Optimal. Leaf size=56 \[ \frac {1}{3} x^3 (a B e+A b e+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} b B e x^4 \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \frac {1}{3} x^3 (a B e+A b e+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} b B e x^4 \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x) (A+B x) (d+e x) \, dx &=\int \left (a A d+(A b d+a B d+a A e) x+(b B d+A b e+a B e) x^2+b B e x^3\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 b e+a B e) x^3+\frac {1}{4} b B e x^4\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.95 \[ \frac {1}{12} x \left (4 x^2 (a B e+A b e+b B d)+6 x (a A e+a B d+A b d)+12 a A d+3 b B e x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 62, normalized size = 1.11 \[ \frac {1}{4} x^{4} e b B + \frac {1}{3} x^{3} d b B + \frac {1}{3} x^{3} e a B + \frac {1}{3} x^{3} e b A + \frac {1}{2} x^{2} d a B + \frac {1}{2} x^{2} d b A + \frac {1}{2} x^{2} e a A + x d a A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 66, normalized size = 1.18 \[ \frac {1}{4} \, B b x^{4} e + \frac {1}{3} \, B b d x^{3} + \frac {1}{3} \, B a x^{3} e + \frac {1}{3} \, A b x^{3} e + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A b d x^{2} + \frac {1}{2} \, A a x^{2} e + A a d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 53, normalized size = 0.95 \[ \frac {B b e \,x^{4}}{4}+A a d x +\frac {\left (B b d +\left (A b +B a \right ) e \right ) x^{3}}{3}+\frac {\left (A a e +\left (A b +B a \right ) d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 52, normalized size = 0.93 \[ \frac {1}{4} \, B b e x^{4} + A a d x + \frac {1}{3} \, {\left (B b d + {\left (B a + A b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a e + {\left (B a + A b\right )} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 54, normalized size = 0.96 \[ \frac {B\,b\,e\,x^4}{4}+\left (\frac {A\,b\,e}{3}+\frac {B\,a\,e}{3}+\frac {B\,b\,d}{3}\right )\,x^3+\left (\frac {A\,a\,e}{2}+\frac {A\,b\,d}{2}+\frac {B\,a\,d}{2}\right )\,x^2+A\,a\,d\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 63, normalized size = 1.12 \[ A a d x + \frac {B b e x^{4}}{4} + x^{3} \left (\frac {A b e}{3} + \frac {B a e}{3} + \frac {B b d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {A b d}{2} + \frac {B a d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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