Optimal. Leaf size=65 \[ \frac {2 d (a+b x)^5 (b c-a d)}{5 b^3}+\frac {(a+b x)^4 (b c-a d)^2}{4 b^3}+\frac {d^2 (a+b x)^6}{6 b^3} \]
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Rubi [A] time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} \frac {2 d (a+b x)^5 (b c-a d)}{5 b^3}+\frac {(a+b x)^4 (b c-a d)^2}{4 b^3}+\frac {d^2 (a+b x)^6}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int (a+b x)^3 (c+d x)^2 \, dx &=\int \left (\frac {(b c-a d)^2 (a+b x)^3}{b^2}+\frac {2 d (b c-a d) (a+b x)^4}{b^2}+\frac {d^2 (a+b x)^5}{b^2}\right ) \, dx\\ &=\frac {(b c-a d)^2 (a+b x)^4}{4 b^3}+\frac {2 d (b c-a d) (a+b x)^5}{5 b^3}+\frac {d^2 (a+b x)^6}{6 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 122, normalized size = 1.88 \begin {gather*} a^3 c^2 x+\frac {1}{4} b x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac {1}{3} a x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{2} a^2 c x^2 (2 a d+3 b c)+\frac {1}{5} b^2 d x^5 (3 a d+2 b c)+\frac {1}{6} b^3 d^2 x^6 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^3 (c+d x)^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.56, size = 130, normalized size = 2.00 \begin {gather*} \frac {1}{6} x^{6} d^{2} b^{3} + \frac {2}{5} x^{5} d c b^{3} + \frac {3}{5} x^{5} d^{2} b^{2} a + \frac {1}{4} x^{4} c^{2} b^{3} + \frac {3}{2} x^{4} d c b^{2} a + \frac {3}{4} x^{4} d^{2} b a^{2} + x^{3} c^{2} b^{2} a + 2 x^{3} d c b a^{2} + \frac {1}{3} x^{3} d^{2} a^{3} + \frac {3}{2} x^{2} c^{2} b a^{2} + x^{2} d c a^{3} + x c^{2} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 130, normalized size = 2.00 \begin {gather*} \frac {1}{6} \, b^{3} d^{2} x^{6} + \frac {2}{5} \, b^{3} c d x^{5} + \frac {3}{5} \, a b^{2} d^{2} x^{5} + \frac {1}{4} \, b^{3} c^{2} x^{4} + \frac {3}{2} \, a b^{2} c d x^{4} + \frac {3}{4} \, a^{2} b d^{2} x^{4} + a b^{2} c^{2} x^{3} + 2 \, a^{2} b c d x^{3} + \frac {1}{3} \, a^{3} d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{2} x^{2} + a^{3} c d x^{2} + a^{3} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 125, normalized size = 1.92 \begin {gather*} \frac {b^{3} d^{2} x^{6}}{6}+a^{3} c^{2} x +\frac {\left (3 a \,b^{2} d^{2}+2 b^{3} c d \right ) x^{5}}{5}+\frac {\left (3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}\right ) x^{4}}{4}+\frac {\left (a^{3} d^{2}+6 a^{2} b c d +3 a \,b^{2} c^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} c d +3 a^{2} b \,c^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 124, normalized size = 1.91 \begin {gather*} \frac {1}{6} \, b^{3} d^{2} x^{6} + a^{3} c^{2} x + \frac {1}{5} \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 115, normalized size = 1.77 \begin {gather*} x^3\,\left (\frac {a^3\,d^2}{3}+2\,a^2\,b\,c\,d+a\,b^2\,c^2\right )+x^4\,\left (\frac {3\,a^2\,b\,d^2}{4}+\frac {3\,a\,b^2\,c\,d}{2}+\frac {b^3\,c^2}{4}\right )+a^3\,c^2\,x+\frac {b^3\,d^2\,x^6}{6}+\frac {a^2\,c\,x^2\,\left (2\,a\,d+3\,b\,c\right )}{2}+\frac {b^2\,d\,x^5\,\left (3\,a\,d+2\,b\,c\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 133, normalized size = 2.05 \begin {gather*} a^{3} c^{2} x + \frac {b^{3} d^{2} x^{6}}{6} + x^{5} \left (\frac {3 a b^{2} d^{2}}{5} + \frac {2 b^{3} c d}{5}\right ) + x^{4} \left (\frac {3 a^{2} b d^{2}}{4} + \frac {3 a b^{2} c d}{2} + \frac {b^{3} c^{2}}{4}\right ) + x^{3} \left (\frac {a^{3} d^{2}}{3} + 2 a^{2} b c d + a b^{2} c^{2}\right ) + x^{2} \left (a^{3} c d + \frac {3 a^{2} b c^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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