Optimal. Leaf size=51 \[ \frac {2}{7} a^3 x^{7/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{11} a b^2 x^{11/2}+\frac {2}{13} b^3 x^{13/2} \]
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Rubi [A] time = 0.01, antiderivative size = 51, 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} \begin {gather*} \frac {2}{3} a^2 b x^{9/2}+\frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a b^2 x^{11/2}+\frac {2}{13} b^3 x^{13/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^{5/2} (a+b x)^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 b x^{7/2}+3 a b^2 x^{9/2}+b^3 x^{11/2}\right ) \, dx\\ &=\frac {2}{7} a^3 x^{7/2}+\frac {2}{3} a^2 b x^{9/2}+\frac {6}{11} a b^2 x^{11/2}+\frac {2}{13} b^3 x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.76 \begin {gather*} \frac {2 x^{7/2} \left (429 a^3+1001 a^2 b x+819 a b^2 x^2+231 b^3 x^3\right )}{3003} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 47, normalized size = 0.92 \begin {gather*} \frac {2 \left (429 a^3 x^{7/2}+1001 a^2 b x^{9/2}+819 a b^2 x^{11/2}+231 b^3 x^{13/2}\right )}{3003} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 40, normalized size = 0.78 \begin {gather*} \frac {2}{3003} \, {\left (231 \, b^{3} x^{6} + 819 \, a b^{2} x^{5} + 1001 \, a^{2} b x^{4} + 429 \, a^{3} x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {6}{11} \, a b^{2} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.71 \begin {gather*} \frac {2 \left (231 b^{3} x^{3}+819 a \,b^{2} x^{2}+1001 a^{2} b x +429 a^{3}\right ) x^{\frac {7}{2}}}{3003} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{13} \, b^{3} x^{\frac {13}{2}} + \frac {6}{11} \, a b^{2} x^{\frac {11}{2}} + \frac {2}{3} \, a^{2} b x^{\frac {9}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,a^3\,x^{7/2}}{7}+\frac {2\,b^3\,x^{13/2}}{13}+\frac {2\,a^2\,b\,x^{9/2}}{3}+\frac {6\,a\,b^2\,x^{11/2}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.88, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 a^{3} x^{\frac {7}{2}}}{7} + \frac {2 a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 b^{3} x^{\frac {13}{2}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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