Optimal. Leaf size=36 \[ \frac {2}{7} a^2 x^{7/2}+\frac {4}{15} a c x^{15/2}+\frac {2}{23} c^2 x^{23/2} \]
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Rubi [A] time = 0.01, antiderivative size = 36, 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} \[ \frac {2}{7} a^2 x^{7/2}+\frac {4}{15} a c x^{15/2}+\frac {2}{23} c^2 x^{23/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^{5/2} \left (a+c x^4\right )^2 \, dx &=\int \left (a^2 x^{5/2}+2 a c x^{13/2}+c^2 x^{21/2}\right ) \, dx\\ &=\frac {2}{7} a^2 x^{7/2}+\frac {4}{15} a c x^{15/2}+\frac {2}{23} c^2 x^{23/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.83 \[ \frac {2 x^{7/2} \left (345 a^2+322 a c x^4+105 c^2 x^8\right )}{2415} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 29, normalized size = 0.81 \[ \frac {2}{2415} \, {\left (105 \, c^{2} x^{11} + 322 \, a c x^{7} + 345 \, a^{2} x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 24, normalized size = 0.67 \[ \frac {2}{23} \, c^{2} x^{\frac {23}{2}} + \frac {4}{15} \, a c x^{\frac {15}{2}} + \frac {2}{7} \, a^{2} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.75 \[ \frac {2 \left (105 c^{2} x^{8}+322 a c \,x^{4}+345 a^{2}\right ) x^{\frac {7}{2}}}{2415} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 24, normalized size = 0.67 \[ \frac {2}{23} \, c^{2} x^{\frac {23}{2}} + \frac {4}{15} \, a c x^{\frac {15}{2}} + \frac {2}{7} \, a^{2} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 25, normalized size = 0.69 \[ x^{7/2}\,\left (\frac {2\,a^2}{7}+\frac {4\,a\,c\,x^4}{15}+\frac {2\,c^2\,x^8}{23}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.25, size = 34, normalized size = 0.94 \[ \frac {2 a^{2} x^{\frac {7}{2}}}{7} + \frac {4 a c x^{\frac {15}{2}}}{15} + \frac {2 c^{2} x^{\frac {23}{2}}}{23} \]
Verification of antiderivative is not currently implemented for this CAS.
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