Optimal. Leaf size=139 \[ \frac {1}{4} A b^3 d x^4+\frac {1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac {1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac {1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9 \]
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Rubi [A] time = 0.16, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 = {771} \begin {gather*} \frac {1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac {1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac {1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac {1}{4} A b^3 d x^4+\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 d x^3+b^2 (b B d+3 A c d+A b e) x^4+b \left (3 A c^2 d+b^2 B e+3 b c (B d+A e)\right ) x^5+c \left (A c^2 d+3 b^2 B e+3 b c (B d+A 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\\ &=\frac {1}{4} A b^3 d x^4+\frac {1}{5} b^2 (b B d+3 A c d+A b e) x^5+\frac {1}{6} b \left (3 A c^2 d+b^2 B e+3 b c (B d+A e)\right ) x^6+\frac {1}{7} c \left (A c^2 d+3 b^2 B e+3 b c (B d+A 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*}
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Mathematica [A] time = 0.04, size = 141, normalized size = 1.01 \begin {gather*} \frac {1}{4} A b^3 d x^4+\frac {1}{7} c x^7 \left (3 A b c e+A c^2 d+3 b^2 B e+3 b B c d\right )+\frac {1}{6} b x^6 \left (3 A b c e+3 A c^2 d+b^2 B e+3 b B c d\right )+\frac {1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\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 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 169, normalized size = 1.22 \begin {gather*} \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 {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} b A + \frac {1}{2} x^{6} e c b^{2} A + \frac {1}{5} x^{5} d b^{3} B + \frac {3}{5} x^{5} d c b^{2} A + \frac {1}{5} x^{5} e b^{3} A + \frac {1}{4} x^{4} d b^{3} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 177, normalized size = 1.27 \begin {gather*} \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} \, A b c^{2} x^{7} e + \frac {1}{2} \, B b^{2} c d x^{6} + \frac {1}{2} \, A b c^{2} d x^{6} + \frac {1}{6} \, B b^{3} x^{6} e + \frac {1}{2} \, A b^{2} c x^{6} e + \frac {1}{5} \, B b^{3} d x^{5} + \frac {3}{5} \, A b^{2} c d x^{5} + \frac {1}{5} \, A b^{3} x^{5} e + \frac {1}{4} \, A b^{3} d x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 138, normalized size = 0.99 \begin {gather*} \frac {B \,c^{3} e \,x^{9}}{9}+\frac {A \,b^{3} d \,x^{4}}{4}+\frac {\left (3 B b \,c^{2} e +\left (A e +B d \right ) c^{3}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d +3 B \,b^{2} c e +3 \left (A e +B d \right ) b \,c^{2}\right ) x^{7}}{7}+\frac {\left (3 A b \,c^{2} d +B \,b^{3} e +3 \left (A e +B d \right ) b^{2} c \right ) x^{6}}{6}+\frac {\left (3 A \,b^{2} c d +\left (A e +B d \right ) b^{3}\right ) x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 149, normalized size = 1.07 \begin {gather*} \frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{4} \, A b^{3} d x^{4} + \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 + A b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + A b c^{2}\right )} d + {\left (B b^{3} + 3 \, A b^{2} c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (A b^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 147, normalized size = 1.06 \begin {gather*} x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,A\,b^2\,c\,d}{5}\right )+x^8\,\left (\frac {A\,c^3\,e}{8}+\frac {B\,c^3\,d}{8}+\frac {3\,B\,b\,c^2\,e}{8}\right )+x^6\,\left (\frac {B\,b^3\,e}{6}+\frac {A\,b\,c^2\,d}{2}+\frac {A\,b^2\,c\,e}{2}+\frac {B\,b^2\,c\,d}{2}\right )+x^7\,\left (\frac {A\,c^3\,d}{7}+\frac {3\,A\,b\,c^2\,e}{7}+\frac {3\,B\,b\,c^2\,d}{7}+\frac {3\,B\,b^2\,c\,e}{7}\right )+\frac {A\,b^3\,d\,x^4}{4}+\frac {B\,c^3\,e\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 177, normalized size = 1.27 \begin {gather*} \frac {A b^{3} d x^{4}}{4} + \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 b^{2} c e}{7} + \frac {3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac {A b^{2} c e}{2} + \frac {A b c^{2} d}{2} + \frac {B b^{3} e}{6} + \frac {B b^{2} c d}{2}\right ) + x^{5} \left (\frac {A b^{3} e}{5} + \frac {3 A b^{2} c d}{5} + \frac {B b^{3} d}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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