Optimal. Leaf size=143 \[ \frac {1}{6} a^6 A x^6+\frac {1}{7} a^5 x^7 (a B+6 A b)+\frac {3}{8} a^4 b x^8 (2 a B+5 A b)+\frac {5}{9} a^3 b^2 x^9 (3 a B+4 A b)+\frac {1}{2} a^2 b^3 x^{10} (4 a B+3 A b)+\frac {1}{12} b^5 x^{12} (6 a B+A b)+\frac {3}{11} a b^4 x^{11} (5 a B+2 A b)+\frac {1}{13} b^6 B x^{13} \]
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Rubi [A] time = 0.14, antiderivative size = 143, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{2} a^2 b^3 x^{10} (4 a B+3 A b)+\frac {5}{9} a^3 b^2 x^9 (3 a B+4 A b)+\frac {3}{8} a^4 b x^8 (2 a B+5 A b)+\frac {1}{7} a^5 x^7 (a B+6 A b)+\frac {1}{6} a^6 A x^6+\frac {1}{12} b^5 x^{12} (6 a B+A b)+\frac {3}{11} a b^4 x^{11} (5 a B+2 A b)+\frac {1}{13} b^6 B x^{13} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^5 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^5 (a+b x)^6 (A+B x) \, dx\\ &=\int \left (a^6 A x^5+a^5 (6 A b+a B) x^6+3 a^4 b (5 A b+2 a B) x^7+5 a^3 b^2 (4 A b+3 a B) x^8+5 a^2 b^3 (3 A b+4 a B) x^9+3 a b^4 (2 A b+5 a B) x^{10}+b^5 (A b+6 a B) x^{11}+b^6 B x^{12}\right ) \, dx\\ &=\frac {1}{6} a^6 A x^6+\frac {1}{7} a^5 (6 A b+a B) x^7+\frac {3}{8} a^4 b (5 A b+2 a B) x^8+\frac {5}{9} a^3 b^2 (4 A b+3 a B) x^9+\frac {1}{2} a^2 b^3 (3 A b+4 a B) x^{10}+\frac {3}{11} a b^4 (2 A b+5 a B) x^{11}+\frac {1}{12} b^5 (A b+6 a B) x^{12}+\frac {1}{13} b^6 B x^{13}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 143, normalized size = 1.00 \[ \frac {1}{6} a^6 A x^6+\frac {1}{7} a^5 x^7 (a B+6 A b)+\frac {3}{8} a^4 b x^8 (2 a B+5 A b)+\frac {5}{9} a^3 b^2 x^9 (3 a B+4 A b)+\frac {1}{2} a^2 b^3 x^{10} (4 a B+3 A b)+\frac {1}{12} b^5 x^{12} (6 a B+A b)+\frac {3}{11} a b^4 x^{11} (5 a B+2 A b)+\frac {1}{13} b^6 B x^{13} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 149, normalized size = 1.04 \[ \frac {1}{13} x^{13} b^{6} B + \frac {1}{2} x^{12} b^{5} a B + \frac {1}{12} x^{12} b^{6} A + \frac {15}{11} x^{11} b^{4} a^{2} B + \frac {6}{11} x^{11} b^{5} a A + 2 x^{10} b^{3} a^{3} B + \frac {3}{2} x^{10} b^{4} a^{2} A + \frac {5}{3} x^{9} b^{2} a^{4} B + \frac {20}{9} x^{9} b^{3} a^{3} A + \frac {3}{4} x^{8} b a^{5} B + \frac {15}{8} x^{8} b^{2} a^{4} A + \frac {1}{7} x^{7} a^{6} B + \frac {6}{7} x^{7} b a^{5} A + \frac {1}{6} x^{6} a^{6} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 149, normalized size = 1.04 \[ \frac {1}{13} \, B b^{6} x^{13} + \frac {1}{2} \, B a b^{5} x^{12} + \frac {1}{12} \, A b^{6} x^{12} + \frac {15}{11} \, B a^{2} b^{4} x^{11} + \frac {6}{11} \, A a b^{5} x^{11} + 2 \, B a^{3} b^{3} x^{10} + \frac {3}{2} \, A a^{2} b^{4} x^{10} + \frac {5}{3} \, B a^{4} b^{2} x^{9} + \frac {20}{9} \, A a^{3} b^{3} x^{9} + \frac {3}{4} \, B a^{5} b x^{8} + \frac {15}{8} \, A a^{4} b^{2} x^{8} + \frac {1}{7} \, B a^{6} x^{7} + \frac {6}{7} \, A a^{5} b x^{7} + \frac {1}{6} \, A a^{6} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 148, normalized size = 1.03 \[ \frac {B \,b^{6} x^{13}}{13}+\frac {A \,a^{6} x^{6}}{6}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{12}}{12}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{11}}{11}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{10}}{10}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{9}}{9}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{8}}{8}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{7}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 147, normalized size = 1.03 \[ \frac {1}{13} \, B b^{6} x^{13} + \frac {1}{6} \, A a^{6} x^{6} + \frac {1}{12} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{10} + \frac {5}{9} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{9} + \frac {3}{8} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 131, normalized size = 0.92 \[ x^7\,\left (\frac {B\,a^6}{7}+\frac {6\,A\,b\,a^5}{7}\right )+x^{12}\,\left (\frac {A\,b^6}{12}+\frac {B\,a\,b^5}{2}\right )+\frac {A\,a^6\,x^6}{6}+\frac {B\,b^6\,x^{13}}{13}+\frac {5\,a^3\,b^2\,x^9\,\left (4\,A\,b+3\,B\,a\right )}{9}+\frac {a^2\,b^3\,x^{10}\,\left (3\,A\,b+4\,B\,a\right )}{2}+\frac {3\,a^4\,b\,x^8\,\left (5\,A\,b+2\,B\,a\right )}{8}+\frac {3\,a\,b^4\,x^{11}\,\left (2\,A\,b+5\,B\,a\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 162, normalized size = 1.13 \[ \frac {A a^{6} x^{6}}{6} + \frac {B b^{6} x^{13}}{13} + x^{12} \left (\frac {A b^{6}}{12} + \frac {B a b^{5}}{2}\right ) + x^{11} \left (\frac {6 A a b^{5}}{11} + \frac {15 B a^{2} b^{4}}{11}\right ) + x^{10} \left (\frac {3 A a^{2} b^{4}}{2} + 2 B a^{3} b^{3}\right ) + x^{9} \left (\frac {20 A a^{3} b^{3}}{9} + \frac {5 B a^{4} b^{2}}{3}\right ) + x^{8} \left (\frac {15 A a^{4} b^{2}}{8} + \frac {3 B a^{5} b}{4}\right ) + x^{7} \left (\frac {6 A a^{5} b}{7} + \frac {B a^{6}}{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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