Optimal. Leaf size=135 \[ \frac{1}{13} e^2 x^{13} \left (e (a e+4 b d)+6 c d^2\right )+\frac{1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{5} d e x^{10} \left (e (2 a e+3 b d)+2 c d^2\right )+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19} \]
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Rubi [A] time = 0.125073, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1407} \[ \frac{1}{13} e^2 x^{13} \left (e (a e+4 b d)+6 c d^2\right )+\frac{1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{5} d e x^{10} \left (e (2 a e+3 b d)+2 c d^2\right )+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19} \]
Antiderivative was successfully verified.
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Rule 1407
Rubi steps
\begin{align*} \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx &=\int \left (a d^4+d^3 (b d+4 a e) x^3+d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^6+2 d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^9+e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{12}+e^3 (4 c d+b e) x^{15}+c e^4 x^{18}\right ) \, dx\\ &=a d^4 x+\frac{1}{4} d^3 (b d+4 a e) x^4+\frac{1}{7} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^7+\frac{1}{5} d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^{10}+\frac{1}{13} e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{13}+\frac{1}{16} e^3 (4 c d+b e) x^{16}+\frac{1}{19} c e^4 x^{19}\\ \end{align*}
Mathematica [A] time = 0.0361796, size = 135, normalized size = 1. \[ \frac{1}{13} e^2 x^{13} \left (a e^2+4 b d e+6 c d^2\right )+\frac{1}{5} d e x^{10} \left (2 a e^2+3 b d e+2 c d^2\right )+\frac{1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac{1}{16} e^3 x^{16} (b e+4 c d)+\frac{1}{19} c e^4 x^{19} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 136, normalized size = 1. \begin{align*}{\frac{c{e}^{4}{x}^{19}}{19}}+{\frac{ \left ({e}^{4}b+4\,d{e}^{3}c \right ){x}^{16}}{16}}+{\frac{ \left ({e}^{4}a+4\,d{e}^{3}b+6\,{e}^{2}{d}^{2}c \right ){x}^{13}}{13}}+{\frac{ \left ( 4\,d{e}^{3}a+6\,{e}^{2}{d}^{2}b+4\,{d}^{3}ec \right ){x}^{10}}{10}}+{\frac{ \left ( 6\,{e}^{2}{d}^{2}a+4\,{d}^{3}eb+c{d}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,{d}^{3}ea+{d}^{4}b \right ){x}^{4}}{4}}+a{d}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98309, size = 182, normalized size = 1.35 \begin{align*} \frac{1}{19} \, c e^{4} x^{19} + \frac{1}{16} \,{\left (4 \, c d e^{3} + b e^{4}\right )} x^{16} + \frac{1}{13} \,{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{13} + \frac{1}{5} \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{10} + \frac{1}{7} \,{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{7} + a d^{4} x + \frac{1}{4} \,{\left (b d^{4} + 4 \, a d^{3} e\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16381, size = 360, normalized size = 2.67 \begin{align*} \frac{1}{19} x^{19} e^{4} c + \frac{1}{4} x^{16} e^{3} d c + \frac{1}{16} x^{16} e^{4} b + \frac{6}{13} x^{13} e^{2} d^{2} c + \frac{4}{13} x^{13} e^{3} d b + \frac{1}{13} x^{13} e^{4} a + \frac{2}{5} x^{10} e d^{3} c + \frac{3}{5} x^{10} e^{2} d^{2} b + \frac{2}{5} x^{10} e^{3} d a + \frac{1}{7} x^{7} d^{4} c + \frac{4}{7} x^{7} e d^{3} b + \frac{6}{7} x^{7} e^{2} d^{2} a + \frac{1}{4} x^{4} d^{4} b + x^{4} e d^{3} a + x d^{4} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.089108, size = 151, normalized size = 1.12 \begin{align*} a d^{4} x + \frac{c e^{4} x^{19}}{19} + x^{16} \left (\frac{b e^{4}}{16} + \frac{c d e^{3}}{4}\right ) + x^{13} \left (\frac{a e^{4}}{13} + \frac{4 b d e^{3}}{13} + \frac{6 c d^{2} e^{2}}{13}\right ) + x^{10} \left (\frac{2 a d e^{3}}{5} + \frac{3 b d^{2} e^{2}}{5} + \frac{2 c d^{3} e}{5}\right ) + x^{7} \left (\frac{6 a d^{2} e^{2}}{7} + \frac{4 b d^{3} e}{7} + \frac{c d^{4}}{7}\right ) + x^{4} \left (a d^{3} e + \frac{b d^{4}}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11388, size = 190, normalized size = 1.41 \begin{align*} \frac{1}{19} \, c x^{19} e^{4} + \frac{1}{4} \, c d x^{16} e^{3} + \frac{1}{16} \, b x^{16} e^{4} + \frac{6}{13} \, c d^{2} x^{13} e^{2} + \frac{4}{13} \, b d x^{13} e^{3} + \frac{1}{13} \, a x^{13} e^{4} + \frac{2}{5} \, c d^{3} x^{10} e + \frac{3}{5} \, b d^{2} x^{10} e^{2} + \frac{2}{5} \, a d x^{10} e^{3} + \frac{1}{7} \, c d^{4} x^{7} + \frac{4}{7} \, b d^{3} x^{7} e + \frac{6}{7} \, a d^{2} x^{7} e^{2} + \frac{1}{4} \, b d^{4} x^{4} + a d^{3} x^{4} e + a d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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