Optimal. Leaf size=118 \[ -\frac {(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac {(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {d (d+e x)^4 (B d-A e) (c d-b e)}{4 e^4}+\frac {B c (d+e x)^7}{7 e^4} \]
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Rubi [A] time = 0.14, antiderivative size = 118, 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} \[ -\frac {(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac {(d+e x)^5 (B d (3 c d-2 b e)-A e (2 c d-b e))}{5 e^4}-\frac {d (d+e x)^4 (B d-A e) (c d-b e)}{4 e^4}+\frac {B c (d+e x)^7}{7 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^3 \left (b x+c x^2\right ) \, dx &=\int \left (-\frac {d (B d-A e) (c d-b e) (d+e x)^3}{e^3}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^4}{e^3}+\frac {(-3 B c d+b B e+A c e) (d+e x)^5}{e^3}+\frac {B c (d+e x)^6}{e^3}\right ) \, dx\\ &=-\frac {d (B d-A e) (c d-b e) (d+e x)^4}{4 e^4}+\frac {(B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^5}{5 e^4}-\frac {(3 B c d-b B e-A c e) (d+e x)^6}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 135, normalized size = 1.14 \[ \frac {1}{3} d^2 x^3 (3 A b e+A c d+b B d)+\frac {1}{6} e^2 x^6 (A c e+b B e+3 B c d)+\frac {1}{5} e x^5 (A e (b e+3 c d)+3 B d (b e+c d))+\frac {1}{4} d x^4 (3 A e (b e+c d)+B d (3 b e+c d))+\frac {1}{2} A b d^3 x^2+\frac {1}{7} B c e^3 x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 168, normalized size = 1.42 \[ \frac {1}{7} x^{7} e^{3} c B + \frac {1}{2} x^{6} e^{2} d c B + \frac {1}{6} x^{6} e^{3} b B + \frac {1}{6} x^{6} e^{3} c A + \frac {3}{5} x^{5} e d^{2} c B + \frac {3}{5} x^{5} e^{2} d b B + \frac {3}{5} x^{5} e^{2} d c A + \frac {1}{5} x^{5} e^{3} b A + \frac {1}{4} x^{4} d^{3} c B + \frac {3}{4} x^{4} e d^{2} b B + \frac {3}{4} x^{4} e d^{2} c A + \frac {3}{4} x^{4} e^{2} d b A + \frac {1}{3} x^{3} d^{3} b B + \frac {1}{3} x^{3} d^{3} c A + x^{3} e d^{2} b A + \frac {1}{2} x^{2} d^{3} b A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 164, normalized size = 1.39 \[ \frac {1}{7} \, B c x^{7} e^{3} + \frac {1}{2} \, B c d x^{6} e^{2} + \frac {3}{5} \, B c d^{2} x^{5} e + \frac {1}{4} \, B c d^{3} x^{4} + \frac {1}{6} \, B b x^{6} e^{3} + \frac {1}{6} \, A c x^{6} e^{3} + \frac {3}{5} \, B b d x^{5} e^{2} + \frac {3}{5} \, A c d x^{5} e^{2} + \frac {3}{4} \, B b d^{2} x^{4} e + \frac {3}{4} \, A c d^{2} x^{4} e + \frac {1}{3} \, B b d^{3} x^{3} + \frac {1}{3} \, A c d^{3} x^{3} + \frac {1}{5} \, A b x^{5} e^{3} + \frac {3}{4} \, A b d x^{4} e^{2} + A b d^{2} x^{3} e + \frac {1}{2} \, A b d^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 152, normalized size = 1.29 \[ \frac {B c \,e^{3} x^{7}}{7}+\frac {A b \,d^{3} x^{2}}{2}+\frac {\left (B b \,e^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) c \right ) x^{6}}{6}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) b +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b +\left (3 A \,d^{2} e +B \,d^{3}\right ) c \right ) x^{4}}{4}+\frac {\left (A c \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) b \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 137, normalized size = 1.16 \[ \frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, A b d^{3} x^{2} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, B c d^{2} e + A b e^{3} + 3 \, {\left (B b + A c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, A b d e^{2} + 3 \, {\left (B b + A c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A b d^{2} e + {\left (B b + A c\right )} d^{3}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 146, normalized size = 1.24 \[ x^3\,\left (\frac {A\,c\,d^3}{3}+\frac {B\,b\,d^3}{3}+A\,b\,d^2\,e\right )+x^6\,\left (\frac {A\,c\,e^3}{6}+\frac {B\,b\,e^3}{6}+\frac {B\,c\,d\,e^2}{2}\right )+x^4\,\left (\frac {B\,c\,d^3}{4}+\frac {3\,A\,b\,d\,e^2}{4}+\frac {3\,A\,c\,d^2\,e}{4}+\frac {3\,B\,b\,d^2\,e}{4}\right )+x^5\,\left (\frac {A\,b\,e^3}{5}+\frac {3\,A\,c\,d\,e^2}{5}+\frac {3\,B\,b\,d\,e^2}{5}+\frac {3\,B\,c\,d^2\,e}{5}\right )+\frac {A\,b\,d^3\,x^2}{2}+\frac {B\,c\,e^3\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 177, normalized size = 1.50 \[ \frac {A b d^{3} x^{2}}{2} + \frac {B c e^{3} x^{7}}{7} + x^{6} \left (\frac {A c e^{3}}{6} + \frac {B b e^{3}}{6} + \frac {B c d e^{2}}{2}\right ) + x^{5} \left (\frac {A b e^{3}}{5} + \frac {3 A c d e^{2}}{5} + \frac {3 B b d e^{2}}{5} + \frac {3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac {3 A b d e^{2}}{4} + \frac {3 A c d^{2} e}{4} + \frac {3 B b d^{2} e}{4} + \frac {B c d^{3}}{4}\right ) + x^{3} \left (A b d^{2} e + \frac {A c d^{3}}{3} + \frac {B b d^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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