Optimal. Leaf size=36 \[ \frac {2}{13} b^2 x^{13/2}+\frac {4}{17} b c x^{17/2}+\frac {2}{21} c^2 x^{21/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 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}{13} b^2 x^{13/2}+\frac {4}{17} b c x^{17/2}+\frac {2}{21} c^2 x^{21/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 1598
Rubi steps
\begin {align*} \int x^{3/2} \left (b x^2+c x^4\right )^2 \, dx &=\int x^{11/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{11/2}+2 b c x^{15/2}+c^2 x^{19/2}\right ) \, dx\\ &=\frac {2}{13} b^2 x^{13/2}+\frac {4}{17} b c x^{17/2}+\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^{13/2} \left (357 b^2+546 b c x^2+221 c^2 x^4\right )}{4641} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 25, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {2 b^{2} x^{\frac {13}{2}}}{13}+\frac {4 b c \,x^{\frac {17}{2}}}{17}+\frac {2 c^{2} x^{\frac {21}{2}}}{21}\) | \(25\) |
default | \(\frac {2 b^{2} x^{\frac {13}{2}}}{13}+\frac {4 b c \,x^{\frac {17}{2}}}{17}+\frac {2 c^{2} x^{\frac {21}{2}}}{21}\) | \(25\) |
gosper | \(\frac {2 x^{\frac {13}{2}} \left (221 c^{2} x^{4}+546 b c \,x^{2}+357 b^{2}\right )}{4641}\) | \(27\) |
trager | \(\frac {2 x^{\frac {13}{2}} \left (221 c^{2} x^{4}+546 b c \,x^{2}+357 b^{2}\right )}{4641}\) | \(27\) |
risch | \(\frac {2 x^{\frac {13}{2}} \left (221 c^{2} x^{4}+546 b c \,x^{2}+357 b^{2}\right )}{4641}\) | \(27\) |
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}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{17} \, b c x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} x^{\frac {13}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 29, normalized size = 0.81 \begin {gather*} \frac {2}{4641} \, {\left (221 \, c^{2} x^{10} + 546 \, b c x^{8} + 357 \, b^{2} x^{6}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.81, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 b^{2} x^{\frac {13}{2}}}{13} + \frac {4 b c x^{\frac {17}{2}}}{17} + \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 = 3.95, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{17} \, b c x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} x^{\frac {13}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.27, size = 25, normalized size = 0.69 \begin {gather*} x^{13/2}\,\left (\frac {2\,b^2}{13}+\frac {4\,b\,c\,x^2}{17}+\frac {2\,c^2\,x^4}{21}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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