Optimal. Leaf size=54 \[ \frac {a^2 x^3}{3}+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1585, 698} \begin {gather*} \frac {a^2 x^3}{3}+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rule 1585
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx &=\int x^2 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^2+2 a b x^3+\left (b^2+2 a c\right ) x^4+2 b c x^5+c^2 x^6\right ) \, dx\\ &=\frac {a^2 x^3}{3}+\frac {1}{2} a b x^4+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 54, normalized size = 1.00 \begin {gather*} \frac {a^2 x^3}{3}+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.04, size = 58, normalized size = 1.07 \begin {gather*} \frac {a^2 x^3}{3}+\frac {1}{2} a b x^4+\frac {2}{5} a c x^5+\frac {b^2 x^5}{5}+\frac {1}{3} b c x^6+\frac {c^2 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.48, size = 46, normalized size = 0.85 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{5} \, b^{2} x^{5} + \frac {2}{5} \, a c x^{5} + \frac {1}{2} \, a b x^{4} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 45, normalized size = 0.83 \begin {gather*} \frac {c^{2} x^{7}}{7}+\frac {b c \,x^{6}}{3}+\frac {a b \,x^{4}}{2}+\frac {a^{2} x^{3}}{3}+\frac {\left (2 a c +b^{2}\right ) x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} + \frac {1}{3} \, b c x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {1}{3} \, a^{2} x^{3} \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^5\,\left (\frac {b^2}{5}+\frac {2\,a\,c}{5}\right )+\frac {a^2\,x^3}{3}+\frac {c^2\,x^7}{7}+\frac {a\,b\,x^4}{2}+\frac {b\,c\,x^6}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 48, normalized size = 0.89 \begin {gather*} \frac {a^{2} x^{3}}{3} + \frac {a b x^{4}}{2} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{7}}{7} + x^{5} \left (\frac {2 a c}{5} + \frac {b^{2}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________