Optimal. Leaf size=87 \[ \frac{a^2 (a+b x)^6 (A b-a B)}{6 b^4}+\frac{(a+b x)^8 (A b-3 a B)}{8 b^4}-\frac{a (a+b x)^7 (2 A b-3 a B)}{7 b^4}+\frac{B (a+b x)^9}{9 b^4} \]
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Rubi [A] time = 0.0561752, antiderivative size = 87, 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 = {76} \[ \frac{a^2 (a+b x)^6 (A b-a B)}{6 b^4}+\frac{(a+b x)^8 (A b-3 a B)}{8 b^4}-\frac{a (a+b x)^7 (2 A b-3 a B)}{7 b^4}+\frac{B (a+b x)^9}{9 b^4} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^2 (a+b x)^5 (A+B x) \, dx &=\int \left (-\frac{a^2 (-A b+a B) (a+b x)^5}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^6}{b^3}+\frac{(A b-3 a B) (a+b x)^7}{b^3}+\frac{B (a+b x)^8}{b^3}\right ) \, dx\\ &=\frac{a^2 (A b-a B) (a+b x)^6}{6 b^4}-\frac{a (2 A b-3 a B) (a+b x)^7}{7 b^4}+\frac{(A b-3 a B) (a+b x)^8}{8 b^4}+\frac{B (a+b x)^9}{9 b^4}\\ \end{align*}
Mathematica [A] time = 0.0146353, size = 114, normalized size = 1.31 \[ \frac{5}{3} a^2 b^2 x^6 (a B+A b)+a^3 b x^5 (a B+2 A b)+\frac{1}{4} a^4 x^4 (a B+5 A b)+\frac{1}{3} a^5 A x^3+\frac{1}{8} b^4 x^8 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{9} b^5 B x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 124, normalized size = 1.4 \begin{align*}{\frac{{b}^{5}B{x}^{9}}{9}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{8}}{8}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{7}}{7}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{6}}{6}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{5}}{5}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{4}}{4}}+{\frac{{a}^{5}A{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00918, size = 159, normalized size = 1.83 \begin{align*} \frac{1}{9} \, B b^{5} x^{9} + \frac{1}{3} \, A a^{5} x^{3} + \frac{1}{8} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + \frac{5}{7} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac{5}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} +{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39074, size = 284, normalized size = 3.26 \begin{align*} \frac{1}{9} x^{9} b^{5} B + \frac{5}{8} x^{8} b^{4} a B + \frac{1}{8} x^{8} b^{5} A + \frac{10}{7} x^{7} b^{3} a^{2} B + \frac{5}{7} x^{7} b^{4} a A + \frac{5}{3} x^{6} b^{2} a^{3} B + \frac{5}{3} x^{6} b^{3} a^{2} A + x^{5} b a^{4} B + 2 x^{5} b^{2} a^{3} A + \frac{1}{4} x^{4} a^{5} B + \frac{5}{4} x^{4} b a^{4} A + \frac{1}{3} x^{3} a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.102232, size = 133, normalized size = 1.53 \begin{align*} \frac{A a^{5} x^{3}}{3} + \frac{B b^{5} x^{9}}{9} + x^{8} \left (\frac{A b^{5}}{8} + \frac{5 B a b^{4}}{8}\right ) + x^{7} \left (\frac{5 A a b^{4}}{7} + \frac{10 B a^{2} b^{3}}{7}\right ) + x^{6} \left (\frac{5 A a^{2} b^{3}}{3} + \frac{5 B a^{3} b^{2}}{3}\right ) + x^{5} \left (2 A a^{3} b^{2} + B a^{4} b\right ) + x^{4} \left (\frac{5 A a^{4} b}{4} + \frac{B a^{5}}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2337, size = 167, normalized size = 1.92 \begin{align*} \frac{1}{9} \, B b^{5} x^{9} + \frac{5}{8} \, B a b^{4} x^{8} + \frac{1}{8} \, A b^{5} x^{8} + \frac{10}{7} \, B a^{2} b^{3} x^{7} + \frac{5}{7} \, A a b^{4} x^{7} + \frac{5}{3} \, B a^{3} b^{2} x^{6} + \frac{5}{3} \, A a^{2} b^{3} x^{6} + B a^{4} b x^{5} + 2 \, A a^{3} b^{2} x^{5} + \frac{1}{4} \, B a^{5} x^{4} + \frac{5}{4} \, A a^{4} b x^{4} + \frac{1}{3} \, A a^{5} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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