Optimal. Leaf size=139 \[ \frac{2 a^2 (a+b x)^9 (3 A b-5 a B)}{9 b^6}-\frac{a^3 (a+b x)^8 (4 A b-5 a B)}{8 b^6}+\frac{a^4 (a+b x)^7 (A b-a B)}{7 b^6}+\frac{(a+b x)^{11} (A b-5 a B)}{11 b^6}-\frac{a (a+b x)^{10} (2 A b-5 a B)}{5 b^6}+\frac{B (a+b x)^{12}}{12 b^6} \]
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Rubi [A] time = 0.0975585, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac{2 a^2 (a+b x)^9 (3 A b-5 a B)}{9 b^6}-\frac{a^3 (a+b x)^8 (4 A b-5 a B)}{8 b^6}+\frac{a^4 (a+b x)^7 (A b-a B)}{7 b^6}+\frac{(a+b x)^{11} (A b-5 a B)}{11 b^6}-\frac{a (a+b x)^{10} (2 A b-5 a B)}{5 b^6}+\frac{B (a+b x)^{12}}{12 b^6} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^4 (a+b x)^6 (A+B x) \, dx\\ &=\int \left (-\frac{a^4 (-A b+a B) (a+b x)^6}{b^5}+\frac{a^3 (-4 A b+5 a B) (a+b x)^7}{b^5}-\frac{2 a^2 (-3 A b+5 a B) (a+b x)^8}{b^5}+\frac{2 a (-2 A b+5 a B) (a+b x)^9}{b^5}+\frac{(A b-5 a B) (a+b x)^{10}}{b^5}+\frac{B (a+b x)^{11}}{b^5}\right ) \, dx\\ &=\frac{a^4 (A b-a B) (a+b x)^7}{7 b^6}-\frac{a^3 (4 A b-5 a B) (a+b x)^8}{8 b^6}+\frac{2 a^2 (3 A b-5 a B) (a+b x)^9}{9 b^6}-\frac{a (2 A b-5 a B) (a+b x)^{10}}{5 b^6}+\frac{(A b-5 a B) (a+b x)^{11}}{11 b^6}+\frac{B (a+b x)^{12}}{12 b^6}\\ \end{align*}
Mathematica [A] time = 0.0177665, size = 143, normalized size = 1.03 \[ \frac{5}{9} a^2 b^3 x^9 (4 a B+3 A b)+\frac{5}{8} a^3 b^2 x^8 (3 a B+4 A b)+\frac{3}{7} a^4 b x^7 (2 a B+5 A b)+\frac{1}{6} a^5 x^6 (a B+6 A b)+\frac{1}{5} a^6 A x^5+\frac{1}{11} b^5 x^{11} (6 a B+A b)+\frac{3}{10} a b^4 x^{10} (5 a B+2 A b)+\frac{1}{12} b^6 B x^{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 148, normalized size = 1.1 \begin{align*}{\frac{B{b}^{6}{x}^{12}}{12}}+{\frac{ \left ( A{b}^{6}+6\,Ba{b}^{5} \right ){x}^{11}}{11}}+{\frac{ \left ( 6\,Aa{b}^{5}+15\,B{a}^{2}{b}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 15\,A{a}^{2}{b}^{4}+20\,B{a}^{3}{b}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 20\,A{a}^{3}{b}^{3}+15\,B{a}^{4}{b}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,A{a}^{4}{b}^{2}+6\,B{a}^{5}b \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,A{a}^{5}b+B{a}^{6} \right ){x}^{6}}{6}}+{\frac{A{a}^{6}{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02903, size = 198, normalized size = 1.42 \begin{align*} \frac{1}{12} \, B b^{6} x^{12} + \frac{1}{5} \, A a^{6} x^{5} + \frac{1}{11} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{11} + \frac{3}{10} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{10} + \frac{5}{9} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{9} + \frac{5}{8} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{8} + \frac{3}{7} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35291, size = 354, normalized size = 2.55 \begin{align*} \frac{1}{12} x^{12} b^{6} B + \frac{6}{11} x^{11} b^{5} a B + \frac{1}{11} x^{11} b^{6} A + \frac{3}{2} x^{10} b^{4} a^{2} B + \frac{3}{5} x^{10} b^{5} a A + \frac{20}{9} x^{9} b^{3} a^{3} B + \frac{5}{3} x^{9} b^{4} a^{2} A + \frac{15}{8} x^{8} b^{2} a^{4} B + \frac{5}{2} x^{8} b^{3} a^{3} A + \frac{6}{7} x^{7} b a^{5} B + \frac{15}{7} x^{7} b^{2} a^{4} A + \frac{1}{6} x^{6} a^{6} B + x^{6} b a^{5} A + \frac{1}{5} x^{5} a^{6} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.096977, size = 162, normalized size = 1.17 \begin{align*} \frac{A a^{6} x^{5}}{5} + \frac{B b^{6} x^{12}}{12} + x^{11} \left (\frac{A b^{6}}{11} + \frac{6 B a b^{5}}{11}\right ) + x^{10} \left (\frac{3 A a b^{5}}{5} + \frac{3 B a^{2} b^{4}}{2}\right ) + x^{9} \left (\frac{5 A a^{2} b^{4}}{3} + \frac{20 B a^{3} b^{3}}{9}\right ) + x^{8} \left (\frac{5 A a^{3} b^{3}}{2} + \frac{15 B a^{4} b^{2}}{8}\right ) + x^{7} \left (\frac{15 A a^{4} b^{2}}{7} + \frac{6 B a^{5} b}{7}\right ) + x^{6} \left (A a^{5} b + \frac{B a^{6}}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13472, size = 200, normalized size = 1.44 \begin{align*} \frac{1}{12} \, B b^{6} x^{12} + \frac{6}{11} \, B a b^{5} x^{11} + \frac{1}{11} \, A b^{6} x^{11} + \frac{3}{2} \, B a^{2} b^{4} x^{10} + \frac{3}{5} \, A a b^{5} x^{10} + \frac{20}{9} \, B a^{3} b^{3} x^{9} + \frac{5}{3} \, A a^{2} b^{4} x^{9} + \frac{15}{8} \, B a^{4} b^{2} x^{8} + \frac{5}{2} \, A a^{3} b^{3} x^{8} + \frac{6}{7} \, B a^{5} b x^{7} + \frac{15}{7} \, A a^{4} b^{2} x^{7} + \frac{1}{6} \, B a^{6} x^{6} + A a^{5} b x^{6} + \frac{1}{5} \, A a^{6} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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