Optimal. Leaf size=99 \[ \frac {1}{5} e x^5 (A c e+b B e+2 B c d)+\frac {1}{4} x^4 (A e (b e+2 c d)+B d (2 b e+c d))+\frac {1}{3} d x^3 (2 A b e+A c d+b B d)+\frac {1}{2} A b d^2 x^2+\frac {1}{6} B c e^2 x^6 \]
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Rubi [A] time = 0.10, antiderivative size = 99, 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}{5} e x^5 (A c e+b B e+2 B c d)+\frac {1}{4} x^4 (A e (b e+2 c d)+B d (2 b e+c d))+\frac {1}{3} d x^3 (2 A b e+A c d+b B d)+\frac {1}{2} A b d^2 x^2+\frac {1}{6} B c e^2 x^6 \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx &=\int \left (A b d^2 x+d (b B d+A c d+2 A b e) x^2+(A e (2 c d+b e)+B d (c d+2 b e)) x^3+e (2 B c d+b B e+A c e) x^4+B c e^2 x^5\right ) \, dx\\ &=\frac {1}{2} A b d^2 x^2+\frac {1}{3} d (b B d+A c d+2 A b e) x^3+\frac {1}{4} (A e (2 c d+b e)+B d (c d+2 b e)) x^4+\frac {1}{5} e (2 B c d+b B e+A c e) x^5+\frac {1}{6} B c e^2 x^6\\ \end {align*}
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Mathematica [A] time = 0.04, size = 91, normalized size = 0.92 \begin {gather*} \frac {1}{60} x^2 \left (12 e x^3 (A c e+b B e+2 B c d)+15 x^2 (A e (b e+2 c d)+B d (2 b e+c d))+20 d x (2 A b e+A c d+b B d)+30 A b d^2+10 B c e^2 x^4\right ) \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)^2 \left (b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 117, normalized size = 1.18 \begin {gather*} \frac {1}{6} x^{6} e^{2} c B + \frac {2}{5} x^{5} e d c B + \frac {1}{5} x^{5} e^{2} b B + \frac {1}{5} x^{5} e^{2} c A + \frac {1}{4} x^{4} d^{2} c B + \frac {1}{2} x^{4} e d b B + \frac {1}{2} x^{4} e d c A + \frac {1}{4} x^{4} e^{2} b A + \frac {1}{3} x^{3} d^{2} b B + \frac {1}{3} x^{3} d^{2} c A + \frac {2}{3} x^{3} e d b A + \frac {1}{2} x^{2} d^{2} b A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 117, normalized size = 1.18 \begin {gather*} \frac {1}{6} \, B c x^{6} e^{2} + \frac {2}{5} \, B c d x^{5} e + \frac {1}{4} \, B c d^{2} x^{4} + \frac {1}{5} \, B b x^{5} e^{2} + \frac {1}{5} \, A c x^{5} e^{2} + \frac {1}{2} \, B b d x^{4} e + \frac {1}{2} \, A c d x^{4} e + \frac {1}{3} \, B b d^{2} x^{3} + \frac {1}{3} \, A c d^{2} x^{3} + \frac {1}{4} \, A b x^{4} e^{2} + \frac {2}{3} \, A b d x^{3} e + \frac {1}{2} \, A b d^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 104, normalized size = 1.05 \begin {gather*} \frac {B c \,e^{2} x^{6}}{6}+\frac {A b \,d^{2} x^{2}}{2}+\frac {\left (B b \,e^{2}+\left (A \,e^{2}+2 B d e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (A \,e^{2}+2 B d e \right ) b +\left (2 A d e +B \,d^{2}\right ) c \right ) x^{4}}{4}+\frac {\left (A c \,d^{2}+\left (2 A d e +B \,d^{2}\right ) b \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 96, normalized size = 0.97 \begin {gather*} \frac {1}{6} \, B c e^{2} x^{6} + \frac {1}{2} \, A b d^{2} x^{2} + \frac {1}{5} \, {\left (2 \, B c d e + {\left (B b + A c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{2} + A b e^{2} + 2 \, {\left (B b + A c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A b d e + {\left (B b + A c\right )} d^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 102, normalized size = 1.03 \begin {gather*} x^4\,\left (\frac {A\,b\,e^2}{4}+\frac {B\,c\,d^2}{4}+\frac {A\,c\,d\,e}{2}+\frac {B\,b\,d\,e}{2}\right )+x^3\,\left (\frac {A\,c\,d^2}{3}+\frac {B\,b\,d^2}{3}+\frac {2\,A\,b\,d\,e}{3}\right )+x^5\,\left (\frac {A\,c\,e^2}{5}+\frac {B\,b\,e^2}{5}+\frac {2\,B\,c\,d\,e}{5}\right )+\frac {A\,b\,d^2\,x^2}{2}+\frac {B\,c\,e^2\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 121, normalized size = 1.22 \begin {gather*} \frac {A b d^{2} x^{2}}{2} + \frac {B c e^{2} x^{6}}{6} + x^{5} \left (\frac {A c e^{2}}{5} + \frac {B b e^{2}}{5} + \frac {2 B c d e}{5}\right ) + x^{4} \left (\frac {A b e^{2}}{4} + \frac {A c d e}{2} + \frac {B b d e}{2} + \frac {B c d^{2}}{4}\right ) + x^{3} \left (\frac {2 A b d e}{3} + \frac {A c d^{2}}{3} + \frac {B b d^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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