Optimal. Leaf size=36 \[ \frac {2}{5} a^2 x^{5/2}+\frac {4}{13} a c x^{13/2}+\frac {2}{21} c^2 x^{21/2} \]
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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 = {276}
\begin {gather*} \frac {2}{5} a^2 x^{5/2}+\frac {4}{13} a c x^{13/2}+\frac {2}{21} c^2 x^{21/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int x^{3/2} \left (a+c x^4\right )^2 \, dx &=\int \left (a^2 x^{3/2}+2 a c x^{11/2}+c^2 x^{19/2}\right ) \, dx\\ &=\frac {2}{5} a^2 x^{5/2}+\frac {4}{13} a c x^{13/2}+\frac {2}{21} c^2 x^{21/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.83 \begin {gather*} \frac {2 x^{5/2} \left (273 a^2+210 a c x^4+65 c^2 x^8\right )}{1365} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 25, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a c \,x^{\frac {13}{2}}}{13}+\frac {2 c^{2} x^{\frac {21}{2}}}{21}\) | \(25\) |
default | \(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a c \,x^{\frac {13}{2}}}{13}+\frac {2 c^{2} x^{\frac {21}{2}}}{21}\) | \(25\) |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (65 c^{2} x^{8}+210 a c \,x^{4}+273 a^{2}\right )}{1365}\) | \(27\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (65 c^{2} x^{8}+210 a c \,x^{4}+273 a^{2}\right )}{1365}\) | \(27\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (65 c^{2} x^{8}+210 a c \,x^{4}+273 a^{2}\right )}{1365}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{13} \, a c x^{\frac {13}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 29, normalized size = 0.81 \begin {gather*} \frac {2}{1365} \, {\left (65 \, c^{2} x^{10} + 210 \, a c x^{6} + 273 \, a^{2} x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.74, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a c x^{\frac {13}{2}}}{13} + \frac {2 c^{2} x^{\frac {21}{2}}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{13} \, a c x^{\frac {13}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 26, normalized size = 0.72 \begin {gather*} \frac {2\,x^{5/2}\,\left (273\,a^2+210\,a\,c\,x^4+65\,c^2\,x^8\right )}{1365} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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