Optimal. Leaf size=38 \[ \frac {(a+b x)^4 (A b-a B)}{4 b^2}+\frac {B (a+b x)^5}{5 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ \frac {(a+b x)^4 (A b-a B)}{4 b^2}+\frac {B (a+b x)^5}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^3 (A+B x) \, dx &=\int \left (\frac {(A b-a B) (a+b x)^3}{b}+\frac {B (a+b x)^4}{b}\right ) \, dx\\ &=\frac {(A b-a B) (a+b x)^4}{4 b^2}+\frac {B (a+b x)^5}{5 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 67, normalized size = 1.76 \[ a^3 A x+\frac {1}{2} a^2 x^2 (a B+3 A b)+\frac {1}{4} b^2 x^4 (3 a B+A b)+a b x^3 (a B+A b)+\frac {1}{5} b^3 B x^5 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 72, normalized size = 1.89 \[ \frac {1}{5} x^{5} b^{3} B + \frac {3}{4} x^{4} b^{2} a B + \frac {1}{4} x^{4} b^{3} A + x^{3} b a^{2} B + x^{3} b^{2} a A + \frac {1}{2} x^{2} a^{3} B + \frac {3}{2} x^{2} b a^{2} A + x a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.32, size = 72, normalized size = 1.89 \[ \frac {1}{5} \, B b^{3} x^{5} + \frac {3}{4} \, B a b^{2} x^{4} + \frac {1}{4} \, A b^{3} x^{4} + B a^{2} b x^{3} + A a b^{2} x^{3} + \frac {1}{2} \, B a^{3} x^{2} + \frac {3}{2} \, A a^{2} b x^{2} + A a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 73, normalized size = 1.92 \[ \frac {B \,b^{3} x^{5}}{5}+A \,a^{3} x +\frac {\left (b^{3} A +3 a \,b^{2} B \right ) x^{4}}{4}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{3}}{3}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 69, normalized size = 1.82 \[ \frac {1}{5} \, B b^{3} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + {\left (B a^{2} b + A a b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 65, normalized size = 1.71 \[ x^2\,\left (\frac {B\,a^3}{2}+\frac {3\,A\,b\,a^2}{2}\right )+x^4\,\left (\frac {A\,b^3}{4}+\frac {3\,B\,a\,b^2}{4}\right )+\frac {B\,b^3\,x^5}{5}+A\,a^3\,x+a\,b\,x^3\,\left (A\,b+B\,a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 73, normalized size = 1.92 \[ A a^{3} x + \frac {B b^{3} x^{5}}{5} + x^{4} \left (\frac {A b^{3}}{4} + \frac {3 B a b^{2}}{4}\right ) + x^{3} \left (A a b^{2} + B a^{2} b\right ) + x^{2} \left (\frac {3 A a^{2} b}{2} + \frac {B a^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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