Optimal. Leaf size=29 \[ 2 a \sqrt {x}+\frac {2}{5} b x^{5/2}+\frac {2}{9} c x^{9/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14}
\begin {gather*} 2 a \sqrt {x}+\frac {2}{5} b x^{5/2}+\frac {2}{9} c x^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{\sqrt {x}} \, dx &=\int \left (\frac {a}{\sqrt {x}}+b x^{3/2}+c x^{7/2}\right ) \, dx\\ &=2 a \sqrt {x}+\frac {2}{5} b x^{5/2}+\frac {2}{9} c x^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.86 \begin {gather*} \frac {2}{45} \sqrt {x} \left (45 a+9 b x^2+5 c x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {2 b \,x^{\frac {5}{2}}}{5}+\frac {2 c \,x^{\frac {9}{2}}}{9}+2 a \sqrt {x}\) | \(20\) |
default | \(\frac {2 b \,x^{\frac {5}{2}}}{5}+\frac {2 c \,x^{\frac {9}{2}}}{9}+2 a \sqrt {x}\) | \(20\) |
trager | \(\left (\frac {2}{9} c \,x^{4}+\frac {2}{5} b \,x^{2}+2 a \right ) \sqrt {x}\) | \(21\) |
gosper | \(\frac {2 \sqrt {x}\, \left (5 c \,x^{4}+9 b \,x^{2}+45 a \right )}{45}\) | \(22\) |
risch | \(\frac {2 \sqrt {x}\, \left (5 c \,x^{4}+9 b \,x^{2}+45 a \right )}{45}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{9} \, c x^{\frac {9}{2}} + \frac {2}{5} \, b x^{\frac {5}{2}} + 2 \, a \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 21, normalized size = 0.72 \begin {gather*} \frac {2}{45} \, {\left (5 \, c x^{4} + 9 \, b x^{2} + 45 \, a\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 27, normalized size = 0.93 \begin {gather*} 2 a \sqrt {x} + \frac {2 b x^{\frac {5}{2}}}{5} + \frac {2 c x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.73, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{9} \, c x^{\frac {9}{2}} + \frac {2}{5} \, b x^{\frac {5}{2}} + 2 \, a \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 21, normalized size = 0.72 \begin {gather*} \frac {2\,\sqrt {x}\,\left (5\,c\,x^4+9\,b\,x^2+45\,a\right )}{45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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