Optimal. Leaf size=111 \[ -\frac{3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac{3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac{(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac{c^3 d^3 (d+e x)^9}{9 e^4} \]
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Rubi [A] time = 0.233257, antiderivative size = 111, 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} \[ -\frac{3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac{3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac{(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac{c^3 d^3 (d+e x)^9}{9 e^4} \]
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 )^3 \, dx &=\int (a e+c d x)^3 (d+e x)^5 \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 (d+e x)^5}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{e^3}+\frac{c^3 d^3 (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac{\left (c d^2-a e^2\right )^3 (d+e x)^6}{6 e^4}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^7}{7 e^4}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^8}{8 e^4}+\frac{c^3 d^3 (d+e x)^9}{9 e^4}\\ \end{align*}
Mathematica [B] time = 0.0729191, size = 255, normalized size = 2.3 \[ \frac{1}{504} x \left (36 a^2 c d e^2 x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+84 a^3 e^3 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+9 a c^2 d^2 e x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+c^3 d^3 x^3 \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 801, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.999495, size = 409, normalized size = 3.68 \begin{align*} \frac{1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac{1}{8} \,{\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} +{\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac{1}{4} \,{\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.37211, size = 694, normalized size = 6.25 \begin{align*} \frac{1}{9} x^{9} e^{5} d^{3} c^{3} + \frac{5}{8} x^{8} e^{4} d^{4} c^{3} + \frac{3}{8} x^{8} e^{6} d^{2} c^{2} a + \frac{10}{7} x^{7} e^{3} d^{5} c^{3} + \frac{15}{7} x^{7} e^{5} d^{3} c^{2} a + \frac{3}{7} x^{7} e^{7} d c a^{2} + \frac{5}{3} x^{6} e^{2} d^{6} c^{3} + 5 x^{6} e^{4} d^{4} c^{2} a + \frac{5}{2} x^{6} e^{6} d^{2} c a^{2} + \frac{1}{6} x^{6} e^{8} a^{3} + x^{5} e d^{7} c^{3} + 6 x^{5} e^{3} d^{5} c^{2} a + 6 x^{5} e^{5} d^{3} c a^{2} + x^{5} e^{7} d a^{3} + \frac{1}{4} x^{4} d^{8} c^{3} + \frac{15}{4} x^{4} e^{2} d^{6} c^{2} a + \frac{15}{2} x^{4} e^{4} d^{4} c a^{2} + \frac{5}{2} x^{4} e^{6} d^{2} a^{3} + x^{3} e d^{7} c^{2} a + 5 x^{3} e^{3} d^{5} c a^{2} + \frac{10}{3} x^{3} e^{5} d^{3} a^{3} + \frac{3}{2} x^{2} e^{2} d^{6} c a^{2} + \frac{5}{2} x^{2} e^{4} d^{4} a^{3} + x e^{3} d^{5} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.197032, size = 335, normalized size = 3.02 \begin{align*} a^{3} d^{5} e^{3} x + \frac{c^{3} d^{3} e^{5} x^{9}}{9} + x^{8} \left (\frac{3 a c^{2} d^{2} e^{6}}{8} + \frac{5 c^{3} d^{4} e^{4}}{8}\right ) + x^{7} \left (\frac{3 a^{2} c d e^{7}}{7} + \frac{15 a c^{2} d^{3} e^{5}}{7} + \frac{10 c^{3} d^{5} e^{3}}{7}\right ) + x^{6} \left (\frac{a^{3} e^{8}}{6} + \frac{5 a^{2} c d^{2} e^{6}}{2} + 5 a c^{2} d^{4} e^{4} + \frac{5 c^{3} d^{6} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{7} + 6 a^{2} c d^{3} e^{5} + 6 a c^{2} d^{5} e^{3} + c^{3} d^{7} e\right ) + x^{4} \left (\frac{5 a^{3} d^{2} e^{6}}{2} + \frac{15 a^{2} c d^{4} e^{4}}{2} + \frac{15 a c^{2} d^{6} e^{2}}{4} + \frac{c^{3} d^{8}}{4}\right ) + x^{3} \left (\frac{10 a^{3} d^{3} e^{5}}{3} + 5 a^{2} c d^{5} e^{3} + a c^{2} d^{7} e\right ) + x^{2} \left (\frac{5 a^{3} d^{4} e^{4}}{2} + \frac{3 a^{2} c d^{6} e^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14721, size = 419, normalized size = 3.77 \begin{align*} \frac{1}{9} \, c^{3} d^{3} x^{9} e^{5} + \frac{5}{8} \, c^{3} d^{4} x^{8} e^{4} + \frac{10}{7} \, c^{3} d^{5} x^{7} e^{3} + \frac{5}{3} \, c^{3} d^{6} x^{6} e^{2} + c^{3} d^{7} x^{5} e + \frac{1}{4} \, c^{3} d^{8} x^{4} + \frac{3}{8} \, a c^{2} d^{2} x^{8} e^{6} + \frac{15}{7} \, a c^{2} d^{3} x^{7} e^{5} + 5 \, a c^{2} d^{4} x^{6} e^{4} + 6 \, a c^{2} d^{5} x^{5} e^{3} + \frac{15}{4} \, a c^{2} d^{6} x^{4} e^{2} + a c^{2} d^{7} x^{3} e + \frac{3}{7} \, a^{2} c d x^{7} e^{7} + \frac{5}{2} \, a^{2} c d^{2} x^{6} e^{6} + 6 \, a^{2} c d^{3} x^{5} e^{5} + \frac{15}{2} \, a^{2} c d^{4} x^{4} e^{4} + 5 \, a^{2} c d^{5} x^{3} e^{3} + \frac{3}{2} \, a^{2} c d^{6} x^{2} e^{2} + \frac{1}{6} \, a^{3} x^{6} e^{8} + a^{3} d x^{5} e^{7} + \frac{5}{2} \, a^{3} d^{2} x^{4} e^{6} + \frac{10}{3} \, a^{3} d^{3} x^{3} e^{5} + \frac{5}{2} \, a^{3} d^{4} x^{2} e^{4} + a^{3} d^{5} x e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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