Optimal. Leaf size=83 \[ \frac{3 a^2 b e^3 (c+d x)^7}{7 d}+\frac{a^3 e^3 (c+d x)^4}{4 d}+\frac{3 a b^2 e^3 (c+d x)^{10}}{10 d}+\frac{b^3 e^3 (c+d x)^{13}}{13 d} \]
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Rubi [A] time = 0.0987832, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {372, 270} \[ \frac{3 a^2 b e^3 (c+d x)^7}{7 d}+\frac{a^3 e^3 (c+d x)^4}{4 d}+\frac{3 a b^2 e^3 (c+d x)^{10}}{10 d}+\frac{b^3 e^3 (c+d x)^{13}}{13 d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 270
Rubi steps
\begin{align*} \int (c e+d e x)^3 \left (a+b (c+d x)^3\right )^3 \, dx &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b x^3\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \left (a^3 x^3+3 a^2 b x^6+3 a b^2 x^9+b^3 x^{12}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{a^3 e^3 (c+d x)^4}{4 d}+\frac{3 a^2 b e^3 (c+d x)^7}{7 d}+\frac{3 a b^2 e^3 (c+d x)^{10}}{10 d}+\frac{b^3 e^3 (c+d x)^{13}}{13 d}\\ \end{align*}
Mathematica [B] time = 0.0158128, size = 327, normalized size = 3.94 \[ e^3 \left (\frac{3}{7} b d^6 x^7 \left (a^2+84 a b c^3+308 b^2 c^6\right )+3 b c d^5 x^6 \left (a^2+21 a b c^3+44 b^2 c^6\right )+\frac{9}{5} b c^2 d^4 x^5 \left (5 a^2+42 a b c^3+55 b^2 c^6\right )+\frac{1}{4} d^3 x^4 \left (60 a^2 b c^3+a^3+252 a b^2 c^6+220 b^3 c^9\right )+c d^2 x^3 \left (15 a^2 b c^3+a^3+36 a b^2 c^6+22 b^3 c^9\right )+\frac{1}{10} b^2 d^9 x^{10} \left (3 a+220 b c^3\right )+b^2 c d^8 x^9 \left (3 a+55 b c^3\right )+\frac{9}{2} b^2 c^2 d^7 x^8 \left (3 a+22 b c^3\right )+\frac{3}{2} c^2 d x^2 \left (a+b c^3\right )^2 \left (a+4 b c^3\right )+c^3 x \left (a+b c^3\right )^3+6 b^3 c^2 d^{10} x^{11}+b^3 c d^{11} x^{12}+\frac{1}{13} b^3 d^{12} x^{13}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 2050, normalized size = 24.7 \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 = 1.02245, size = 537, normalized size = 6.47 \begin{align*} \frac{1}{13} \, b^{3} d^{12} e^{3} x^{13} + b^{3} c d^{11} e^{3} x^{12} + 6 \, b^{3} c^{2} d^{10} e^{3} x^{11} + \frac{1}{10} \,{\left (220 \, b^{3} c^{3} + 3 \, a b^{2}\right )} d^{9} e^{3} x^{10} +{\left (55 \, b^{3} c^{4} + 3 \, a b^{2} c\right )} d^{8} e^{3} x^{9} + \frac{9}{2} \,{\left (22 \, b^{3} c^{5} + 3 \, a b^{2} c^{2}\right )} d^{7} e^{3} x^{8} + \frac{3}{7} \,{\left (308 \, b^{3} c^{6} + 84 \, a b^{2} c^{3} + a^{2} b\right )} d^{6} e^{3} x^{7} + 3 \,{\left (44 \, b^{3} c^{7} + 