Optimal. Leaf size=139 \[ \frac {a^4 (a+b x)^7 (A b-a B)}{7 b^6}-\frac {a^3 (a+b x)^8 (4 A b-5 a B)}{8 b^6}+\frac {2 a^2 (a+b x)^9 (3 A b-5 a B)}{9 b^6}+\frac {(a+b x)^{11} (A b-5 a B)}{11 b^6}-\frac {a (a+b x)^{10} (2 A b-5 a B)}{5 b^6}+\frac {B (a+b x)^{12}}{12 b^6} \]
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Rubi [A] time = 0.10, antiderivative size = 139, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [A] time = 0.02, size = 143, normalized size = 1.03 \begin {gather*} \frac {1}{5} a^6 A x^5+\frac {1}{6} a^5 x^6 (a B+6 A b)+\frac {3}{7} a^4 b x^7 (2 a B+5 A b)+\frac {5}{8} a^3 b^2 x^8 (3 a B+4 A b)+\frac {5}{9} a^2 b^3 x^9 (4 a B+3 A b)+\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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 148, normalized size = 1.06 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 148, normalized size = 1.06 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 148, normalized size = 1.06 \begin {gather*} \frac {B \,b^{6} x^{12}}{12}+\frac {A \,a^{6} x^{5}}{5}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{11}}{11}+\frac {\left (6 A a \,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 147, normalized size = 1.06 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 130, normalized size = 0.94 \begin {gather*} x^6\,\left (\frac {B\,a^6}{6}+A\,b\,a^5\right )+x^{11}\,\left (\frac {A\,b^6}{11}+\frac {6\,B\,a\,b^5}{11}\right )+\frac {A\,a^6\,x^5}{5}+\frac {B\,b^6\,x^{12}}{12}+\frac {5\,a^3\,b^2\,x^8\,\left (4\,A\,b+3\,B\,a\right )}{8}+\frac {5\,a^2\,b^3\,x^9\,\left (3\,A\,b+4\,B\,a\right )}{9}+\frac {3\,a^4\,b\,x^7\,\left (5\,A\,b+2\,B\,a\right )}{7}+\frac {3\,a\,b^4\,x^{10}\,\left (2\,A\,b+5\,B\,a\right )}{10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 162, normalized size = 1.17 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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