Optimal. Leaf size=99 \[ \frac {1}{5} a^4 A x^5+\frac {1}{6} a^3 x^6 (a B+4 A b)+\frac {2}{7} a^2 b x^7 (2 a B+3 A b)+\frac {1}{9} b^3 x^9 (4 a B+A b)+\frac {1}{4} a b^2 x^8 (3 a B+2 A b)+\frac {1}{10} b^4 B x^{10} \]
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Rubi [A] time = 0.08, antiderivative size = 99, 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}{7} a^2 b x^7 (2 a B+3 A b)+\frac {1}{6} a^3 x^6 (a B+4 A b)+\frac {1}{5} a^4 A x^5+\frac {1}{9} b^3 x^9 (4 a B+A b)+\frac {1}{4} a b^2 x^8 (3 a B+2 A b)+\frac {1}{10} b^4 B x^{10} \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 )^2 \, dx &=\int x^4 (a+b x)^4 (A+B x) \, dx\\ &=\int \left (a^4 A x^4+a^3 (4 A b+a B) x^5+2 a^2 b (3 A b+2 a B) x^6+2 a b^2 (2 A b+3 a B) x^7+b^3 (A b+4 a B) x^8+b^4 B x^9\right ) \, dx\\ &=\frac {1}{5} a^4 A x^5+\frac {1}{6} a^3 (4 A b+a B) x^6+\frac {2}{7} a^2 b (3 A b+2 a B) x^7+\frac {1}{4} a b^2 (2 A b+3 a B) x^8+\frac {1}{9} b^3 (A b+4 a B) x^9+\frac {1}{10} b^4 B x^{10}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 99, normalized size = 1.00 \begin {gather*} \frac {1}{5} a^4 A x^5+\frac {1}{6} a^3 x^6 (a B+4 A b)+\frac {2}{7} a^2 b x^7 (2 a B+3 A b)+\frac {1}{9} b^3 x^9 (4 a B+A b)+\frac {1}{4} a b^2 x^8 (3 a B+2 A b)+\frac {1}{10} b^4 B x^{10} \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 )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 101, normalized size = 1.02 \begin {gather*} \frac {1}{10} x^{10} b^{4} B + \frac {4}{9} x^{9} b^{3} a B + \frac {1}{9} x^{9} b^{4} A + \frac {3}{4} x^{8} b^{2} a^{2} B + \frac {1}{2} x^{8} b^{3} a A + \frac {4}{7} x^{7} b a^{3} B + \frac {6}{7} x^{7} b^{2} a^{2} A + \frac {1}{6} x^{6} a^{4} B + \frac {2}{3} x^{6} b a^{3} A + \frac {1}{5} x^{5} a^{4} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 101, normalized size = 1.02 \begin {gather*} \frac {1}{10} \, B b^{4} x^{10} + \frac {4}{9} \, B a b^{3} x^{9} + \frac {1}{9} \, A b^{4} x^{9} + \frac {3}{4} \, B a^{2} b^{2} x^{8} + \frac {1}{2} \, A a b^{3} x^{8} + \frac {4}{7} \, B a^{3} b x^{7} + \frac {6}{7} \, A a^{2} b^{2} x^{7} + \frac {1}{6} \, B a^{4} x^{6} + \frac {2}{3} \, A a^{3} b x^{6} + \frac {1}{5} \, A a^{4} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 100, normalized size = 1.01 \begin {gather*} \frac {B \,b^{4} x^{10}}{10}+\frac {A \,a^{4} x^{5}}{5}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{9}}{9}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{8}}{8}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{7}}{7}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 99, normalized size = 1.00 \begin {gather*} \frac {1}{10} \, B b^{4} x^{10} + \frac {1}{5} \, A a^{4} x^{5} + \frac {1}{9} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{9} + \frac {1}{4} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{8} + \frac {2}{7} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 91, normalized size = 0.92 \begin {gather*} x^6\,\left (\frac {B\,a^4}{6}+\frac {2\,A\,b\,a^3}{3}\right )+x^9\,\left (\frac {A\,b^4}{9}+\frac {4\,B\,a\,b^3}{9}\right )+\frac {A\,a^4\,x^5}{5}+\frac {B\,b^4\,x^{10}}{10}+\frac {2\,a^2\,b\,x^7\,\left (3\,A\,b+2\,B\,a\right )}{7}+\frac {a\,b^2\,x^8\,\left (2\,A\,b+3\,B\,a\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 109, normalized size = 1.10 \begin {gather*} \frac {A a^{4} x^{5}}{5} + \frac {B b^{4} x^{10}}{10} + x^{9} \left (\frac {A b^{4}}{9} + \frac {4 B a b^{3}}{9}\right ) + x^{8} \left (\frac {A a b^{3}}{2} + \frac {3 B a^{2} b^{2}}{4}\right ) + x^{7} \left (\frac {6 A a^{2} b^{2}}{7} + \frac {4 B a^{3} b}{7}\right ) + x^{6} \left (\frac {2 A a^{3} b}{3} + \frac {B a^{4}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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