21 \, a b^{2} c^{4} + a^{2} b c\right )} d^{5} e^{3} x^{6} + \frac{9}{5} \,{\left (55 \, b^{3} c^{8} + 42 \, a b^{2} c^{5} + 5 \, a^{2} b c^{2}\right )} d^{4} e^{3} x^{5} + \frac{1}{4} \,{\left (220 \, b^{3} c^{9} + 252 \, a b^{2} c^{6} + 60 \, a^{2} b c^{3} + a^{3}\right )} d^{3} e^{3} x^{4} +{\left (22 \, b^{3} c^{10} + 36 \, a b^{2} c^{7} + 15 \, a^{2} b c^{4} + a^{3} c\right )} d^{2} e^{3} x^{3} + \frac{3}{2} \,{\left (4 \, b^{3} c^{11} + 9 \, a b^{2} c^{8} + 6 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d e^{3} x^{2} +{\left (b^{3} c^{12} + 3 \, a b^{2} c^{9} + 3 \, a^{2} b c^{6} + a^{3} c^{3}\right )} e^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34212, size = 1126, normalized size = 13.57 \begin{align*} \frac{1}{13} x^{13} e^{3} d^{12} b^{3} + x^{12} e^{3} d^{11} c b^{3} + 6 x^{11} e^{3} d^{10} c^{2} b^{3} + 22 x^{10} e^{3} d^{9} c^{3} b^{3} + 55 x^{9} e^{3} d^{8} c^{4} b^{3} + 99 x^{8} e^{3} d^{7} c^{5} b^{3} + 132 x^{7} e^{3} d^{6} c^{6} b^{3} + \frac{3}{10} x^{10} e^{3} d^{9} b^{2} a + 132 x^{6} e^{3} d^{5} c^{7} b^{3} + 3 x^{9} e^{3} d^{8} c b^{2} a + 99 x^{5} e^{3} d^{4} c^{8} b^{3} + \frac{27}{2} x^{8} e^{3} d^{7} c^{2} b^{2} a + 55 x^{4} e^{3} d^{3} c^{9} b^{3} + 36 x^{7} e^{3} d^{6} c^{3} b^{2} a + 22 x^{3} e^{3} d^{2} c^{10} b^{3} + 63 x^{6} e^{3} d^{5} c^{4} b^{2} a + 6 x^{2} e^{3} d c^{11} b^{3} + \frac{378}{5} x^{5} e^{3} d^{4} c^{5} b^{2} a + x e^{3} c^{12} b^{3} + 63 x^{4} e^{3} d^{3} c^{6} b^{2} a + \frac{3}{7} x^{7} e^{3} d^{6} b a^{2} + 36 x^{3} e^{3} d^{2} c^{7} b^{2} a + 3 x^{6} e^{3} d^{5} c b a^{2} + \frac{27}{2} x^{2} e^{3} d c^{8} b^{2} a + 9 x^{5} e^{3} d^{4} c^{2} b a^{2} + 3 x e^{3} c^{9} b^{2} a + 15 x^{4} e^{3} d^{3} c^{3} b a^{2} + 15 x^{3} e^{3} d^{2} c^{4} b a^{2} + 9 x^{2} e^{3} d c^{5} b a^{2} + 3 x e^{3} c^{6} b a^{2} + \frac{1}{4} x^{4} e^{3} d^{3} a^{3} + x^{3} e^{3} d^{2} c a^{3} + \frac{3}{2} x^{2} e^{3} d c^{2} a^{3} + x e^{3} c^{3} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.168819, size = 552, normalized size = 6.65 \begin{align*} 6 b^{3} c^{2} d^{10} e^{3} x^{11} + b^{3} c d^{11} e^{3} x^{12} + \frac{b^{3} d^{12} e^{3} x^{13}}{13} + x^{10} \left (\frac{3 a b^{2} d^{9} e^{3}}{10} + 22 b^{3} c^{3} d^{9} e^{3}\right ) + x^{9} \left (3 a b^{2} c d^{8} e^{3} + 55 b^{3} c^{4} d^{8} e^{3}\right ) + x^{8} \left (\frac{27 a b^{2} c^{2} d^{7} e^{3}}{2} + 99 