Optimal. Leaf size=99 \[ 2 a^2 b x^{3/2}-\frac{2 a^3}{\sqrt{x}}+\frac{2}{5} c x^{15/2} \left (a c+b^2\right )+\frac{2}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac{6}{7} a x^{7/2} \left (a c+b^2\right )+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2} \]
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Rubi [A] time = 0.0425691, antiderivative size = 99, 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} \[ 2 a^2 b x^{3/2}-\frac{2 a^3}{\sqrt{x}}+\frac{2}{5} c x^{15/2} \left (a c+b^2\right )+\frac{2}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac{6}{7} a x^{7/2} \left (a c+b^2\right )+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2} \]
Antiderivative was successfully verified.
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Rule 1108
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^3}{x^{3/2}} \, dx &=\int \left (\frac{a^3}{x^{3/2}}+3 a^2 b \sqrt{x}+3 a \left (b^2+a c\right ) x^{5/2}+b \left (b^2+6 a c\right ) x^{9/2}+3 c \left (b^2+a c\right ) x^{13/2}+3 b c^2 x^{17/2}+c^3 x^{21/2}\right ) \, dx\\ &=-\frac{2 a^3}{\sqrt{x}}+2 a^2 b x^{3/2}+\frac{6}{7} a \left (b^2+a c\right ) x^{7/2}+\frac{2}{11} b \left (b^2+6 a c\right ) x^{11/2}+\frac{2}{5} c \left (b^2+a c\right ) x^{15/2}+\frac{6}{19} b c^2 x^{19/2}+\frac{2}{23} c^3 x^{23/2}\\ \end{align*}
Mathematica [A] time = 0.0810406, size = 100, normalized size = 1.01 \[ 2 \left (a^2 b x^{3/2}-\frac{a^3}{\sqrt{x}}+\frac{1}{5} c x^{15/2} \left (a c+b^2\right )+\frac{1}{11} b x^{11/2} \left (6 a c+b^2\right )+\frac{3}{7} a x^{7/2} \left (a c+b^2\right )+\frac{3}{19} b c^2 x^{19/2}+\frac{1}{23} c^3 x^{23/2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 90, normalized size = 0.9 \begin{align*} -{\frac{-14630\,{c}^{3}{x}^{12}-53130\,b{c}^{2}{x}^{10}-67298\,{x}^{8}a{c}^{2}-67298\,{x}^{8}{b}^{2}c-183540\,{x}^{6}abc-30590\,{x}^{6}{b}^{3}-144210\,{a}^{2}c{x}^{4}-144210\,{x}^{4}{b}^{2}a-336490\,{a}^{2}b{x}^{2}+336490\,{a}^{3}}{168245}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961496, size = 109, normalized size = 1.1 \begin{align*} \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b c^{2} x^{\frac{19}{2}} + \frac{2}{5} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{11}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + \frac{6}{7} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{7}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19299, size = 228, normalized size = 2.3 \begin{align*} \frac{2 \,{\left (7315 \, c^{3} x^{12} + 26565 \, b c^{2} x^{10} + 33649 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 15295 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 168245 \, a^{2} b x^{2} + 72105 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 168245 \, a^{3}\right )}}{168245 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.8602, size = 126, normalized size = 1.27 \begin{align*} - \frac{2 a^{3}}{\sqrt{x}} + 2 a^{2} b x^{\frac{3}{2}} + \frac{6 a^{2} c x^{\frac{7}{2}}}{7} + \frac{6 a b^{2} x^{\frac{7}{2}}}{7} + \frac{12 a b c x^{\frac{11}{2}}}{11} + \frac{2 a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} + \frac{2 b^{2} c x^{\frac{15}{2}}}{5} + \frac{6 b c^{2} x^{\frac{19}{2}}}{19} + \frac{2 c^{3} x^{\frac{23}{2}}}{23} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13689, size = 117, normalized size = 1.18 \begin{align*} \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b c^{2} x^{\frac{19}{2}} + \frac{2}{5} \, b^{2} c x^{\frac{15}{2}} + \frac{2}{5} \, a c^{2} x^{\frac{15}{2}} + \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{12}{11} \, a b c x^{\frac{11}{2}} + \frac{6}{7} \, a b^{2} x^{\frac{7}{2}} + \frac{6}{7} \, a^{2} c x^{\frac{7}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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