Optimal. Leaf size=62 \[ 2 a^2 \sqrt {x}+\frac {4}{5} a b x^{5/2}+\frac {2}{9} \left (b^2+2 a c\right ) x^{9/2}+\frac {4}{13} b c x^{13/2}+\frac {2}{17} c^2 x^{17/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1122}
\begin {gather*} 2 a^2 \sqrt {x}+\frac {2}{9} x^{9/2} \left (2 a c+b^2\right )+\frac {4}{5} a b x^{5/2}+\frac {4}{13} b c x^{13/2}+\frac {2}{17} c^2 x^{17/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1122
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{\sqrt {x}} \, dx &=\int \left (\frac {a^2}{\sqrt {x}}+2 a b x^{3/2}+\left (b^2+2 a c\right ) x^{7/2}+2 b c x^{11/2}+c^2 x^{15/2}\right ) \, dx\\ &=2 a^2 \sqrt {x}+\frac {4}{5} a b x^{5/2}+\frac {2}{9} \left (b^2+2 a c\right ) x^{9/2}+\frac {4}{13} b c x^{13/2}+\frac {2}{17} c^2 x^{17/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {x} \left (9945 a^2+442 a \left (9 b x^2+5 c x^4\right )+5 x^4 \left (221 b^2+306 b c x^2+117 c^2 x^4\right )\right )}{9945} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 45, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {4 a b \,x^{\frac {5}{2}}}{5}+\frac {2 \left (2 a c +b^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {4 b c \,x^{\frac {13}{2}}}{13}+\frac {2 c^{2} x^{\frac {17}{2}}}{17}+2 a^{2} \sqrt {x}\) | \(45\) |
default | \(\frac {4 a b \,x^{\frac {5}{2}}}{5}+\frac {2 \left (2 a c +b^{2}\right ) x^{\frac {9}{2}}}{9}+\frac {4 b c \,x^{\frac {13}{2}}}{13}+\frac {2 c^{2} x^{\frac {17}{2}}}{17}+2 a^{2} \sqrt {x}\) | \(45\) |
trager | \(\left (\frac {2}{17} c^{2} x^{8}+\frac {4}{13} b c \,x^{6}+\frac {4}{9} c \,x^{4} a +\frac {2}{9} b^{2} x^{4}+\frac {4}{5} a b \,x^{2}+2 a^{2}\right ) \sqrt {x}\) | \(48\) |
gosper | \(\frac {2 \sqrt {x}\, \left (585 c^{2} x^{8}+1530 b c \,x^{6}+2210 c \,x^{4} a +1105 b^{2} x^{4}+3978 a b \,x^{2}+9945 a^{2}\right )}{9945}\) | \(49\) |
risch | \(\frac {2 \sqrt {x}\, \left (585 c^{2} x^{8}+1530 b c \,x^{6}+2210 c \,x^{4} a +1105 b^{2} x^{4}+3978 a b \,x^{2}+9945 a^{2}\right )}{9945}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 48, normalized size = 0.77 \begin {gather*} \frac {2}{17} \, c^{2} x^{\frac {17}{2}} + \frac {4}{13} \, b c x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + 2 \, a^{2} \sqrt {x} + \frac {4}{45} \, {\left (5 \, c x^{\frac {9}{2}} + 9 \, b x^{\frac {5}{2}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 46, normalized size = 0.74 \begin {gather*} \frac {2}{9945} \, {\left (585 \, c^{2} x^{8} + 1530 \, b c x^{6} + 1105 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 3978 \, a b x^{2} + 9945 \, a^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 68, normalized size = 1.10 \begin {gather*} 2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {4 a c x^{\frac {9}{2}}}{9} + \frac {2 b^{2} x^{\frac {9}{2}}}{9} + \frac {4 b c x^{\frac {13}{2}}}{13} + \frac {2 c^{2} x^{\frac {17}{2}}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.48, size = 46, normalized size = 0.74 \begin {gather*} \frac {2}{17} \, c^{2} x^{\frac {17}{2}} + \frac {4}{13} \, b c x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, a c x^{\frac {9}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + 2 \, a^{2} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 45, normalized size = 0.73 \begin {gather*} x^{9/2}\,\left (\frac {2\,b^2}{9}+\frac {4\,a\,c}{9}\right )+2\,a^2\,\sqrt {x}+\frac {2\,c^2\,x^{17/2}}{17}+\frac {4\,a\,b\,x^{5/2}}{5}+\frac {4\,b\,c\,x^{13/2}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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