left ( a{e}^{2}+c{d}^{2} \right ) \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) \right ) +e \left ( ade \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}ade+{d}^{3}{e}^{3}c{a}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( ade \left ( 2\,ac{d}^{2}{e}^{2}+ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \right ) +2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}ade+{d}^{3}{e}^{3}c{a}^{2} \right ) +3\,{e}^{3}{a}^{2}{d}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{3}{a}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) +{a}^{3}{d}^{3}{e}^{4} \right ){x}^{2}}{2}}+{a}^{3}{d}^{4}{e}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01722, size = 339, normalized size = 3.05 \begin{align*} \frac{1}{8} \, c^{3} d^{3} e^{4} x^{8} + a^{3} d^{4} e^{3} x + \frac{1}{7} \,{\left (4 \, c^{3} d^{4} e^{3} + 3 \, a c^{2} d^{2} e^{5}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, c^{3} d^{5} e^{2} + 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, c^{3} d^{6} e + 18 \, a c^{2} d^{4} e^{3} + 12 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{7} + 12 \, a c^{2} d^{5} e^{2} + 18 \, a^{2} c d^{3} e^{4} + 4 \, a^{3} d e^{6}\right )} x^{4} +{\left (a c^{2} d^{6} e + 4 \, a^{2} c d^{4} e^{3} + 2 \, a^{3} d^{2} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{5} e^{2} + 4 \, a^{3} d^{3} e^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.30077, size = 563, normalized size = 5.07 \begin{align*} \frac{1}{8} x^{8} e^{4} d^{3} c^{3} + \frac{4}{7} x^{7} e^{3} d^{4} c^{3} + \frac{3}{7} x^{7} e^{5} d^{2} c^{2} a + x^{6} e^{2} d^{5} c^{3} + 2 x^{6} e^{4} d^{3} c^{2} a + \frac{1}{2} x^{6} e^{6} d c a^{2} + \frac{4}{5} x^{5} e d^{6} c^{3} + \frac{18}{5} x^{5} e^{3} d^{4} c^{2} a + \frac{12}{5} x^{5} e^{5} d^{2} c a^{2} + \frac{1}{5} x^{5} e^{7} a^{3} + \frac{1}{4} x^{4} d^{7} c^{3} + 3 x^{4} e^{2} d^{5} c^{2} a + \frac{9}{2} x^{4} e^{4} d^{3} c a^{2} + x^{4} e^{6} d a^{3} + x^{3} e d^{6} c^{2} a + 4 x^{3} e^{3} d^{4} c a^{2} + 2 x^{3} e^{5} d^{2} a^{3} + \frac{3}{2} x^{2} e^{2} d^{5} c a^{2} + 2 x^{2} e^{4} d^{3} a^{3} + x e^{3} d^{4} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.179508, size = 270, normalized size = 2.43 \begin{align*} a^{3} d^{4} e^{3} x + \frac{c^{3} d^{3} e^{4} x^{8}}{8} + x^{7} \left (\frac{3 a c^{2} d^{2} e^{5}}{7} + \frac{4 c^{3} d^{4} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} c d e^{6}}{2} + 2 a c^{2} d^{3} e^{4} + c^{3} d^{5} e^{2}\right ) + x^{5} \left (\frac{a^{3} e^{7}}{5} + \frac{12 a^{2} c d^{2} e^{5}}{5} + \frac{18 a c^{2} d^{4} e^{3}}{5} + \frac{4 c^{3} d^{6} e}{5}\right ) + x^{4} \left (a^{3} d e^{6} + \frac{9 a^{2} c d^{3} e^{4}}{2} + 3 a c^{2} d^{5} e^{2} + \frac{c^{3} d^{7}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{5} + 4 a^{2} c d^{4} e^{3} + a c^{2} d^{6} e\right ) + x^{2} \left (2 a^{3} d^{3} e^{4} + \frac{3 a^{2} c d^{5} e^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21402, size = 346, normalized size = 3.12 \begin{align*} \frac{1}{8} \, c^{3} d^{3} x^{8} e^{4} + \frac{4}{7} \, c^{3} d^{4} x^{7} e^{3} + c^{3} d^{5} x^{6} e^{2} + \frac{4}{5} \, c^{3} d^{6} x^{5} e + \frac{1}{4} \, c^{3} d^{7} x^{4} + \frac{3}{7} \, a c^{2} d^{2} x^{7} e^{5} + 2 \, a c^{2} d^{3} x^{6} e^{4} + \frac{18}{5} \, a c^{2} d^{4} x^{5} e^{3} + 3 \, a c^{2} d^{5} x^{4} e^{2} + a c^{2} d^{6} x^{3} e + \frac{1}{2} \, a^{2} c d x^{6} e^{6} + \frac{12}{5} \, a^{2} c d^{2} x^{5} e^{5} + \frac{9}{2} \, a^{2} c d^{3} x^{4} e^{4} + 4 \, a^{2} c d^{4} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{5} x^{2} e^{2} + \frac{1}{5} \, a^{3} x^{5} e^{7} + a^{3} d x^{4} e^{6} + 2 \, a^{3} d^{2} x^{3} e^{5} + 2 \, a^{3} d^{3} x^{2} e^{4} + a^{3} d^{4} x e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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