Optimal. Leaf size=80 \[ \frac {1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} x^4 (A c e+b B e+B c d)+\frac {1}{5} B c e x^5 \]
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Rubi [A] time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {771} \begin {gather*} \frac {1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} x^4 (A c e+b B e+B c d)+\frac {1}{5} B c e x^5 \end {gather*}
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 ) \, dx &=\int \left (a A d+(A b d+a B d+a A e) x+(b B d+A c d+A b e+a B e) x^2+(B c d+b B e+A c e) x^3+B c e x^4\right ) \, dx\\ &=a A d x+\frac {1}{2} (A b d+a B d+a A e) x^2+\frac {1}{3} (b B d+A c d+A b e+a B e) x^3+\frac {1}{4} (B c d+b B e+A c e) x^4+\frac {1}{5} B c e x^5\\ \end {align*}
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Mathematica [A] time = 0.03, size = 80, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} x^4 (A c e+b B e+B c d)+\frac {1}{5} B c e x^5 \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 (a+b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 94, normalized size = 1.18 \begin {gather*} \frac {1}{5} x^{5} e c B + \frac {1}{4} x^{4} d c B + \frac {1}{4} x^{4} e b B + \frac {1}{4} x^{4} e c A + \frac {1}{3} x^{3} d b B + \frac {1}{3} x^{3} e a B + \frac {1}{3} x^{3} d c A + \frac {1}{3} x^{3} e b A + \frac {1}{2} x^{2} d a B + \frac {1}{2} x^{2} d b A + \frac {1}{2} x^{2} e a A + x d a A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 100, normalized size = 1.25 \begin {gather*} \frac {1}{5} \, B c x^{5} e + \frac {1}{4} \, B c d x^{4} + \frac {1}{4} \, B b x^{4} e + \frac {1}{4} \, A c x^{4} e + \frac {1}{3} \, B b d x^{3} + \frac {1}{3} \, A c d x^{3} + \frac {1}{3} \, B a x^{3} e + \frac {1}{3} \, A b x^{3} e + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A b d x^{2} + \frac {1}{2} \, A a x^{2} e + A a d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 0.95 \begin {gather*} \frac {B c e \,x^{5}}{5}+A a d x +\frac {\left (B b e +\left (A e +B d \right ) c \right ) x^{4}}{4}+\frac {\left (A c d +B a e +\left (A e +B d \right ) b \right ) x^{3}}{3}+\frac {\left (A b d +\left (A e +B d \right ) a \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 76, normalized size = 0.95 \begin {gather*} \frac {1}{5} \, B c e x^{5} + \frac {1}{4} \, {\left (B c d + {\left (B b + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left ({\left (B b + A c\right )} d + {\left (B a + A b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a e + {\left (B a + A b\right )} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 79, normalized size = 0.99 \begin {gather*} \frac {B\,c\,e\,x^5}{5}+\left (\frac {A\,c\,e}{4}+\frac {B\,b\,e}{4}+\frac {B\,c\,d}{4}\right )\,x^4+\left (\frac {A\,b\,e}{3}+\frac {A\,c\,d}{3}+\frac {B\,a\,e}{3}+\frac {B\,b\,d}{3}\right )\,x^3+\left (\frac {A\,a\,e}{2}+\frac {A\,b\,d}{2}+\frac {B\,a\,d}{2}\right )\,x^2+A\,a\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 94, normalized size = 1.18 \begin {gather*} A a d x + \frac {B c e x^{5}}{5} + x^{4} \left (\frac {A c e}{4} + \frac {B b e}{4} + \frac {B c d}{4}\right ) + x^{3} \left (\frac {A b e}{3} + \frac {A c d}{3} + \frac {B a e}{3} + \frac {B b d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {A b d}{2} + \frac {B a d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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