Optimal. Leaf size=61 \[ \frac {(a+b x)^8 (A b-2 a B)}{8 b^3}-\frac {a (a+b x)^7 (A b-a B)}{7 b^3}+\frac {B (a+b x)^9}{9 b^3} \]
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Rubi [A] time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 76} \[ \frac {(a+b x)^8 (A b-2 a B)}{8 b^3}-\frac {a (a+b x)^7 (A b-a B)}{7 b^3}+\frac {B (a+b x)^9}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x (a+b x)^6 (A+B x) \, dx\\ &=\int \left (\frac {a (-A b+a B) (a+b x)^6}{b^2}+\frac {(A b-2 a B) (a+b x)^7}{b^2}+\frac {B (a+b x)^8}{b^2}\right ) \, dx\\ &=-\frac {a (A b-a B) (a+b x)^7}{7 b^3}+\frac {(A b-2 a B) (a+b x)^8}{8 b^3}+\frac {B (a+b x)^9}{9 b^3}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 140, normalized size = 2.30 \[ \frac {1}{2} a^6 A x^2+\frac {1}{3} a^5 x^3 (a B+6 A b)+\frac {3}{4} a^4 b x^4 (2 a B+5 A b)+a^3 b^2 x^5 (3 a B+4 A b)+\frac {5}{6} a^2 b^3 x^6 (4 a B+3 A b)+\frac {1}{8} b^5 x^8 (6 a B+A b)+\frac {3}{7} a b^4 x^7 (5 a B+2 A b)+\frac {1}{9} b^6 B x^9 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 149, normalized size = 2.44 \[ \frac {1}{9} x^{9} b^{6} B + \frac {3}{4} x^{8} b^{5} a B + \frac {1}{8} x^{8} b^{6} A + \frac {15}{7} x^{7} b^{4} a^{2} B + \frac {6}{7} x^{7} b^{5} a A + \frac {10}{3} x^{6} b^{3} a^{3} B + \frac {5}{2} x^{6} b^{4} a^{2} A + 3 x^{5} b^{2} a^{4} B + 4 x^{5} b^{3} a^{3} A + \frac {3}{2} x^{4} b a^{5} B + \frac {15}{4} x^{4} b^{2} a^{4} A + \frac {1}{3} x^{3} a^{6} B + 2 x^{3} b a^{5} A + \frac {1}{2} x^{2} a^{6} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 149, normalized size = 2.44 \[ \frac {1}{9} \, B b^{6} x^{9} + \frac {3}{4} \, B a b^{5} x^{8} + \frac {1}{8} \, A b^{6} x^{8} + \frac {15}{7} \, B a^{2} b^{4} x^{7} + \frac {6}{7} \, A a b^{5} x^{7} + \frac {10}{3} \, B a^{3} b^{3} x^{6} + \frac {5}{2} \, A a^{2} b^{4} x^{6} + 3 \, B a^{4} b^{2} x^{5} + 4 \, A a^{3} b^{3} x^{5} + \frac {3}{2} \, B a^{5} b x^{4} + \frac {15}{4} \, A a^{4} b^{2} x^{4} + \frac {1}{3} \, B a^{6} x^{3} + 2 \, A a^{5} b x^{3} + \frac {1}{2} \, A a^{6} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 148, normalized size = 2.43 \[ \frac {B \,b^{6} x^{9}}{9}+\frac {A \,a^{6} x^{2}}{2}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{8}}{8}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{7}}{7}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{6}}{6}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{5}}{5}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{4}}{4}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 146, normalized size = 2.39 \[ \frac {1}{9} \, B b^{6} x^{9} + \frac {1}{2} \, A a^{6} x^{2} + \frac {1}{8} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{8} + \frac {3}{7} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{7} + \frac {5}{6} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{6} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{5} + \frac {3}{4} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 130, normalized size = 2.13 \[ x^3\,\left (\frac {B\,a^6}{3}+2\,A\,b\,a^5\right )+x^8\,\left (\frac {A\,b^6}{8}+\frac {3\,B\,a\,b^5}{4}\right )+\frac {A\,a^6\,x^2}{2}+\frac {B\,b^6\,x^9}{9}+a^3\,b^2\,x^5\,\left (4\,A\,b+3\,B\,a\right )+\frac {5\,a^2\,b^3\,x^6\,\left (3\,A\,b+4\,B\,a\right )}{6}+\frac {3\,a^4\,b\,x^4\,\left (5\,A\,b+2\,B\,a\right )}{4}+\frac {3\,a\,b^4\,x^7\,\left (2\,A\,b+5\,B\,a\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 160, normalized size = 2.62 \[ \frac {A a^{6} x^{2}}{2} + \frac {B b^{6} x^{9}}{9} + x^{8} \left (\frac {A b^{6}}{8} + \frac {3 B a b^{5}}{4}\right ) + x^{7} \left (\frac {6 A a b^{5}}{7} + \frac {15 B a^{2} b^{4}}{7}\right ) + x^{6} \left (\frac {5 A a^{2} b^{4}}{2} + \frac {10 B a^{3} b^{3}}{3}\right ) + x^{5} \left (4 A a^{3} b^{3} + 3 B a^{4} b^{2}\right ) + x^{4} \left (\frac {15 A a^{4} b^{2}}{4} + \frac {3 B a^{5} b}{2}\right ) + x^{3} \left (2 A a^{5} b + \frac {B a^{6}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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