Optimal. Leaf size=54 \[ \frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {1}{8} \left (b^2+2 a c\right ) x^8+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1599, 712}
\begin {gather*} \frac {a^2 x^6}{6}+\frac {1}{8} x^8 \left (2 a c+b^2\right )+\frac {2}{7} a b x^7+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 712
Rule 1599
Rubi steps
\begin {align*} \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx &=\int x^5 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^5+2 a b x^6+\left (b^2+2 a c\right ) x^7+2 b c x^8+c^2 x^9\right ) \, dx\\ &=\frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {1}{8} \left (b^2+2 a c\right ) x^8+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 54, normalized size = 1.00 \begin {gather*} \frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {1}{8} \left (b^2+2 a c\right ) x^8+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, size = 45, normalized size = 0.83
method | result | size |
default | \(\frac {a^{2} x^{6}}{6}+\frac {2 a b \,x^{7}}{7}+\frac {\left (2 a c +b^{2}\right ) x^{8}}{8}+\frac {2 b c \,x^{9}}{9}+\frac {c^{2} x^{10}}{10}\) | \(45\) |
norman | \(\frac {c^{2} x^{10}}{10}+\frac {2 b c \,x^{9}}{9}+\left (\frac {a c}{4}+\frac {b^{2}}{8}\right ) x^{8}+\frac {2 a b \,x^{7}}{7}+\frac {a^{2} x^{6}}{6}\) | \(46\) |
gosper | \(\frac {x^{6} \left (252 c^{2} x^{4}+560 b c \,x^{3}+630 a c \,x^{2}+315 b^{2} x^{2}+720 a b x +420 a^{2}\right )}{2520}\) | \(47\) |
risch | \(\frac {1}{6} a^{2} x^{6}+\frac {2}{7} a b \,x^{7}+\frac {1}{4} x^{8} a c +\frac {1}{8} b^{2} x^{8}+\frac {2}{9} b c \,x^{9}+\frac {1}{10} c^{2} x^{10}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{10} \, c^{2} x^{10} + \frac {2}{9} \, b c x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{8} \, {\left (b^{2} + 2 \, a c\right )} x^{8} + \frac {1}{6} \, a^{2} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.32, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{10} \, c^{2} x^{10} + \frac {2}{9} \, b c x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{8} \, {\left (b^{2} + 2 \, a c\right )} x^{8} + \frac {1}{6} \, a^{2} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 49, normalized size = 0.91 \begin {gather*} \frac {a^{2} x^{6}}{6} + \frac {2 a b x^{7}}{7} + \frac {2 b c x^{9}}{9} + \frac {c^{2} x^{10}}{10} + x^{8} \left (\frac {a c}{4} + \frac {b^{2}}{8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.99, size = 46, normalized size = 0.85 \begin {gather*} \frac {1}{10} \, c^{2} x^{10} + \frac {2}{9} \, b c x^{9} + \frac {1}{8} \, b^{2} x^{8} + \frac {1}{4} \, a c x^{8} + \frac {2}{7} \, a b x^{7} + \frac {1}{6} \, a^{2} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.02, size = 45, normalized size = 0.83 \begin {gather*} x^8\,\left (\frac {b^2}{8}+\frac {a\,c}{4}\right )+\frac {a^2\,x^6}{6}+\frac {c^2\,x^{10}}{10}+\frac {2\,a\,b\,x^7}{7}+\frac {2\,b\,c\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________