Optimal. Leaf size=38 \[ \frac{(a+b x)^6 (A b-a B)}{6 b^2}+\frac{B (a+b x)^7}{7 b^2} \]
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Rubi [A] time = 0.0141093, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{(a+b x)^6 (A b-a B)}{6 b^2}+\frac{B (a+b x)^7}{7 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^5 (A+B x) \, dx &=\int \left (\frac{(A b-a B) (a+b x)^5}{b}+\frac{B (a+b x)^6}{b}\right ) \, dx\\ &=\frac{(A b-a B) (a+b x)^6}{6 b^2}+\frac{B (a+b x)^7}{7 b^2}\\ \end{align*}
Mathematica [B] time = 0.014118, size = 109, normalized size = 2.87 \[ \frac{5}{2} a^2 b^2 x^4 (a B+A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\frac{1}{2} a^4 x^2 (a B+5 A b)+a^5 A x+\frac{1}{6} b^4 x^6 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 121, normalized size = 3.2 \begin{align*}{\frac{{b}^{5}B{x}^{7}}{7}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{2}}{2}}+{a}^{5}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01734, size = 155, normalized size = 4.08 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} + A a^{5} x + \frac{1}{6} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} +{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac{5}{2} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45262, size = 275, normalized size = 7.24 \begin{align*} \frac{1}{7} x^{7} b^{5} B + \frac{5}{6} x^{6} b^{4} a B + \frac{1}{6} x^{6} b^{5} A + 2 x^{5} b^{3} a^{2} B + x^{5} b^{4} a A + \frac{5}{2} x^{4} b^{2} a^{3} B + \frac{5}{2} x^{4} b^{3} a^{2} A + \frac{5}{3} x^{3} b a^{4} B + \frac{10}{3} x^{3} b^{2} a^{3} A + \frac{1}{2} x^{2} a^{5} B + \frac{5}{2} x^{2} b a^{4} A + x a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.08903, size = 129, normalized size = 3.39 \begin{align*} A a^{5} x + \frac{B b^{5} x^{7}}{7} + x^{6} \left (\frac{A b^{5}}{6} + \frac{5 B a b^{4}}{6}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{4} \left (\frac{5 A a^{2} b^{3}}{2} + \frac{5 B a^{3} b^{2}}{2}\right ) + x^{3} \left (\frac{10 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{3}\right ) + x^{2} \left (\frac{5 A a^{4} b}{2} + \frac{B a^{5}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19099, size = 163, normalized size = 4.29 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} + \frac{5}{6} \, B a b^{4} x^{6} + \frac{1}{6} \, A b^{5} x^{6} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + \frac{5}{2} \, B a^{3} b^{2} x^{4} + \frac{5}{2} \, A a^{2} b^{3} x^{4} + \frac{5}{3} \, B a^{4} b x^{3} + \frac{10}{3} \, A a^{3} b^{2} x^{3} + \frac{1}{2} \, B a^{5} x^{2} + \frac{5}{2} \, A a^{4} b x^{2} + A a^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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