Optimal. Leaf size=99 \[ \frac {1}{4} a^4 A x^4+\frac {1}{5} a^3 x^5 (a B+4 A b)+\frac {1}{3} a^2 b x^6 (2 a B+3 A b)+\frac {1}{8} b^3 x^8 (4 a B+A b)+\frac {2}{7} a b^2 x^7 (3 a B+2 A b)+\frac {1}{9} b^4 B x^9 \]
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Rubi [A] time = 0.06, 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 {1}{3} a^2 b x^6 (2 a B+3 A b)+\frac {1}{5} a^3 x^5 (a B+4 A b)+\frac {1}{4} a^4 A x^4+\frac {1}{8} b^3 x^8 (4 a B+A b)+\frac {2}{7} a b^2 x^7 (3 a B+2 A b)+\frac {1}{9} b^4 B x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x^3 (a+b x)^4 (A+B x) \, dx\\ &=\int \left (a^4 A x^3+a^3 (4 A b+a B) x^4+2 a^2 b (3 A b+2 a B) x^5+2 a b^2 (2 A b+3 a B) x^6+b^3 (A b+4 a B) x^7+b^4 B x^8\right ) \, dx\\ &=\frac {1}{4} a^4 A x^4+\frac {1}{5} a^3 (4 A b+a B) x^5+\frac {1}{3} a^2 b (3 A b+2 a B) x^6+\frac {2}{7} a b^2 (2 A b+3 a B) x^7+\frac {1}{8} b^3 (A b+4 a B) x^8+\frac {1}{9} b^4 B x^9\\ \end {align*}
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Mathematica [A] time = 0.01, size = 99, normalized size = 1.00 \begin {gather*} \frac {1}{4} a^4 A x^4+\frac {1}{5} a^3 x^5 (a B+4 A b)+\frac {1}{3} a^2 b x^6 (2 a B+3 A b)+\frac {1}{8} b^3 x^8 (4 a B+A b)+\frac {2}{7} a b^2 x^7 (3 a B+2 A b)+\frac {1}{9} b^4 B x^9 \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^3 (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 = 0.36, size = 100, normalized size = 1.01 \begin {gather*} \frac {1}{9} x^{9} b^{4} B + \frac {1}{2} x^{8} b^{3} a B + \frac {1}{8} x^{8} b^{4} A + \frac {6}{7} x^{7} b^{2} a^{2} B + \frac {4}{7} x^{7} b^{3} a A + \frac {2}{3} x^{6} b a^{3} B + x^{6} b^{2} a^{2} A + \frac {1}{5} x^{5} a^{4} B + \frac {4}{5} x^{5} b a^{3} A + \frac {1}{4} x^{4} a^{4} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 100, normalized size = 1.01 \begin {gather*} \frac {1}{9} \, B b^{4} x^{9} + \frac {1}{2} \, B a b^{3} x^{8} + \frac {1}{8} \, A b^{4} x^{8} + \frac {6}{7} \, B a^{2} b^{2} x^{7} + \frac {4}{7} \, A a b^{3} x^{7} + \frac {2}{3} \, B a^{3} b x^{6} + A a^{2} b^{2} x^{6} + \frac {1}{5} \, B a^{4} x^{5} + \frac {4}{5} \, A a^{3} b x^{5} + \frac {1}{4} \, A a^{4} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 100, normalized size = 1.01 \begin {gather*} \frac {B \,b^{4} x^{9}}{9}+\frac {A \,a^{4} x^{4}}{4}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{8}}{8}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{7}}{7}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{6}}{6}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 99, normalized size = 1.00 \begin {gather*} \frac {1}{9} \, B b^{4} x^{9} + \frac {1}{4} \, A a^{4} x^{4} + \frac {1}{8} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{8} + \frac {2}{7} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{7} + \frac {1}{3} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 91, normalized size = 0.92 \begin {gather*} x^5\,\left (\frac {B\,a^4}{5}+\frac {4\,A\,b\,a^3}{5}\right )+x^8\,\left (\frac {A\,b^4}{8}+\frac {B\,a\,b^3}{2}\right )+\frac {A\,a^4\,x^4}{4}+\frac {B\,b^4\,x^9}{9}+\frac {a^2\,b\,x^6\,\left (3\,A\,b+2\,B\,a\right )}{3}+\frac {2\,a\,b^2\,x^7\,\left (2\,A\,b+3\,B\,a\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 105, normalized size = 1.06 \begin {gather*} \frac {A a^{4} x^{4}}{4} + \frac {B b^{4} x^{9}}{9} + x^{8} \left (\frac {A b^{4}}{8} + \frac {B a b^{3}}{2}\right ) + x^{7} \left (\frac {4 A a b^{3}}{7} + \frac {6 B a^{2} b^{2}}{7}\right ) + x^{6} \left (A a^{2} b^{2} + \frac {2 B a^{3} b}{3}\right ) + x^{5} \left (\frac {4 A a^{3} b}{5} + \frac {B a^{4}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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