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 = 0.0445928, antiderivative size = 56, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0209038, 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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Maple [A] time = 0., size = 53, normalized size = 1. \begin{align*}{\frac{bBe{x}^{4}}{4}}+{\frac{ \left ( \left ( Ab+Ba \right ) e+Bbd \right ){x}^{3}}{3}}+{\frac{ \left ( Aae+ \left ( Ab+Ba \right ) d \right ){x}^{2}}{2}}+aAdx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.33237, size = 70, normalized size = 1.25 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58343, size = 163, normalized size = 2.91 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083221, size = 63, normalized size = 1.12 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.06777, size = 89, normalized size = 1.59 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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