Optimal. Leaf size=62 \[ -\frac {2 a^2}{3 x^{3/2}}+4 a b \sqrt {x}+\frac {2}{5} \left (b^2+2 a c\right ) x^{5/2}+\frac {4}{9} b c x^{9/2}+\frac {2}{13} c^2 x^{13/2} \]
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Rubi [A]
time = 0.02, 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*} -\frac {2 a^2}{3 x^{3/2}}+\frac {2}{5} x^{5/2} \left (2 a c+b^2\right )+4 a b \sqrt {x}+\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 1122
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^{5/2}} \, dx &=\int \left (\frac {a^2}{x^{5/2}}+\frac {2 a b}{\sqrt {x}}+\left (b^2+2 a c\right ) x^{3/2}+2 b c x^{7/2}+c^2 x^{11/2}\right ) \, dx\\ &=-\frac {2 a^2}{3 x^{3/2}}+4 a b \sqrt {x}+\frac {2}{5} \left (b^2+2 a c\right ) 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 = 0.03, size = 52, normalized size = 0.84 \begin {gather*} -\frac {2 \left (195 a^2-1170 a b x^2-117 b^2 x^4-234 a c x^4-130 b c x^6-45 c^2 x^8\right )}{585 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 47, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {2 c^{2} x^{\frac {13}{2}}}{13}+\frac {4 b c \,x^{\frac {9}{2}}}{9}+\frac {4 a c \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} x^{\frac {5}{2}}}{5}+4 a b \sqrt {x}-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}\) | \(47\) |
default | \(\frac {2 c^{2} x^{\frac {13}{2}}}{13}+\frac {4 b c \,x^{\frac {9}{2}}}{9}+\frac {4 a c \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} x^{\frac {5}{2}}}{5}+4 a b \sqrt {x}-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}\) | \(47\) |
gosper | \(-\frac {2 \left (-45 c^{2} x^{8}-130 b c \,x^{6}-234 c \,x^{4} a -117 b^{2} x^{4}-1170 a b \,x^{2}+195 a^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(49\) |
trager | \(-\frac {2 \left (-45 c^{2} x^{8}-130 b c \,x^{6}-234 c \,x^{4} a -117 b^{2} x^{4}-1170 a b \,x^{2}+195 a^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(49\) |
risch | \(-\frac {2 \left (-45 c^{2} x^{8}-130 b c \,x^{6}-234 c \,x^{4} a -117 b^{2} x^{4}-1170 a b \,x^{2}+195 a^{2}\right )}{585 x^{\frac {3}{2}}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 44, normalized size = 0.71 \begin {gather*} \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{\frac {5}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 46, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (45 \, c^{2} x^{8} + 130 \, b c x^{6} + 117 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 1170 \, a b x^{2} - 195 \, a^{2}\right )}}{585 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.70, size = 68, normalized size = 1.10 \begin {gather*} - \frac {2 a^{2}}{3 x^{\frac {3}{2}}} + 4 a b \sqrt {x} + \frac {4 a c x^{\frac {5}{2}}}{5} + \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 = 3.66, size = 46, normalized size = 0.74 \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}} + \frac {4}{5} \, a c x^{\frac {5}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, a^{2}}{3 \, x^{\frac {3}{2}}} \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^{5/2}\,\left (\frac {2\,b^2}{5}+\frac {4\,a\,c}{5}\right )-\frac {2\,a^2}{3\,x^{3/2}}+\frac {2\,c^2\,x^{13/2}}{13}+4\,a\,b\,\sqrt {x}+\frac {4\,b\,c\,x^{9/2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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