Optimal. Leaf size=51 \[ \frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a^2 b x^{11/2}+\frac {2}{5} a b^2 x^{15/2}+\frac {2}{19} b^3 x^{19/2} \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \frac {6}{11} a^2 b x^{11/2}+\frac {2}{7} a^3 x^{7/2}+\frac {2}{5} a b^2 x^{15/2}+\frac {2}{19} b^3 x^{19/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^{5/2} \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 b x^{9/2}+3 a b^2 x^{13/2}+b^3 x^{17/2}\right ) \, dx\\ &=\frac {2}{7} a^3 x^{7/2}+\frac {6}{11} a^2 b x^{11/2}+\frac {2}{5} a b^2 x^{15/2}+\frac {2}{19} b^3 x^{19/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.80 \[ \frac {2 x^{7/2} \left (1045 a^3+1995 a^2 b x^2+1463 a b^2 x^4+385 b^3 x^6\right )}{7315} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 40, normalized size = 0.78 \[ \frac {2}{7315} \, {\left (385 \, b^{3} x^{9} + 1463 \, a b^{2} x^{7} + 1995 \, a^{2} b x^{5} + 1045 \, a^{3} x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 35, normalized size = 0.69 \[ \frac {2}{19} \, b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, a b^{2} x^{\frac {15}{2}} + \frac {6}{11} \, a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.75 \[ \frac {2 \left (385 b^{3} x^{6}+1463 a \,b^{2} x^{4}+1995 a^{2} b \,x^{2}+1045 a^{3}\right ) x^{\frac {7}{2}}}{7315} \]
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 \[ \frac {2}{19} \, b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, a b^{2} x^{\frac {15}{2}} + \frac {6}{11} \, a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
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 \[ \frac {2\,a^3\,x^{7/2}}{7}+\frac {2\,b^3\,x^{19/2}}{19}+\frac {6\,a^2\,b\,x^{11/2}}{11}+\frac {2\,a\,b^2\,x^{15/2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.62, size = 49, normalized size = 0.96 \[ \frac {2 a^{3} x^{\frac {7}{2}}}{7} + \frac {6 a^{2} b x^{\frac {11}{2}}}{11} + \frac {2 a b^{2} x^{\frac {15}{2}}}{5} + \frac {2 b^{3} x^{\frac {19}{2}}}{19} \]
Verification of antiderivative is not currently implemented for this CAS.
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