Optimal. Leaf size=64 \[ \frac {2}{3} a^2 x^{3/2}+\frac {2}{11} x^{11/2} \left (2 a c+b^2\right )+\frac {4}{7} a b x^{7/2}+\frac {4}{15} b c x^{15/2}+\frac {2}{19} c^2 x^{19/2} \]
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Rubi [A] time = 0.02, antiderivative size = 64, 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 = {1108} \[ \frac {2}{3} a^2 x^{3/2}+\frac {2}{11} x^{11/2} \left (2 a c+b^2\right )+\frac {4}{7} a b x^{7/2}+\frac {4}{15} b c x^{15/2}+\frac {2}{19} c^2 x^{19/2} \]
Antiderivative was successfully verified.
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Rule 1108
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 \sqrt {x}+2 a b x^{5/2}+\left (b^2+2 a c\right ) x^{9/2}+2 b c x^{13/2}+c^2 x^{17/2}\right ) \, dx\\ &=\frac {2}{3} a^2 x^{3/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{11} \left (b^2+2 a c\right ) x^{11/2}+\frac {4}{15} b c x^{15/2}+\frac {2}{19} c^2 x^{19/2}\\ \end {align*}
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Mathematica [A] time = 3.38, size = 50, normalized size = 0.78 \[ \frac {2 x^{3/2} \left (7315 a^2+1995 x^4 \left (2 a c+b^2\right )+6270 a b x^2+2926 b c x^6+1155 c^2 x^8\right )}{21945} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 47, normalized size = 0.73 \[ \frac {2}{21945} \, {\left (1155 \, c^{2} x^{9} + 2926 \, b c x^{7} + 1995 \, {\left (b^{2} + 2 \, a c\right )} x^{5} + 6270 \, a b x^{3} + 7315 \, a^{2} x\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 46, normalized size = 0.72 \[ \frac {2}{19} \, c^{2} x^{\frac {19}{2}} + \frac {4}{15} \, b c x^{\frac {15}{2}} + \frac {2}{11} \, b^{2} x^{\frac {11}{2}} + \frac {4}{11} \, a c x^{\frac {11}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.77 \[ \frac {2 \left (1155 c^{2} x^{8}+2926 b c \,x^{6}+3990 a c \,x^{4}+1995 b^{2} x^{4}+6270 a b \,x^{2}+7315 a^{2}\right ) x^{\frac {3}{2}}}{21945} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 44, normalized size = 0.69 \[ \frac {2}{19} \, c^{2} x^{\frac {19}{2}} + \frac {4}{15} \, b c x^{\frac {15}{2}} + \frac {2}{11} \, {\left (b^{2} + 2 \, a c\right )} x^{\frac {11}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 45, normalized size = 0.70 \[ x^{11/2}\,\left (\frac {2\,b^2}{11}+\frac {4\,a\,c}{11}\right )+\frac {2\,a^2\,x^{3/2}}{3}+\frac {2\,c^2\,x^{19/2}}{19}+\frac {4\,a\,b\,x^{7/2}}{7}+\frac {4\,b\,c\,x^{15/2}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.45, size = 63, normalized size = 0.98 \[ \frac {2 a^{2} x^{\frac {3}{2}}}{3} + \frac {4 a b x^{\frac {7}{2}}}{7} + \frac {4 b c x^{\frac {15}{2}}}{15} + \frac {2 c^{2} x^{\frac {19}{2}}}{19} + \frac {2 x^{\frac {11}{2}} \left (2 a c + b^{2}\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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