Optimal. Leaf size=161 \[ \frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+a^3 d x+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right )+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8 \]
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Rubi [A] time = 0.157483, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {631} \[ \frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+a^3 d x+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right )+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 d+a^2 (3 b d+a e) x+3 a \left (b^2 d+a c d+a b e\right ) x^2+\left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^3+\left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^4+3 c \left (b c d+b^2 e+a c e\right ) x^5+c^2 (c d+3 b e) x^6+c^3 e x^7\right ) \, dx\\ &=a^3 d x+\frac{1}{2} a^2 (3 b d+a e) x^2+a \left (b^2 d+a c d+a b e\right ) x^3+\frac{1}{4} \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^4+\frac{1}{5} \left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^5+\frac{1}{2} c \left (b c d+b^2 e+a c e\right ) x^6+\frac{1}{7} c^2 (c d+3 b e) x^7+\frac{1}{8} c^3 e x^8\\ \end{align*}
Mathematica [A] time = 0.037271, size = 161, normalized size = 1. \[ \frac{1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac{1}{2} a^2 x^2 (a e+3 b d)+a^3 d x+\frac{1}{5} x^5 \left (6 a b c e+3 a c^2 d+3 b^2 c d+b^3 e\right )+\frac{1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{8} c^3 e x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 223, normalized size = 1.4 \begin{align*}{\frac{{c}^{3}e{x}^{8}}{8}}+{\frac{ \left ( 3\,eb{c}^{2}+d{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,db{c}^{2}+e \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( d \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( d \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +e \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( d \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +3\,{a}^{2}be \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{3}+3\,db{a}^{2} \right ){x}^{2}}{2}}+{a}^{3}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00462, size = 220, normalized size = 1.37 \begin{align*} \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{7} \,{\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d +{\left (b^{2} c + a c^{2}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (3 \,{\left (b^{2} c + a c^{2}\right )} d +{\left (b^{3} + 6 \, a b c\right )} e\right )} x^{5} + a^{3} d x + \frac{1}{4} \,{\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \,{\left (a b^{2} + a^{2} c\right )} e\right )} x^{4} +{\left (a^{2} b e +{\left (a b^{2} + a^{2} c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78398, size = 444, normalized size = 2.76 \begin{align*} \frac{1}{8} x^{8} e c^{3} + \frac{1}{7} x^{7} d c^{3} + \frac{3}{7} x^{7} e c^{2} b + \frac{1}{2} x^{6} d c^{2} b + \frac{1}{2} x^{6} e c b^{2} + \frac{1}{2} x^{6} e c^{2} a + \frac{3}{5} x^{5} d c b^{2} + \frac{1}{5} x^{5} e b^{3} + \frac{3}{5} x^{5} d c^{2} a + \frac{6}{5} x^{5} e c b a + \frac{1}{4} x^{4} d b^{3} + \frac{3}{2} x^{4} d c b a + \frac{3}{4} x^{4} e b^{2} a + \frac{3}{4} x^{4} e c a^{2} + x^{3} d b^{2} a + x^{3} d c a^{2} + x^{3} e b a^{2} + \frac{3}{2} x^{2} d b a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.097579, size = 190, normalized size = 1.18 \begin{align*} a^{3} d x + \frac{c^{3} e x^{8}}{8} + x^{7} \left (\frac{3 b c^{2} e}{7} + \frac{c^{3} d}{7}\right ) + x^{6} \left (\frac{a c^{2} e}{2} + \frac{b^{2} c e}{2} + \frac{b c^{2} d}{2}\right ) + x^{5} \left (\frac{6 a b c e}{5} + \frac{3 a c^{2} d}{5} + \frac{b^{3} e}{5} + \frac{3 b^{2} c d}{5}\right ) + x^{4} \left (\frac{3 a^{2} c e}{4} + \frac{3 a b^{2} e}{4} + \frac{3 a b c d}{2} + \frac{b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a^{2} c d + a b^{2} d\right ) + x^{2} \left (\frac{a^{3} e}{2} + \frac{3 a^{2} b d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07771, size = 266, normalized size = 1.65 \begin{align*} \frac{1}{8} \, c^{3} x^{8} e + \frac{1}{7} \, c^{3} d x^{7} + \frac{3}{7} \, b c^{2} x^{7} e + \frac{1}{2} \, b c^{2} d x^{6} + \frac{1}{2} \, b^{2} c x^{6} e + \frac{1}{2} \, a c^{2} x^{6} e + \frac{3}{5} \, b^{2} c d x^{5} + \frac{3}{5} \, a c^{2} d x^{5} + \frac{1}{5} \, b^{3} x^{5} e + \frac{6}{5} \, a b c x^{5} e + \frac{1}{4} \, b^{3} d x^{4} + \frac{3}{2} \, a b c d x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + \frac{3}{4} \, a^{2} c x^{4} e + a b^{2} d x^{3} + a^{2} c d x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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