b^{3} c^{5} d^{7} e^{3}\right ) + x^{7} \left (\frac{3 a^{2} b d^{6} e^{3}}{7} + 36 a b^{2} c^{3} d^{6} e^{3} + 132 b^{3} c^{6} d^{6} e^{3}\right ) + x^{6} \left (3 a^{2} b c d^{5} e^{3} + 63 a b^{2} c^{4} d^{5} e^{3} + 132 b^{3} c^{7} d^{5} e^{3}\right ) + x^{5} \left (9 a^{2} b c^{2} d^{4} e^{3} + \frac{378 a b^{2} c^{5} d^{4} e^{3}}{5} + 99 b^{3} c^{8} d^{4} e^{3}\right ) + x^{4} \left (\frac{a^{3} d^{3} e^{3}}{4} + 15 a^{2} b c^{3} d^{3} e^{3} + 63 a b^{2} c^{6} d^{3} e^{3} + 55 b^{3} c^{9} d^{3} e^{3}\right ) + x^{3} \left (a^{3} c d^{2} e^{3} + 15 a^{2} b c^{4} d^{2} e^{3} + 36 a b^{2} c^{7} d^{2} e^{3} + 22 b^{3} c^{10} d^{2} e^{3}\right ) + x^{2} \left (\frac{3 a^{3} c^{2} d e^{3}}{2} + 9 a^{2} b c^{5} d e^{3} + \frac{27 a b^{2} c^{8} d e^{3}}{2} + 6 b^{3} c^{11} d e^{3}\right ) + x \left (a^{3} c^{3} e^{3} + 3 a^{2} b c^{6} e^{3} + 3 a b^{2} c^{9} e^{3} + b^{3} c^{12} e^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1112, size = 689, normalized size = 8.3 \begin{align*} \frac{1}{13} \, b^{3} d^{12} x^{13} e^{3} + b^{3} c d^{11} x^{12} e^{3} + 6 \, b^{3} c^{2} d^{10} x^{11} e^{3} + 22 \, b^{3} c^{3} d^{9} x^{10} e^{3} + 55 \, b^{3} c^{4} d^{8} x^{9} e^{3} + 99 \, b^{3} c^{5} d^{7} x^{8} e^{3} + 132 \, b^{3} c^{6} d^{6} x^{7} e^{3} + \frac{3}{10} \, a b^{2} d^{9} x^{10} e^{3} + 132 \, b^{3} c^{7} d^{5} x^{6} e^{3} + 3 \, a b^{2} c d^{8} x^{9} e^{3} + 99 \, b^{3} c^{8} d^{4} x^{5} e^{3} + \frac{27}{2} \, a b^{2} c^{2} d^{7} x^{8} e^{3} + 55 \, b^{3} c^{9} d^{3} x^{4} e^{3} + 36 \, a b^{2} c^{3} d^{6} x^{7} e^{3} + 22 \, b^{3} c^{10} d^{2} x^{3} e^{3} + 63 \, a b^{2} c^{4} d^{5} x^{6} e^{3} + 6 \, b^{3} c^{11} d x^{2} e^{3} + \frac{378}{5} \, a b^{2} c^{5} d^{4} x^{5} e^{3} + b^{3} c^{12} x e^{3} + 63 \, a b^{2} c^{6} d^{3} x^{4} e^{3} + \frac{3}{7} \, a^{2} b d^{6} x^{7} e^{3} + 36 \, a b^{2} c^{7} d^{2} x^{3} e^{3} + 3 \, a^{2} b c d^{5} x^{6} e^{3} + \frac{27}{2} \, a b^{2} c^{8} d x^{2} e^{3} + 9 \, a^{2} b c^{2} d^{4} x^{5} e^{3} + 3 \, a b^{2} c^{9} x e^{3} + 15 \, a^{2} b c^{3} d^{3} x^{4} e^{3} + 15 \, a^{2} b c^{4} d^{2} x^{3} e^{3} + 9 \, a^{2} b c^{5} d x^{2} e^{3} + 3 \, a^{2} b c^{6} x e^{3} + \frac{1}{4} \, a^{3} d^{3} x^{4} e^{3} + a^{3} c d^{2} x^{3} e^{3} + \frac{3}{2} \, a^{3} c^{2} d x^{2} e^{3} + a^{3} c^{3} x e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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