Optimal. Leaf size=61 \[ \frac{(a+b x)^5 (A b-2 a B)}{5 b^3}-\frac{a (a+b x)^4 (A b-a B)}{4 b^3}+\frac{B (a+b x)^6}{6 b^3} \]
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Rubi [A] time = 0.0373683, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {76} \[ \frac{(a+b x)^5 (A b-2 a B)}{5 b^3}-\frac{a (a+b x)^4 (A b-a B)}{4 b^3}+\frac{B (a+b x)^6}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x (a+b x)^3 (A+B x) \, dx &=\int \left (\frac{a (-A b+a B) (a+b x)^3}{b^2}+\frac{(A b-2 a B) (a+b x)^4}{b^2}+\frac{B (a+b x)^5}{b^2}\right ) \, dx\\ &=-\frac{a (A b-a B) (a+b x)^4}{4 b^3}+\frac{(A b-2 a B) (a+b x)^5}{5 b^3}+\frac{B (a+b x)^6}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.0136092, size = 69, normalized size = 1.13 \[ \frac{1}{60} x^2 \left (15 a^2 b x (4 A+3 B x)+10 a^3 (3 A+2 B x)+9 a b^2 x^2 (5 A+4 B x)+2 b^3 x^3 (6 A+5 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 76, normalized size = 1.3 \begin{align*}{\frac{{b}^{3}B{x}^{6}}{6}}+{\frac{ \left ({b}^{3}A+3\,a{b}^{2}B \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{2}bA+{a}^{3}B \right ){x}^{3}}{3}}+{\frac{{a}^{3}A{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01319, size = 99, normalized size = 1.62 \begin{align*} \frac{1}{6} \, B b^{3} x^{6} + \frac{1}{2} \, A a^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42487, size = 177, normalized size = 2.9 \begin{align*} \frac{1}{6} x^{6} b^{3} B + \frac{3}{5} x^{5} b^{2} a B + \frac{1}{5} x^{5} b^{3} A + \frac{3}{4} x^{4} b a^{2} B + \frac{3}{4} x^{4} b^{2} a A + \frac{1}{3} x^{3} a^{3} B + x^{3} b a^{2} A + \frac{1}{2} x^{2} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.077233, size = 80, normalized size = 1.31 \begin{align*} \frac{A a^{3} x^{2}}{2} + \frac{B b^{3} x^{6}}{6} + x^{5} \left (\frac{A b^{3}}{5} + \frac{3 B a b^{2}}{5}\right ) + x^{4} \left (\frac{3 A a b^{2}}{4} + \frac{3 B a^{2} b}{4}\right ) + x^{3} \left (A a^{2} b + \frac{B a^{3}}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2018, size = 103, normalized size = 1.69 \begin{align*} \frac{1}{6} \, B b^{3} x^{6} + \frac{3}{5} \, B a b^{2} x^{5} + \frac{1}{5} \, A b^{3} x^{5} + \frac{3}{4} \, B a^{2} b x^{4} + \frac{3}{4} \, A a b^{2} x^{4} + \frac{1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + \frac{1}{2} \, A a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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