Optimal. Leaf size=51 \[ \frac {2}{5} a^3 x^{5/2}+\frac {6}{13} a^2 c x^{13/2}+\frac {2}{7} a c^2 x^{21/2}+\frac {2}{29} c^3 x^{29/2} \]
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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} \[ \frac {6}{13} a^2 c x^{13/2}+\frac {2}{5} a^3 x^{5/2}+\frac {2}{7} a c^2 x^{21/2}+\frac {2}{29} c^3 x^{29/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int x^{3/2} \left (a+c x^4\right )^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 c x^{11/2}+3 a c^2 x^{19/2}+c^3 x^{27/2}\right ) \, dx\\ &=\frac {2}{5} a^3 x^{5/2}+\frac {6}{13} a^2 c x^{13/2}+\frac {2}{7} a c^2 x^{21/2}+\frac {2}{29} c^3 x^{29/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.00 \[ \frac {2}{5} a^3 x^{5/2}+\frac {6}{13} a^2 c x^{13/2}+\frac {2}{7} a c^2 x^{21/2}+\frac {2}{29} c^3 x^{29/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 40, normalized size = 0.78 \[ \frac {2}{13195} \, {\left (455 \, c^{3} x^{14} + 1885 \, a c^{2} x^{10} + 3045 \, a^{2} c x^{6} + 2639 \, a^{3} x^{2}\right )} \sqrt {x} \]
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 \[ \frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {2}{7} \, a c^{2} x^{\frac {21}{2}} + \frac {6}{13} \, a^{2} c x^{\frac {13}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \]
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 \[ \frac {2 \left (455 c^{3} x^{12}+1885 a \,c^{2} x^{8}+3045 a^{2} c \,x^{4}+2639 a^{3}\right ) x^{\frac {5}{2}}}{13195} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 35, normalized size = 0.69 \[ \frac {2}{29} \, c^{3} x^{\frac {29}{2}} + \frac {2}{7} \, a c^{2} x^{\frac {21}{2}} + \frac {6}{13} \, a^{2} c x^{\frac {13}{2}} + \frac {2}{5} \, a^{3} x^{\frac {5}{2}} \]
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 \[ \frac {2\,a^3\,x^{5/2}}{5}+\frac {2\,c^3\,x^{29/2}}{29}+\frac {6\,a^2\,c\,x^{13/2}}{13}+\frac {2\,a\,c^2\,x^{21/2}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 48.12, size = 49, normalized size = 0.96 \[ \frac {2 a^{3} x^{\frac {5}{2}}}{5} + \frac {6 a^{2} c x^{\frac {13}{2}}}{13} + \frac {2 a c^{2} x^{\frac {21}{2}}}{7} + \frac {2 c^{3} x^{\frac {29}{2}}}{29} \]
Verification of antiderivative is not currently implemented for this CAS.
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