Optimal. Leaf size=75 \[ \frac{(a+b x)^5 (-2 a B e+A b e+b B d)}{5 b^3}+\frac{(a+b x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac{B e (a+b x)^6}{6 b^3} \]
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Rubi [A] time = 0.0968763, antiderivative size = 75, 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} \[ \frac{(a+b x)^5 (-2 a B e+A b e+b B d)}{5 b^3}+\frac{(a+b x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac{B e (a+b x)^6}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int (a+b x)^3 (A+B x) (d+e x) \, dx &=\int \left (\frac{(A b-a B) (b d-a e) (a+b x)^3}{b^2}+\frac{(b B d+A b e-2 a B e) (a+b x)^4}{b^2}+\frac{B e (a+b x)^5}{b^2}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e) (a+b x)^4}{4 b^3}+\frac{(b B d+A b e-2 a B e) (a+b x)^5}{5 b^3}+\frac{B e (a+b x)^6}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.0409614, size = 130, normalized size = 1.73 \[ \frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+a^3 A d x+\frac{1}{5} b^2 x^5 (3 a B e+A b e+b B d)+\frac{1}{4} b x^4 (A b (3 a e+b d)+3 a B (a e+b d))+\frac{1}{3} a x^3 (3 A b (a e+b d)+a B (a e+3 b d))+\frac{1}{6} b^3 B e x^6 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 149, normalized size = 2. \begin{align*}{\frac{{b}^{3}Be{x}^{6}}{6}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ) e+{b}^{3}Bd \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) e+ \left ({b}^{3}A+3\,a{b}^{2}B \right ) d \right ){x}^{4}}{4}}+{\frac{ \left ( \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ) e+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) d \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{3}Ae+ \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ) d \right ){x}^{2}}{2}}+{a}^{3}Adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04902, size = 197, normalized size = 2.63 \begin{align*} \frac{1}{6} \, B b^{3} e x^{6} + A a^{3} d x + \frac{1}{5} \,{\left (B b^{3} d +{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \,{\left (B a^{2} b + A a b^{2}\right )} d +{\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.87132, size = 385, normalized size = 5.13 \begin{align*} \frac{1}{6} x^{6} e b^{3} B + \frac{1}{5} x^{5} d b^{3} B + \frac{3}{5} x^{5} e b^{2} a B + \frac{1}{5} x^{5} e b^{3} A + \frac{3}{4} x^{4} d b^{2} a B + \frac{3}{4} x^{4} e b a^{2} B + \frac{1}{4} x^{4} d b^{3} A + \frac{3}{4} x^{4} e b^{2} a A + x^{3} d b a^{2} B + \frac{1}{3} x^{3} e a^{3} B + x^{3} d b^{2} a A + x^{3} e b a^{2} A + \frac{1}{2} x^{2} d a^{3} B + \frac{3}{2} x^{2} d b a^{2} A + \frac{1}{2} x^{2} e a^{3} A + x d a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.105798, size = 168, normalized size = 2.24 \begin{align*} A a^{3} d x + \frac{B b^{3} e x^{6}}{6} + x^{5} \left (\frac{A b^{3} e}{5} + \frac{3 B a b^{2} e}{5} + \frac{B b^{3} d}{5}\right ) + x^{4} \left (\frac{3 A a b^{2} e}{4} + \frac{A b^{3} d}{4} + \frac{3 B a^{2} b e}{4} + \frac{3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a b^{2} d + \frac{B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac{A a^{3} e}{2} + \frac{3 A a^{2} b d}{2} + \frac{B a^{3} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.46421, size = 231, normalized size = 3.08 \begin{align*} \frac{1}{6} \, B b^{3} x^{6} e + \frac{1}{5} \, B b^{3} d x^{5} + \frac{3}{5} \, B a b^{2} x^{5} e + \frac{1}{5} \, A b^{3} x^{5} e + \frac{3}{4} \, B a b^{2} d x^{4} + \frac{1}{4} \, A b^{3} d x^{4} + \frac{3}{4} \, B a^{2} b x^{4} e + \frac{3}{4} \, A a b^{2} x^{4} e + B a^{2} b d x^{3} + A a b^{2} d x^{3} + \frac{1}{3} \, B a^{3} x^{3} e + A a^{2} b x^{3} e + \frac{1}{2} \, B a^{3} d x^{2} + \frac{3}{2} \, A a^{2} b d x^{2} + \frac{1}{2} \, A a^{3} x^{2} e + A a^{3} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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