Optimal. Leaf size=111 \[ -\frac{c^2 d^2 (d+e x)^6 \left (c d^2-a e^2\right )}{2 e^4}+\frac{3 c d (d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{(d+e x)^4 \left (c d^2-a e^2\right )^3}{4 e^4}+\frac{c^3 d^3 (d+e x)^7}{7 e^4} \]
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Rubi [A] time = 0.148613, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {610, 43} \[ -\frac{c^2 d^2 (d+e x)^6 \left (c d^2-a e^2\right )}{2 e^4}+\frac{3 c d (d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{(d+e x)^4 \left (c d^2-a e^2\right )^3}{4 e^4}+\frac{c^3 d^3 (d+e x)^7}{7 e^4} \]
Antiderivative was successfully verified.
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Rule 610
Rule 43
Rubi steps
\begin{align*} \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx &=\frac{\int \left (c d^2+c d e x\right )^3 \left (a e^2+c d e x\right )^3 \, dx}{c^3 d^3 e^3}\\ &=\frac{\int \left (-\left (c d^2-a e^2\right )^3 \left (c d^2+c d e x\right )^3+3 \left (c d^2-a e^2\right )^2 \left (c d^2+c d e x\right )^4-3 \left (c d^2-a e^2\right ) \left (c d^2+c d e x\right )^5+\left (c d^2+c d e x\right )^6\right ) \, dx}{c^3 d^3 e^3}\\ &=-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^4}{4 e^4}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^5}{5 e^4}-\frac{c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^6}{2 e^4}+\frac{c^3 d^3 (d+e x)^7}{7 e^4}\\ \end{align*}
Mathematica [A] time = 0.0485335, size = 167, normalized size = 1.5 \[ \frac{1}{140} x \left (21 a^2 c d e^2 x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+35 a^3 e^3 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+7 a c^2 d^2 e x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+c^3 d^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 266, normalized size = 2.4 \begin{align*}{\frac{{d}^{3}{e}^{3}{c}^{3}{x}^{7}}{7}}+{\frac{ \left ( a{e}^{2}+c{d}^{2} \right ){d}^{2}{e}^{2}{c}^{2}{x}^{6}}{2}}+{\frac{ \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 ){x}^{5}}{5}}+{\frac{ \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 ){x}^{4}}{4}}+{\frac{ \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 ){x}^{3}}{3}}+{\frac{3\,{a}^{2}{d}^{2}{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{3}{e}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10195, size = 247, normalized size = 2.23 \begin{align*} \frac{1}{7} \, c^{3} d^{3} e^{3} x^{7} + \frac{1}{2} \,{\left (c d^{2} + a e^{2}\right )} c^{2} d^{2} e^{2} x^{6} + a^{3} d^{3} e^{3} x + \frac{3}{5} \,{\left (c d^{2} + a e^{2}\right )}^{2} c d e x^{5} + \frac{1}{2} \,{\left (2 \, c d e x^{3} + 3 \,{\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a^{2} d^{2} e^{2} + \frac{1}{4} \,{\left (c d^{2} + a e^{2}\right )}^{3} x^{4} + \frac{1}{10} \,{\left (6 \, c^{2} d^{2} e^{2} x^{5} + 15 \,{\left (c d^{2} + a e^{2}\right )} c d e x^{4} + 10 \,{\left (c d^{2} + a e^{2}\right )}^{2} x^{3}\right )} a d e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.37993, size = 452, normalized size = 4.07 \begin{align*} \frac{1}{7} x^{7} e^{3} d^{3} c^{3} + \frac{1}{2} x^{6} e^{2} d^{4} c^{3} + \frac{1}{2} x^{6} e^{4} d^{2} c^{2} a + \frac{3}{5} x^{5} e d^{5} c^{3} + \frac{9}{5} x^{5} e^{3} d^{3} c^{2} a + \frac{3}{5} x^{5} e^{5} d c a^{2} + \frac{1}{4} x^{4} d^{6} c^{3} + \frac{9}{4} x^{4} e^{2} d^{4} c^{2} a + \frac{9}{4} x^{4} e^{4} d^{2} c a^{2} + \frac{1}{4} x^{4} e^{6} a^{3} + x^{3} e d^{5} c^{2} a + 3 x^{3} e^{3} d^{3} c a^{2} + x^{3} e^{5} d a^{3} + \frac{3}{2} x^{2} e^{2} d^{4} c a^{2} + \frac{3}{2} x^{2} e^{4} d^{2} a^{3} + x e^{3} d^{3} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.145143, size = 218, normalized size = 1.96 \begin{align*} a^{3} d^{3} e^{3} x + \frac{c^{3} d^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac{a c^{2} d^{2} e^{4}}{2} + \frac{c^{3} d^{4} e^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} c d e^{5}}{5} + \frac{9 a c^{2} d^{3} e^{3}}{5} + \frac{3 c^{3} d^{5} e}{5}\right ) + x^{4} \left (\frac{a^{3} e^{6}}{4} + \frac{9 a^{2} c d^{2} e^{4}}{4} + \frac{9 a c^{2} d^{4} e^{2}}{4} + \frac{c^{3} d^{6}}{4}\right ) + x^{3} \left (a^{3} d e^{5} + 3 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e\right ) + x^{2} \left (\frac{3 a^{3} d^{2} e^{4}}{2} + \frac{3 a^{2} c d^{4} e^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16336, size = 274, normalized size = 2.47 \begin{align*} \frac{1}{7} \, c^{3} d^{3} x^{7} e^{3} + \frac{1}{2} \, c^{3} d^{4} x^{6} e^{2} + \frac{3}{5} \, c^{3} d^{5} x^{5} e + \frac{1}{4} \, c^{3} d^{6} x^{4} + \frac{1}{2} \, a c^{2} d^{2} x^{6} e^{4} + \frac{9}{5} \, a c^{2} d^{3} x^{5} e^{3} + \frac{9}{4} \, a c^{2} d^{4} x^{4} e^{2} + a c^{2} d^{5} x^{3} e + \frac{3}{5} \, a^{2} c d x^{5} e^{5} + \frac{9}{4} \, a^{2} c d^{2} x^{4} e^{4} + 3 \, a^{2} c d^{3} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{4} x^{2} e^{2} + \frac{1}{4} \, a^{3} x^{4} e^{6} + a^{3} d x^{3} e^{5} + \frac{3}{2} \, a^{3} d^{2} x^{2} e^{4} + a^{3} d^{3} x e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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