Optimal. Leaf size=103 \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{7} c x^{21/2} \left (a c+b^2\right )+\frac{2}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac{6}{13} a x^{13/2} \left (a c+b^2\right )+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0434847, 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} \[ \frac{2}{3} a^2 b x^{9/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{7} c x^{21/2} \left (a c+b^2\right )+\frac{2}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac{6}{13} a x^{13/2} \left (a c+b^2\right )+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1108
Rubi steps
\begin{align*} \int x^{3/2} \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 b x^{7/2}+3 a \left (b^2+a c\right ) x^{11/2}+b \left (b^2+6 a c\right ) x^{15/2}+3 c \left (b^2+a c\right ) x^{19/2}+3 b c^2 x^{23/2}+c^3 x^{27/2}\right ) \, dx\\ &=\frac{2}{5} a^3 x^{5/2}+\frac{2}{3} a^2 b x^{9/2}+\frac{6}{13} a \left (b^2+a c\right ) x^{13/2}+\frac{2}{17} b \left (b^2+6 a c\right ) x^{17/2}+\frac{2}{7} c \left (b^2+a c\right ) x^{21/2}+\frac{6}{25} b c^2 x^{25/2}+\frac{2}{29} c^3 x^{29/2}\\ \end{align*}
Mathematica [A] time = 0.09244, size = 105, normalized size = 1.02 \[ 2 \left (\frac{1}{3} a^2 b x^{9/2}+\frac{1}{5} a^3 x^{5/2}+\frac{1}{7} c x^{21/2} \left (a c+b^2\right )+\frac{1}{17} b x^{17/2} \left (6 a c+b^2\right )+\frac{3}{13} a x^{13/2} \left (a c+b^2\right )+\frac{3}{25} b c^2 x^{25/2}+\frac{1}{29} c^3 x^{29/2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 90, normalized size = 0.9 \begin{align*}{\frac{232050\,{c}^{3}{x}^{12}+807534\,b{c}^{2}{x}^{10}+961350\,{x}^{8}a{c}^{2}+961350\,{x}^{8}{b}^{2}c+2375100\,{x}^{6}abc+395850\,{x}^{6}{b}^{3}+1552950\,{a}^{2}c{x}^{4}+1552950\,{x}^{4}{b}^{2}a+2243150\,{a}^{2}b{x}^{2}+1345890\,{a}^{3}}{3364725}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.976441, size = 109, normalized size = 1.06 \begin{align*} \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{6}{25} \, b c^{2} x^{\frac{25}{2}} + \frac{2}{7} \,{\left (b^{2} c + a c^{2}\right )} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (b^{3} + 6 \, a b c\right )} x^{\frac{17}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{6}{13} \,{\left (a b^{2} + a^{2} c\right )} x^{\frac{13}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.19322, size = 246, normalized size = 2.39 \begin{align*} \frac{2}{3364725} \,{\left (116025 \, c^{3} x^{14} + 403767 \, b c^{2} x^{12} + 480675 \,{\left (b^{2} c + a c^{2}\right )} x^{10} + 197925 \,{\left (b^{3} + 6 \, a b c\right )} x^{8} + 1121575 \, a^{2} b x^{4} + 776475 \,{\left (a b^{2} + a^{2} c\right )} x^{6} + 672945 \, a^{3} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 36.9578, size = 129, normalized size = 1.25 \begin{align*} \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 a b^{2} x^{\frac{13}{2}}}{13} + \frac{12 a b c x^{\frac{17}{2}}}{17} + \frac{2 a c^{2} x^{\frac{21}{2}}}{7} + \frac{2 b^{3} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c x^{\frac{21}{2}}}{7} + \frac{6 b c^{2} x^{\frac{25}{2}}}{25} + \frac{2 c^{3} x^{\frac{29}{2}}}{29} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14891, size = 117, normalized size = 1.14 \begin{align*} \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{6}{25} \, b c^{2} x^{\frac{25}{2}} + \frac{2}{7} \, b^{2} c x^{\frac{21}{2}} + \frac{2}{7} \, a c^{2} x^{\frac{21}{2}} + \frac{2}{17} \, b^{3} x^{\frac{17}{2}} + \frac{12}{17} \, a b c x^{\frac{17}{2}} + \frac{6}{13} \, a b^{2} x^{\frac{13}{2}} + \frac{6}{13} \, a^{2} c x^{\frac{13}{2}} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{2}{5} \, a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]