Optimal. Leaf size=54 \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]
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Rubi [A] time = 0.0324322, antiderivative size = 54, normalized size of antiderivative = 1., 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} \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 698
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 = 0.0071731, size = 54, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{ab{x}^{4}}{2}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{5}}{5}}+{\frac{bc{x}^{6}}{3}}+{\frac{{c}^{2}{x}^{7}}{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15267, size = 59, normalized size = 1.09 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45505, size = 107, normalized size = 1.98 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072056, size = 48, normalized size = 0.89 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07228, size = 62, normalized size = 1.15 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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