Optimal. Leaf size=310 \[ \frac{1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+a^3 A d x+\frac{1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac{1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac{1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac{1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]
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Rubi [A] time = 0.63988, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ \frac{1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+a^3 A d x+\frac{1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac{1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac{1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac{1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A d+a^2 (3 A b d+a B d+a A e) x+a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^2+\left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^3+\left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^4+\left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^5+c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^6+c^2 (B c d+3 b B e+A c e) x^7+B c^3 e x^8\right ) \, dx\\ &=a^3 A d x+\frac{1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac{1}{3} a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^3+\frac{1}{4} \left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^4+\frac{1}{5} \left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^5+\frac{1}{6} \left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^6+\frac{1}{7} c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^7+\frac{1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac{1}{9} B c^3 e x^9\\ \end{align*}
Mathematica [A] time = 0.168886, size = 310, normalized size = 1. \[ \frac{1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+a^3 A d x+\frac{1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac{1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac{1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac{1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 375, normalized size = 1.2 \begin{align*}{\frac{B{c}^{3}e{x}^{9}}{9}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{3}+3\,Beb{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( Ad{c}^{3}+3\, \left ( Ae+Bd \right ) b{c}^{2}+Be \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Adb{c}^{2}+ \left ( Ae+Bd \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +Be \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( Ad \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( Ae+Bd \right ) \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +Be \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( Ad \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( Ae+Bd \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\,Be{a}^{2}b \right ){x}^{4}}{4}}+{\frac{ \left ( Ad \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\, \left ( Ae+Bd \right ){a}^{2}b+Be{a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,Ad{a}^{2}b+ \left ( Ae+Bd \right ){a}^{3} \right ){x}^{2}}{2}}+{a}^{3}Adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99288, size = 451, normalized size = 1.45 \begin{align*} \frac{1}{9} \, B c^{3} e x^{9} + \frac{1}{8} \,{\left (B c^{3} d +{\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac{1}{7} \,{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} d +{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} e\right )} x^{6} + A a^{3} d x + \frac{1}{5} \,{\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} d +{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} e\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} d + 3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d +{\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.921716, size = 1011, normalized size = 3.26 \begin{align*} \frac{1}{9} x^{9} e c^{3} B + \frac{1}{8} x^{8} d c^{3} B + \frac{3}{8} x^{8} e c^{2} b B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} d c^{2} b B + \frac{3}{7} x^{7} e c b^{2} B + \frac{3}{7} x^{7} e c^{2} a B + \frac{1}{7} x^{7} d c^{3} A + \frac{3}{7} x^{7} e c^{2} b A + \frac{1}{2} x^{6} d c b^{2} B + \frac{1}{6} x^{6} e b^{3} B + \frac{1}{2} x^{6} d c^{2} a B + x^{6} e c b a B + \frac{1}{2} x^{6} d c^{2} b A + \frac{1}{2} x^{6} e c b^{2} A + \frac{1}{2} x^{6} e c^{2} a A + \frac{1}{5} x^{5} d b^{3} B + \frac{6}{5} x^{5} d c b a B + \frac{3}{5} x^{5} e b^{2} a B + \frac{3}{5} x^{5} e c a^{2} B + \frac{3}{5} x^{5} d c b^{2} A + \frac{1}{5} x^{5} e b^{3} A + \frac{3}{5} x^{5} d c^{2} a A + \frac{6}{5} x^{5} e c b a A + \frac{3}{4} x^{4} d b^{2} a B + \frac{3}{4} x^{4} d c a^{2} B + \frac{3}{4} x^{4} e b a^{2} B + \frac{1}{4} x^{4} d b^{3} A + \frac{3}{2} x^{4} d c b a A + \frac{3}{4} x^{4} e b^{2} a A + \frac{3}{4} x^{4} e c a^{2} A + x^{3} d b a^{2} B + \frac{1}{3} x^{3} e a^{3} B + x^{3} d b^{2} a A + x^{3} d c a^{2} A + x^{3} e b a^{2} A + \frac{1}{2} x^{2} d a^{3} B + \frac{3}{2} x^{2} d b a^{2} A + \frac{1}{2} x^{2} e a^{3} A + x d a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.118688, size = 435, normalized size = 1.4 \begin{align*} A a^{3} d x + \frac{B c^{3} e x^{9}}{9} + x^{8} \left (\frac{A c^{3} e}{8} + \frac{3 B b c^{2} e}{8} + \frac{B c^{3} d}{8}\right ) + x^{7} \left (\frac{3 A b c^{2} e}{7} + \frac{A c^{3} d}{7} + \frac{3 B a c^{2} e}{7} + \frac{3 B b^{2} c e}{7} + \frac{3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac{A a c^{2} e}{2} + \frac{A b^{2} c e}{2} + \frac{A b c^{2} d}{2} + B a b c e + \frac{B a c^{2} d}{2} + \frac{B b^{3} e}{6} + \frac{B b^{2} c d}{2}\right ) + x^{5} \left (\frac{6 A a b c e}{5} + \frac{3 A a c^{2} d}{5} + \frac{A b^{3} e}{5} + \frac{3 A b^{2} c d}{5} + \frac{3 B a^{2} c e}{5} + \frac{3 B a b^{2} e}{5} + \frac{6 B a b c d}{5} + \frac{B b^{3} d}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c e}{4} + \frac{3 A a b^{2} e}{4} + \frac{3 A a b c d}{2} + \frac{A b^{3} d}{4} + \frac{3 B a^{2} b e}{4} + \frac{3 B a^{2} c d}{4} + \frac{3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a^{2} c d + A a b^{2} d + \frac{B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac{A a^{3} e}{2} + \frac{3 A a^{2} b d}{2} + \frac{B a^{3} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1117, size = 590, normalized size = 1.9 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{3}{8} \, B b c^{2} x^{8} e + \frac{1}{8} \, A c^{3} x^{8} e + \frac{3}{7} \, B b c^{2} d x^{7} + \frac{1}{7} \, A c^{3} d x^{7} + \frac{3}{7} \, B b^{2} c x^{7} e + \frac{3}{7} \, B a c^{2} x^{7} e + \frac{3}{7} \, A b c^{2} x^{7} e + \frac{1}{2} \, B b^{2} c d x^{6} + \frac{1}{2} \, B a c^{2} d x^{6} + \frac{1}{2} \, A b c^{2} d x^{6} + \frac{1}{6} \, B b^{3} x^{6} e + B a b c x^{6} e + \frac{1}{2} \, A b^{2} c x^{6} e + \frac{1}{2} \, A a c^{2} x^{6} e + \frac{1}{5} \, B b^{3} d x^{5} + \frac{6}{5} \, B a b c d x^{5} + \frac{3}{5} \, A b^{2} c d x^{5} + \frac{3}{5} \, A a c^{2} d x^{5} + \frac{3}{5} \, B a b^{2} x^{5} e + \frac{1}{5} \, A b^{3} x^{5} e + \frac{3}{5} \, B a^{2} c x^{5} e + \frac{6}{5} \, A a b c x^{5} e + \frac{3}{4} \, B a b^{2} d x^{4} + \frac{1}{4} \, A b^{3} d x^{4} + \frac{3}{4} \, B a^{2} c d x^{4} + \frac{3}{2} \, A a b c d x^{4} + \frac{3}{4} \, B a^{2} b x^{4} e + \frac{3}{4} \, A a b^{2} x^{4} e + \frac{3}{4} \, A a^{2} c x^{4} e + B a^{2} b d x^{3} + A a b^{2} d x^{3} + A a^{2} c d x^{3} + \frac{1}{3} \, B a^{3} x^{3} e + A a^{2} b x^{3} e + \frac{1}{2} \, B a^{3} d x^{2} + \frac{3}{2} \, A a^{2} b d x^{2} + \frac{1}{2} \, A a^{3} x^{2} e + A a^{3} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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