Optimal. Leaf size=36 \[ \frac {2}{5} b^2 x^{5/2}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1598, 276}
\begin {gather*} \frac {2}{5} b^2 x^{5/2}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^{5/2}} \, dx &=\int x^{3/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{3/2}+2 b c x^{7/2}+c^2 x^{11/2}\right ) \, dx\\ &=\frac {2}{5} b^2 x^{5/2}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.83 \begin {gather*} \frac {2}{585} x^{5/2} \left (117 b^2+130 b c x^2+45 c^2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 25, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {2 b^{2} x^{\frac {5}{2}}}{5}+\frac {4 b c \,x^{\frac {9}{2}}}{9}+\frac {2 c^{2} x^{\frac {13}{2}}}{13}\) | \(25\) |
default | \(\frac {2 b^{2} x^{\frac {5}{2}}}{5}+\frac {4 b c \,x^{\frac {9}{2}}}{9}+\frac {2 c^{2} x^{\frac {13}{2}}}{13}\) | \(25\) |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (45 c^{2} x^{4}+130 b c \,x^{2}+117 b^{2}\right )}{585}\) | \(27\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (45 c^{2} x^{4}+130 b c \,x^{2}+117 b^{2}\right )}{585}\) | \(27\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (45 c^{2} x^{4}+130 b c \,x^{2}+117 b^{2}\right )}{585}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 29, normalized size = 0.81 \begin {gather*} \frac {2}{585} \, {\left (45 \, c^{2} x^{6} + 130 \, b c x^{4} + 117 \, b^{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 = 0.67, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 b^{2} x^{\frac {5}{2}}}{5} + \frac {4 b c x^{\frac {9}{2}}}{9} + \frac {2 c^{2} x^{\frac {13}{2}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.77, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, b^{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 (117\,b^2+130\,b\,c\,x^2+45\,c^2\,x^4\right )}{585} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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