Optimal. Leaf size=48 \[ -\frac {a^2}{x}+\frac {1}{3} x^3 \left (2 a c+b^2\right )+2 a b x+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \]
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Rubi [A] time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1108} \[ -\frac {a^2}{x}+\frac {1}{3} x^3 \left (2 a c+b^2\right )+2 a b x+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \]
Antiderivative was successfully verified.
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Rule 1108
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^2} \, dx &=\int \left (2 a b+\frac {a^2}{x^2}+\left (b^2+2 a c\right ) x^2+2 b c x^4+c^2 x^6\right ) \, dx\\ &=-\frac {a^2}{x}+2 a b x+\frac {1}{3} \left (b^2+2 a c\right ) x^3+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 1.00 \[ -\frac {a^2}{x}+\frac {1}{3} x^3 \left (2 a c+b^2\right )+2 a b x+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 46, normalized size = 0.96 \[ \frac {15 \, c^{2} x^{8} + 42 \, b c x^{6} + 35 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 210 \, a b x^{2} - 105 \, a^{2}}{105 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 44, normalized size = 0.92 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {2}{5} \, b c x^{5} + \frac {1}{3} \, b^{2} x^{3} + \frac {2}{3} \, a c x^{3} + 2 \, a b x - \frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 0.94 \[ \frac {c^{2} x^{7}}{7}+\frac {2 b c \,x^{5}}{5}+\frac {2 a c \,x^{3}}{3}+\frac {b^{2} x^{3}}{3}+2 a b x -\frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 42, normalized size = 0.88 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {2}{5} \, b c x^{5} + \frac {1}{3} \, {\left (b^{2} + 2 \, a c\right )} x^{3} + 2 \, a b x - \frac {a^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 43, normalized size = 0.90 \[ x^3\,\left (\frac {b^2}{3}+\frac {2\,a\,c}{3}\right )-\frac {a^2}{x}+\frac {c^2\,x^7}{7}+2\,a\,b\,x+\frac {2\,b\,c\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 44, normalized size = 0.92 \[ - \frac {a^{2}}{x} + 2 a b x + \frac {2 b c x^{5}}{5} + \frac {c^{2} x^{7}}{7} + x^{3} \left (\frac {2 a c}{3} + \frac {b^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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