Optimal. Leaf size=54 \[ \frac {a^2 x^2}{2}+\frac {1}{6} x^6 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{4} b c x^8+\frac {c^2 x^{10}}{10} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1585, 1107, 611} \begin {gather*} \frac {a^2 x^2}{2}+\frac {1}{6} x^6 \left (2 a c+b^2\right )+\frac {1}{2} a b x^4+\frac {1}{4} b c x^8+\frac {c^2 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
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Rule 611
Rule 1107
Rule 1585
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^3+c x^5\right )^2}{x} \, dx &=\int x \left (a+b x^2+c x^4\right )^2 \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (a^2+2 a b x+b^2 \left (1+\frac {2 a c}{b^2}\right ) x^2+2 b c x^3+c^2 x^4\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^2}{2}+\frac {1}{2} a b x^4+\frac {1}{6} \left (b^2+2 a c\right ) x^6+\frac {1}{4} b c x^8+\frac {c^2 x^{10}}{10}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 48, normalized size = 0.89 \begin {gather*} \frac {1}{60} x^2 \left (30 a^2+10 x^4 \left (2 a c+b^2\right )+30 a b x^2+15 b c x^6+6 c^2 x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 50, normalized size = 0.93 \begin {gather*} \frac {1}{60} x^2 \left (30 a^2+30 a b x^2+20 a c x^4+10 b^2 x^4+15 b c x^6+6 c^2 x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 44, normalized size = 0.81 \begin {gather*} \frac {1}{10} \, c^{2} x^{10} + \frac {1}{4} \, b c x^{8} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 46, normalized size = 0.85 \begin {gather*} \frac {1}{10} \, c^{2} x^{10} + \frac {1}{4} \, b c x^{8} + \frac {1}{6} \, b^{2} x^{6} + \frac {1}{3} \, a c x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \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^{10}}{10}+\frac {b c \,x^{8}}{4}+\frac {a b \,x^{4}}{2}+\frac {\left (2 a c +b^{2}\right ) x^{6}}{6}+\frac {a^{2} x^{2}}{2} \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}{10} \, c^{2} x^{10} + \frac {1}{4} \, b c x^{8} + \frac {1}{6} \, {\left (b^{2} + 2 \, a c\right )} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \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^2}{2}+\frac {c^2\,x^{10}}{10}+\frac {a\,b\,x^4}{2}+\frac {b\,c\,x^8}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 46, normalized size = 0.85 \begin {gather*} \frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b c x^{8}}{4} + \frac {c^{2} x^{10}}{10} + 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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