Optimal. Leaf size=30 \[ \frac {b^2 x^3}{3}+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1584, 270} \[ \frac {b^2 x^3}{3}+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^2} \, dx &=\int x^2 \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^2+2 b c x^4+c^2 x^6\right ) \, dx\\ &=\frac {b^2 x^3}{3}+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \[ \frac {b^2 x^3}{3}+\frac {2}{5} b c x^5+\frac {c^2 x^7}{7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 24, normalized size = 0.80 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {2}{5} \, b c x^{5} + \frac {1}{3} \, b^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 24, normalized size = 0.80 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {2}{5} \, b c x^{5} + \frac {1}{3} \, b^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.83 \[ \frac {1}{7} c^{2} x^{7}+\frac {2}{5} b c \,x^{5}+\frac {1}{3} b^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 24, normalized size = 0.80 \[ \frac {1}{7} \, c^{2} x^{7} + \frac {2}{5} \, b c x^{5} + \frac {1}{3} \, b^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.80 \[ \frac {b^2\,x^3}{3}+\frac {2\,b\,c\,x^5}{5}+\frac {c^2\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 26, normalized size = 0.87 \[ \frac {b^{2} x^{3}}{3} + \frac {2 b c x^{5}}{5} + \frac {c^{2} x^{7}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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