Optimal. Leaf size=37 \[ \frac{a e^3 (c+d x)^4}{4 d}+\frac{b e^3 (c+d x)^7}{7 d} \]
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Rubi [A] time = 0.0291845, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {372, 14} \[ \frac{a e^3 (c+d x)^4}{4 d}+\frac{b e^3 (c+d x)^7}{7 d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 14
Rubi steps
\begin{align*} \int (c e+d e x)^3 \left (a+b (c+d x)^3\right ) \, dx &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \left (a x^3+b x^6\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{a e^3 (c+d x)^4}{4 d}+\frac{b e^3 (c+d x)^7}{7 d}\\ \end{align*}
Mathematica [B] time = 0.0081209, size = 102, normalized size = 2.76 \[ e^3 \left (\frac{1}{4} d^3 x^4 \left (a+20 b c^3\right )+c d^2 x^3 \left (a+5 b c^3\right )+\frac{3}{2} c^2 d x^2 \left (a+2 b c^3\right )+c^3 x \left (a+b c^3\right )+3 b c^2 d^4 x^5+b c d^5 x^6+\frac{1}{7} b d^6 x^7\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 154, normalized size = 4.2 \begin{align*}{\frac{{d}^{6}{e}^{3}b{x}^{7}}{7}}+c{e}^{3}{d}^{5}b{x}^{6}+3\,{c}^{2}{e}^{3}{d}^{4}b{x}^{5}+{\frac{ \left ( 19\,{c}^{3}{e}^{3}b{d}^{3}+{d}^{3}{e}^{3} \left ( b{c}^{3}+a \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 12\,{c}^{4}{e}^{3}b{d}^{2}+3\,c{e}^{3}{d}^{2} \left ( b{c}^{3}+a \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{c}^{5}{e}^{3}bd+3\,{c}^{2}{e}^{3}d \left ( b{c}^{3}+a \right ) \right ){x}^{2}}{2}}+{c}^{3}{e}^{3} \left ( b{c}^{3}+a \right ) x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15691, size = 157, normalized size = 4.24 \begin{align*} \frac{1}{7} \, b d^{6} e^{3} x^{7} + b c d^{5} e^{3} x^{6} + 3 \, b c^{2} d^{4} e^{3} x^{5} + \frac{1}{4} \,{\left (20 \, b c^{3} + a\right )} d^{3} e^{3} x^{4} +{\left (5 \, b c^{4} + a c\right )} d^{2} e^{3} x^{3} + \frac{3}{2} \,{\left (2 \, b c^{5} + a c^{2}\right )} d e^{3} x^{2} +{\left (b c^{6} + a 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 = 1.29751, size = 285, normalized size = 7.7 \begin{align*} \frac{1}{7} x^{7} e^{3} d^{6} b + x^{6} e^{3} d^{5} c b + 3 x^{5} e^{3} d^{4} c^{2} b + 5 x^{4} e^{3} d^{3} c^{3} b + 5 x^{3} e^{3} d^{2} c^{4} b + 3 x^{2} e^{3} d c^{5} b + x e^{3} c^{6} b + \frac{1}{4} x^{4} e^{3} d^{3} a + x^{3} e^{3} d^{2} c a + \frac{3}{2} x^{2} e^{3} d c^{2} a + x e^{3} c^{3} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.08839, size = 144, normalized size = 3.89 \begin{align*} 3 b c^{2} d^{4} e^{3} x^{5} + b c d^{5} e^{3} x^{6} + \frac{b d^{6} e^{3} x^{7}}{7} + x^{4} \left (\frac{a d^{3} e^{3}}{4} + 5 b c^{3} d^{3} e^{3}\right ) + x^{3} \left (a c d^{2} e^{3} + 5 b c^{4} d^{2} e^{3}\right ) + x^{2} \left (\frac{3 a c^{2} d e^{3}}{2} + 3 b c^{5} d e^{3}\right ) + x \left (a c^{3} e^{3} + b c^{6} e^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08927, size = 171, normalized size = 4.62 \begin{align*} \frac{1}{7} \, b d^{6} x^{7} e^{3} + b c d^{5} x^{6} e^{3} + 3 \, b c^{2} d^{4} x^{5} e^{3} + 5 \, b c^{3} d^{3} x^{4} e^{3} + 5 \, b c^{4} d^{2} x^{3} e^{3} + 3 \, b c^{5} d x^{2} e^{3} + b c^{6} x e^{3} + \frac{1}{4} \, a d^{3} x^{4} e^{3} + a c d^{2} x^{3} e^{3} + \frac{3}{2} \, a c^{2} d x^{2} e^{3} + a c^{3} x e^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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