Optimal. Leaf size=51 \[ \frac {2}{7} a^3 x^{7/2}+\frac {2}{5} a^2 c x^{15/2}+\frac {6}{23} a c^2 x^{23/2}+\frac {2}{31} c^3 x^{31/2} \]
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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} \[ \frac {2}{5} a^2 c x^{15/2}+\frac {2}{7} a^3 x^{7/2}+\frac {6}{23} a c^2 x^{23/2}+\frac {2}{31} c^3 x^{31/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^{5/2} \left (a+c x^4\right )^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 c x^{13/2}+3 a c^2 x^{21/2}+c^3 x^{29/2}\right ) \, dx\\ &=\frac {2}{7} a^3 x^{7/2}+\frac {2}{5} a^2 c x^{15/2}+\frac {6}{23} a c^2 x^{23/2}+\frac {2}{31} c^3 x^{31/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.00 \[ \frac {2}{7} a^3 x^{7/2}+\frac {2}{5} a^2 c x^{15/2}+\frac {6}{23} a c^2 x^{23/2}+\frac {2}{31} c^3 x^{31/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 40, normalized size = 0.78 \[ \frac {2}{24955} \, {\left (805 \, c^{3} x^{15} + 3255 \, a c^{2} x^{11} + 4991 \, a^{2} c x^{7} + 3565 \, a^{3} x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 0.69 \[ \frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {6}{23} \, a c^{2} x^{\frac {23}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {15}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.75 \[ \frac {2 \left (805 c^{3} x^{12}+3255 a \,c^{2} x^{8}+4991 a^{2} c \,x^{4}+3565 a^{3}\right ) x^{\frac {7}{2}}}{24955} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 35, normalized size = 0.69 \[ \frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {6}{23} \, a c^{2} x^{\frac {23}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {15}{2}} + \frac {2}{7} \, a^{3} x^{\frac {7}{2}} \]
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 \[ \frac {2\,a^3\,x^{7/2}}{7}+\frac {2\,c^3\,x^{31/2}}{31}+\frac {2\,a^2\,c\,x^{15/2}}{5}+\frac {6\,a\,c^2\,x^{23/2}}{23} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 70.18, size = 49, normalized size = 0.96 \[ \frac {2 a^{3} x^{\frac {7}{2}}}{7} + \frac {2 a^{2} c x^{\frac {15}{2}}}{5} + \frac {6 a c^{2} x^{\frac {23}{2}}}{23} + \frac {2 c^{3} x^{\frac {31}{2}}}{31} \]
Verification of antiderivative is not currently implemented for this CAS.
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