Optimal. Leaf size=61 \[ \frac {(a+b x)^5 (A b-2 a B)}{5 b^3}-\frac {a (a+b x)^4 (A b-a B)}{4 b^3}+\frac {B (a+b x)^6}{6 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {76} \begin {gather*} \frac {(a+b x)^5 (A b-2 a B)}{5 b^3}-\frac {a (a+b x)^4 (A b-a B)}{4 b^3}+\frac {B (a+b x)^6}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int x (a+b x)^3 (A+B x) \, dx &=\int \left (\frac {a (-A b+a B) (a+b x)^3}{b^2}+\frac {(A b-2 a B) (a+b x)^4}{b^2}+\frac {B (a+b x)^5}{b^2}\right ) \, dx\\ &=-\frac {a (A b-a B) (a+b x)^4}{4 b^3}+\frac {(A b-2 a B) (a+b x)^5}{5 b^3}+\frac {B (a+b x)^6}{6 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 69, normalized size = 1.13 \begin {gather*} \frac {1}{60} x^2 \left (10 a^3 (3 A+2 B x)+15 a^2 b x (4 A+3 B x)+9 a b^2 x^2 (5 A+4 B x)+2 b^3 x^3 (6 A+5 B x)\right ) \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 (a+b x)^3 (A+B x) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.76, size = 76, normalized size = 1.25 \begin {gather*} \frac {1}{6} x^{6} b^{3} B + \frac {3}{5} x^{5} b^{2} a B + \frac {1}{5} x^{5} b^{3} A + \frac {3}{4} x^{4} b a^{2} B + \frac {3}{4} x^{4} b^{2} a A + \frac {1}{3} x^{3} a^{3} B + x^{3} b a^{2} A + \frac {1}{2} x^{2} a^{3} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 76, normalized size = 1.25 \begin {gather*} \frac {1}{6} \, B b^{3} x^{6} + \frac {3}{5} \, B a b^{2} x^{5} + \frac {1}{5} \, A b^{3} x^{5} + \frac {3}{4} \, B a^{2} b x^{4} + \frac {3}{4} \, A a b^{2} x^{4} + \frac {1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + \frac {1}{2} \, A a^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 1.25 \begin {gather*} \frac {B \,b^{3} x^{6}}{6}+\frac {A \,a^{3} x^{2}}{2}+\frac {\left (b^{3} A +3 a \,b^{2} B \right ) x^{5}}{5}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{4}}{4}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 73, normalized size = 1.20 \begin {gather*} \frac {1}{6} \, B b^{3} x^{6} + \frac {1}{2} \, A a^{3} x^{2} + \frac {1}{5} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{5} + \frac {3}{4} \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 68, normalized size = 1.11 \begin {gather*} x^3\,\left (\frac {B\,a^3}{3}+A\,b\,a^2\right )+x^5\,\left (\frac {A\,b^3}{5}+\frac {3\,B\,a\,b^2}{5}\right )+\frac {A\,a^3\,x^2}{2}+\frac {B\,b^3\,x^6}{6}+\frac {3\,a\,b\,x^4\,\left (A\,b+B\,a\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 80, normalized size = 1.31 \begin {gather*} \frac {A a^{3} x^{2}}{2} + \frac {B b^{3} x^{6}}{6} + x^{5} \left (\frac {A b^{3}}{5} + \frac {3 B a b^{2}}{5}\right ) + x^{4} \left (\frac {3 A a b^{2}}{4} + \frac {3 B a^{2} b}{4}\right ) + x^{3} \left (A a^{2} b + \frac {B a^{3}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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