Optimal. Leaf size=89 \[ \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{11} c x^{11} \left (a c+b^2\right )+\frac {1}{9} b x^9 \left (6 a c+b^2\right )+\frac {3}{7} a x^7 \left (a c+b^2\right )+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1108} \begin {gather*} \frac {3}{5} a^2 b x^5+\frac {a^3 x^3}{3}+\frac {3}{11} c x^{11} \left (a c+b^2\right )+\frac {1}{9} b x^9 \left (6 a c+b^2\right )+\frac {3}{7} a x^7 \left (a c+b^2\right )+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1108
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 x^2+3 a^2 b x^4+3 a \left (b^2+a c\right ) x^6+b \left (b^2+6 a c\right ) x^8+3 c \left (b^2+a c\right ) x^{10}+3 b c^2 x^{12}+c^3 x^{14}\right ) \, dx\\ &=\frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a \left (b^2+a c\right ) x^7+\frac {1}{9} b \left (b^2+6 a c\right ) x^9+\frac {3}{11} c \left (b^2+a c\right ) x^{11}+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 89, normalized size = 1.00 \begin {gather*} \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{11} c x^{11} \left (a c+b^2\right )+\frac {1}{9} b x^9 \left (6 a c+b^2\right )+\frac {3}{7} a x^7 \left (a c+b^2\right )+\frac {3}{13} b c^2 x^{13}+\frac {c^3 x^{15}}{15} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b x^2+c x^4\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.26, size = 87, normalized size = 0.98 \begin {gather*} \frac {1}{15} x^{15} c^{3} + \frac {3}{13} x^{13} c^{2} b + \frac {3}{11} x^{11} c b^{2} + \frac {3}{11} x^{11} c^{2} a + \frac {1}{9} x^{9} b^{3} + \frac {2}{3} x^{9} c b a + \frac {3}{7} x^{7} b^{2} a + \frac {3}{7} x^{7} c a^{2} + \frac {3}{5} x^{5} b a^{2} + \frac {1}{3} x^{3} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 87, normalized size = 0.98 \begin {gather*} \frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, b^{2} c x^{11} + \frac {3}{11} \, a c^{2} x^{11} + \frac {1}{9} \, b^{3} x^{9} + \frac {2}{3} \, a b c x^{9} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{7} \, a^{2} c x^{7} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 111, normalized size = 1.25 \begin {gather*} \frac {c^{3} x^{15}}{15}+\frac {3 b \,c^{2} x^{13}}{13}+\frac {\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) x^{11}}{11}+\frac {\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) x^{9}}{9}+\frac {3 a^{2} b \,x^{5}}{5}+\frac {\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) x^{7}}{7}+\frac {a^{3} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 81, normalized size = 0.91 \begin {gather*} \frac {1}{15} \, c^{3} x^{15} + \frac {3}{13} \, b c^{2} x^{13} + \frac {3}{11} \, {\left (b^{2} c + a c^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{3} + 6 \, a b c\right )} x^{9} + \frac {3}{5} \, a^{2} b x^{5} + \frac {3}{7} \, {\left (a b^{2} + a^{2} c\right )} x^{7} + \frac {1}{3} \, a^{3} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 76, normalized size = 0.85 \begin {gather*} x^9\,\left (\frac {b^3}{9}+\frac {2\,a\,c\,b}{3}\right )+\frac {a^3\,x^3}{3}+\frac {c^3\,x^{15}}{15}+\frac {3\,a^2\,b\,x^5}{5}+\frac {3\,b\,c^2\,x^{13}}{13}+\frac {3\,a\,x^7\,\left (b^2+a\,c\right )}{7}+\frac {3\,c\,x^{11}\,\left (b^2+a\,c\right )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 97, normalized size = 1.09 \begin {gather*} \frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{5}}{5} + \frac {3 b c^{2} x^{13}}{13} + \frac {c^{3} x^{15}}{15} + x^{11} \left (\frac {3 a c^{2}}{11} + \frac {3 b^{2} c}{11}\right ) + x^{9} \left (\frac {2 a b c}{3} + \frac {b^{3}}{9}\right ) + x^{7} \left (\frac {3 a^{2} c}{7} + \frac {3 a b^{2}}{7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________