Optimal. Leaf size=49 \[ a^2 x+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {2}{3} a b x^3+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \]
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Rubi [A] time = 0.03, antiderivative size = 49, 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 = {1585, 1090} \[ a^2 x+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {2}{3} a b x^3+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \]
Antiderivative was successfully verified.
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Rule 1090
Rule 1585
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^3+c x^5\right )^2}{x^2} \, dx &=\int \left (a+b x^2+c x^4\right )^2 \, dx\\ &=\int \left (a^2+2 a b x^2+b^2 \left (1+\frac {2 a c}{b^2}\right ) x^4+2 b c x^6+c^2 x^8\right ) \, dx\\ &=a^2 x+\frac {2}{3} a b x^3+\frac {1}{5} \left (b^2+2 a c\right ) x^5+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.00 \[ a^2 x+\frac {1}{5} x^5 \left (2 a c+b^2\right )+\frac {2}{3} a b x^3+\frac {2}{7} b c x^7+\frac {c^2 x^9}{9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 41, normalized size = 0.84 \[ \frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 43, normalized size = 0.88 \[ \frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, b^{2} x^{5} + \frac {2}{5} \, a c x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 42, normalized size = 0.86 \[ \frac {c^{2} x^{9}}{9}+\frac {2 b c \,x^{7}}{7}+\frac {2 a b \,x^{3}}{3}+\frac {\left (2 a c +b^{2}\right ) x^{5}}{5}+a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 41, normalized size = 0.84 \[ \frac {1}{9} \, c^{2} x^{9} + \frac {2}{7} \, b c x^{7} + \frac {1}{5} \, {\left (b^{2} + 2 \, a c\right )} x^{5} + \frac {2}{3} \, a b x^{3} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 42, normalized size = 0.86 \[ a^2\,x+x^5\,\left (\frac {b^2}{5}+\frac {2\,a\,c}{5}\right )+\frac {c^2\,x^9}{9}+\frac {2\,a\,b\,x^3}{3}+\frac {2\,b\,c\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 48, normalized size = 0.98 \[ a^{2} x + \frac {2 a b x^{3}}{3} + \frac {2 b c x^{7}}{7} + \frac {c^{2} x^{9}}{9} + x^{5} \left (\frac {2 a c}{5} + \frac {b^{2}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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