Optimal. Leaf size=25 \[ b^2 x+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]
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Rubi [A] time = 0.0131654, antiderivative size = 25, normalized size of antiderivative = 1., 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, 194} \[ b^2 x+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 194
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^2}{x^4} \, dx &=\int \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2+2 b c x^2+c^2 x^4\right ) \, dx\\ &=b^2 x+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5}\\ \end{align*}
Mathematica [A] time = 0.0014582, size = 25, normalized size = 1. \[ b^2 x+\frac{2}{3} b c x^3+\frac{c^2 x^5}{5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 22, normalized size = 0.9 \begin{align*}{b}^{2}x+{\frac{2\,bc{x}^{3}}{3}}+{\frac{{c}^{2}{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989572, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, b c x^{3} + b^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15821, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, b c x^{3} + b^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.065741, size = 22, normalized size = 0.88 \begin{align*} b^{2} x + \frac{2 b c x^{3}}{3} + \frac{c^{2} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19576, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{5} \, c^{2} x^{5} + \frac{2}{3} \, b c x^{3} + b^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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