Optimal. Leaf size=77 \[ -\frac{(d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3}+\frac{(d+e x)^4 (b d-a e) (B d-A e)}{4 e^3}+\frac{b B (d+e x)^6}{6 e^3} \]
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Rubi [A] time = 0.0947771, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{(d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3}+\frac{(d+e x)^4 (b d-a e) (B d-A e)}{4 e^3}+\frac{b B (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int (a+b x) (A+B x) (d+e x)^3 \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e) (d+e x)^3}{e^2}+\frac{(-2 b B d+A b e+a B e) (d+e x)^4}{e^2}+\frac{b B (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac{(b d-a e) (B d-A e) (d+e x)^4}{4 e^3}-\frac{(2 b B d-A b e-a B e) (d+e x)^5}{5 e^3}+\frac{b B (d+e x)^6}{6 e^3}\\ \end{align*}
Mathematica [A] time = 0.0438463, size = 130, normalized size = 1.69 \[ \frac{1}{2} d^2 x^2 (3 a A e+a B d+A b d)+\frac{1}{5} e^2 x^5 (a B e+A b e+3 b B d)+\frac{1}{4} e x^4 (a e (A e+3 B d)+3 b d (A e+B d))+\frac{1}{3} d x^3 (3 a e (A e+B d)+b d (3 A e+B d))+a A d^3 x+\frac{1}{6} b B e^3 x^6 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 135, normalized size = 1.8 \begin{align*}{\frac{bB{e}^{3}{x}^{6}}{6}}+{\frac{ \left ( \left ( Ab+Ba \right ){e}^{3}+3\,bBd{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( aA{e}^{3}+3\, \left ( Ab+Ba \right ) d{e}^{2}+3\,bB{d}^{2}e \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,aAd{e}^{2}+3\, \left ( Ab+Ba \right ){d}^{2}e+bB{d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,aA{d}^{2}e+ \left ( Ab+Ba \right ){d}^{3} \right ){x}^{2}}{2}}+aA{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10231, size = 181, normalized size = 2.35 \begin{align*} \frac{1}{6} \, B b e^{3} x^{6} + A a d^{3} x + \frac{1}{5} \,{\left (3 \, B b d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B b d^{2} e + A a e^{3} + 3 \,{\left (B a + A b\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{3} + 3 \, A a d e^{2} + 3 \,{\left (B a + A b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a d^{2} e +{\left (B a + A b\right )} d^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50019, size = 385, normalized size = 5. \begin{align*} \frac{1}{6} x^{6} e^{3} b B + \frac{3}{5} x^{5} e^{2} d b B + \frac{1}{5} x^{5} e^{3} a B + \frac{1}{5} x^{5} e^{3} b A + \frac{3}{4} x^{4} e d^{2} b B + \frac{3}{4} x^{4} e^{2} d a B + \frac{3}{4} x^{4} e^{2} d b A + \frac{1}{4} x^{4} e^{3} a A + \frac{1}{3} x^{3} d^{3} b B + x^{3} e d^{2} a B + x^{3} e d^{2} b A + x^{3} e^{2} d a A + \frac{1}{2} x^{2} d^{3} a B + \frac{1}{2} x^{2} d^{3} b A + \frac{3}{2} x^{2} e d^{2} a A + x d^{3} a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.103564, size = 168, normalized size = 2.18 \begin{align*} A a d^{3} x + \frac{B b e^{3} x^{6}}{6} + x^{5} \left (\frac{A b e^{3}}{5} + \frac{B a e^{3}}{5} + \frac{3 B b d e^{2}}{5}\right ) + x^{4} \left (\frac{A a e^{3}}{4} + \frac{3 A b d e^{2}}{4} + \frac{3 B a d e^{2}}{4} + \frac{3 B b d^{2} e}{4}\right ) + x^{3} \left (A a d e^{2} + A b d^{2} e + B a d^{2} e + \frac{B b d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a d^{2} e}{2} + \frac{A b d^{3}}{2} + \frac{B a d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.62946, size = 215, normalized size = 2.79 \begin{align*} \frac{1}{6} \, B b x^{6} e^{3} + \frac{3}{5} \, B b d x^{5} e^{2} + \frac{3}{4} \, B b d^{2} x^{4} e + \frac{1}{3} \, B b d^{3} x^{3} + \frac{1}{5} \, B a x^{5} e^{3} + \frac{1}{5} \, A b x^{5} e^{3} + \frac{3}{4} \, B a d x^{4} e^{2} + \frac{3}{4} \, A b d x^{4} e^{2} + B a d^{2} x^{3} e + A b d^{2} x^{3} e + \frac{1}{2} \, B a d^{3} x^{2} + \frac{1}{2} \, A b d^{3} x^{2} + \frac{1}{4} \, A a x^{4} e^{3} + A a d x^{3} e^{2} + \frac{3}{2} \, A a d^{2} x^{2} e + A a d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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