Optimal. Leaf size=54 \[ \frac {a^2 x^4}{4}+\frac {1}{6} x^6 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \]
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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^4}{4}+\frac {1}{6} x^6 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rule 1585
Rubi steps
\begin {align*} \int \frac {\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx &=\int x^3 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^3+2 a b x^4+\left (b^2+2 a c\right ) x^5+2 b c x^6+c^2 x^7\right ) \, dx\\ &=\frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {1}{6} \left (b^2+2 a c\right ) x^6+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 1.00 \begin {gather*} \frac {a^2 x^4}{4}+\frac {1}{6} x^6 \left (2 a c+b^2\right )+\frac {2}{5} a b x^5+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 58, normalized size = 1.07 \begin {gather*} \frac {a^2 x^4}{4}+\frac {2}{5} a b x^5+\frac {1}{3} a c x^6+\frac {b^2 x^6}{6}+\frac {2}{7} b c x^7+\frac {c^2 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {2}{7} \, b c x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 46, normalized size = 0.85 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {2}{7} \, b c x^{7} + \frac {1}{6} \, b^{2} x^{6} + \frac {1}{3} \, a c x^{6} + \frac {2}{5} \, a b x^{5} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 0.83 \begin {gather*} \frac {c^{2} x^{8}}{8}+\frac {2 b c \,x^{7}}{7}+\frac {2 a b \,x^{5}}{5}+\frac {a^{2} x^{4}}{4}+\frac {\left (2 a c +b^{2}\right ) x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {2}{7} \, b c x^{7} + \frac {2}{5} \, a b x^{5} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 45, normalized size = 0.83 \begin {gather*} x^6\,\left (\frac {b^2}{6}+\frac {a\,c}{3}\right )+\frac {a^2\,x^4}{4}+\frac {c^2\,x^8}{8}+\frac {2\,a\,b\,x^5}{5}+\frac {2\,b\,c\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 49, normalized size = 0.91 \begin {gather*} \frac {a^{2} x^{4}}{4} + \frac {2 a b x^{5}}{5} + \frac {2 b c x^{7}}{7} + \frac {c^{2} x^{8}}{8} + x^{6} \left (\frac {a c}{3} + \frac {b^{2}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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