Optimal. Leaf size=87 \[ \frac {a^2 (a+b x)^7 (A b-a B)}{7 b^4}+\frac {(a+b x)^9 (A b-3 a B)}{9 b^4}-\frac {a (a+b x)^8 (2 A b-3 a B)}{8 b^4}+\frac {B (a+b x)^{10}}{10 b^4} \]
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Rubi [A] time = 0.06, antiderivative size = 87, 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 {a^2 (a+b x)^7 (A b-a B)}{7 b^4}+\frac {(a+b x)^9 (A b-3 a B)}{9 b^4}-\frac {a (a+b x)^8 (2 A b-3 a B)}{8 b^4}+\frac {B (a+b x)^{10}}{10 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^2 (a+b x)^6 (A+B x) \, dx\\ &=\int \left (-\frac {a^2 (-A b+a B) (a+b x)^6}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^7}{b^3}+\frac {(A b-3 a B) (a+b x)^8}{b^3}+\frac {B (a+b x)^9}{b^3}\right ) \, dx\\ &=\frac {a^2 (A b-a B) (a+b x)^7}{7 b^4}-\frac {a (2 A b-3 a B) (a+b x)^8}{8 b^4}+\frac {(A b-3 a B) (a+b x)^9}{9 b^4}+\frac {B (a+b x)^{10}}{10 b^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 143, normalized size = 1.64 \begin {gather*} \frac {1}{3} a^6 A x^3+\frac {1}{4} a^5 x^4 (a B+6 A b)+\frac {3}{5} a^4 b x^5 (2 a B+5 A b)+\frac {5}{6} a^3 b^2 x^6 (3 a B+4 A b)+\frac {5}{7} a^2 b^3 x^7 (4 a B+3 A b)+\frac {1}{9} b^5 x^9 (6 a B+A b)+\frac {3}{8} a b^4 x^8 (5 a B+2 A b)+\frac {1}{10} b^6 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^2 (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 = 0.38, size = 149, normalized size = 1.71 \begin {gather*} \frac {1}{10} x^{10} b^{6} B + \frac {2}{3} x^{9} b^{5} a B + \frac {1}{9} x^{9} b^{6} A + \frac {15}{8} x^{8} b^{4} a^{2} B + \frac {3}{4} x^{8} b^{5} a A + \frac {20}{7} x^{7} b^{3} a^{3} B + \frac {15}{7} x^{7} b^{4} a^{2} A + \frac {5}{2} x^{6} b^{2} a^{4} B + \frac {10}{3} x^{6} b^{3} a^{3} A + \frac {6}{5} x^{5} b a^{5} B + 3 x^{5} b^{2} a^{4} A + \frac {1}{4} x^{4} a^{6} B + \frac {3}{2} x^{4} b a^{5} A + \frac {1}{3} x^{3} a^{6} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 149, normalized size = 1.71 \begin {gather*} \frac {1}{10} \, B b^{6} x^{10} + \frac {2}{3} \, B a b^{5} x^{9} + \frac {1}{9} \, A b^{6} x^{9} + \frac {15}{8} \, B a^{2} b^{4} x^{8} + \frac {3}{4} \, A a b^{5} x^{8} + \frac {20}{7} \, B a^{3} b^{3} x^{7} + \frac {15}{7} \, A a^{2} b^{4} x^{7} + \frac {5}{2} \, B a^{4} b^{2} x^{6} + \frac {10}{3} \, A a^{3} b^{3} x^{6} + \frac {6}{5} \, B a^{5} b x^{5} + 3 \, A a^{4} b^{2} x^{5} + \frac {1}{4} \, B a^{6} x^{4} + \frac {3}{2} \, A a^{5} b x^{4} + \frac {1}{3} \, A a^{6} x^{3} \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 = 1.70 \begin {gather*} \frac {B \,b^{6} x^{10}}{10}+\frac {A \,a^{6} x^{3}}{3}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{9}}{9}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{8}}{8}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{7}}{7}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{6}}{6}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{5}}{5}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{4}}{4} \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 = 1.69 \begin {gather*} \frac {1}{10} \, B b^{6} x^{10} + \frac {1}{3} \, A a^{6} x^{3} + \frac {1}{9} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{9} + \frac {3}{8} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{8} + \frac {5}{7} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{7} + \frac {5}{6} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{6} + \frac {3}{5} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 131, normalized size = 1.51 \begin {gather*} x^4\,\left (\frac {B\,a^6}{4}+\frac {3\,A\,b\,a^5}{2}\right )+x^9\,\left (\frac {A\,b^6}{9}+\frac {2\,B\,a\,b^5}{3}\right )+\frac {A\,a^6\,x^3}{3}+\frac {B\,b^6\,x^{10}}{10}+\frac {5\,a^3\,b^2\,x^6\,\left (4\,A\,b+3\,B\,a\right )}{6}+\frac {5\,a^2\,b^3\,x^7\,\left (3\,A\,b+4\,B\,a\right )}{7}+\frac {3\,a^4\,b\,x^5\,\left (5\,A\,b+2\,B\,a\right )}{5}+\frac {3\,a\,b^4\,x^8\,\left (2\,A\,b+5\,B\,a\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 163, normalized size = 1.87 \begin {gather*} \frac {A a^{6} x^{3}}{3} + \frac {B b^{6} x^{10}}{10} + x^{9} \left (\frac {A b^{6}}{9} + \frac {2 B a b^{5}}{3}\right ) + x^{8} \left (\frac {3 A a b^{5}}{4} + \frac {15 B a^{2} b^{4}}{8}\right ) + x^{7} \left (\frac {15 A a^{2} b^{4}}{7} + \frac {20 B a^{3} b^{3}}{7}\right ) + x^{6} \left (\frac {10 A a^{3} b^{3}}{3} + \frac {5 B a^{4} b^{2}}{2}\right ) + x^{5} \left (3 A a^{4} b^{2} + \frac {6 B a^{5} b}{5}\right ) + x^{4} \left (\frac {3 A a^{5} b}{2} + \frac {B a^{6}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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