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 = 0.0230685, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}
Mathematica [A] time = 3.32368, 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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Maple [A] time = 0.045, size = 49, normalized size = 0.8 \begin{align*}{\frac{2310\,{c}^{2}{x}^{8}+5852\,bc{x}^{6}+7980\,{x}^{4}ac+3990\,{b}^{2}{x}^{4}+12540\,ab{x}^{2}+14630\,{a}^{2}}{21945}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956327, size = 59, normalized size = 0.92 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24976, size = 135, normalized size = 2.11 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.66458, size = 63, normalized size = 0.98 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14697, size = 62, normalized size = 0.97 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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