Optimal. Leaf size=103 \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{19} c x^{19/2} \left (a c+b^2\right )+\frac{2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac{6}{11} a x^{11/2} \left (a c+b^2\right )+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]
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Rubi [A] time = 0.0416628, antiderivative size = 103, 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} \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{19} c x^{19/2} \left (a c+b^2\right )+\frac{2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac{6}{11} a x^{11/2} \left (a c+b^2\right )+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]
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 )^3 \, dx &=\int \left (a^3 \sqrt{x}+3 a^2 b x^{5/2}+3 a \left (b^2+a c\right ) x^{9/2}+b \left (b^2+6 a c\right ) x^{13/2}+3 c \left (b^2+a c\right ) x^{17/2}+3 b c^2 x^{21/2}+c^3 x^{25/2}\right ) \, dx\\ &=\frac{2}{3} a^3 x^{3/2}+\frac{6}{7} a^2 b x^{7/2}+\frac{6}{11} a \left (b^2+a c\right ) x^{11/2}+\frac{2}{15} b \left (b^2+6 a c\right ) x^{15/2}+\frac{6}{19} c \left (b^2+a c\right ) x^{19/2}+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2}\\ \end{align*}
Mathematica [A] time = 3.31634, size = 103, normalized size = 1. \[ \frac{6}{7} a^2 b x^{7/2}+\frac{2}{3} a^3 x^{3/2}+\frac{6}{19} c x^{19/2} \left (a c+b^2\right )+\frac{2}{15} b x^{15/2} \left (6 a c+b^2\right )+\frac{6}{11} a x^{11/2} \left (a c+b^2\right )+\frac{6}{23} b c^2 x^{23/2}+\frac{2}{27} c^3 x^{27/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 90, normalized size = 0.9 \begin{align*}{\frac{336490\,{c}^{3}{x}^{12}+1185030\,b{c}^{2}{x}^{10}+1434510\,{x}^{8}a{c}^{2}+1434510\,{x}^{8}{b}^{2}c+3634092\,{x}^{6}abc+605682\,{x}^{6}{b}^{3}+2477790\,{a}^{2}c{x}^{4}+2477790\,{x}^{4}{b}^{2}a+3893670\,{a}^{2}b{x}^{2}+3028410\,{a}^{3}}{4542615}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97738, size = 109, normalized size = 1.06 \begin{align*} \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{23} \, b c^{2} x^{\frac{23}{2}} + \frac{6}{19} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{15}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{6}{11} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{11}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29559, size = 244, normalized size = 2.37 \begin{align*} \frac{2}{4542615} \,{\left (168245 \, c^{3} x^{13} + 592515 \, b c^{2} x^{11} + 717255 \,{\left (b^{2} c + a c^{2}\right )} x^{9} + 302841 \,{\left (b^{3} + 6 \, a b c\right )} x^{7} + 1946835 \, a^{2} b x^{3} + 1238895 \,{\left (a b^{2} + a^{2} c\right )} x^{5} + 1514205 \, a^{3} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.61993, size = 112, normalized size = 1.09 \begin{align*} \frac{2 a^{3} x^{\frac{3}{2}}}{3} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{6 b c^{2} x^{\frac{23}{2}}}{23} + \frac{2 c^{3} x^{\frac{27}{2}}}{27} + \frac{2 x^{\frac{19}{2}} \left (3 a c^{2} + 3 b^{2} c\right )}{19} + \frac{2 x^{\frac{15}{2}} \left (6 a b c + b^{3}\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (3 a^{2} c + 3 a b^{2}\right )}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15765, size = 117, normalized size = 1.14 \begin{align*} \frac{2}{27} \, c^{3} x^{\frac{27}{2}} + \frac{6}{23} \, b c^{2} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c x^{\frac{19}{2}} + \frac{6}{19} \, a c^{2} x^{\frac{19}{2}} + \frac{2}{15} \, b^{3} x^{\frac{15}{2}} + \frac{4}{5} \, a b c x^{\frac{15}{2}} + \frac{6}{11} \, a b^{2} x^{\frac{11}{2}} + \frac{6}{11} \, a^{2} c x^{\frac{11}{2}} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{2}{3} \, a^{3} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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