Optimal. Leaf size=43 \[ \frac{3}{7} b^2 c x^7+\frac{b^3 x^5}{5}+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11} \]
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Rubi [A] time = 0.023521, antiderivative size = 43, 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, 270} \[ \frac{3}{7} b^2 c x^7+\frac{b^3 x^5}{5}+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^3}{x^2} \, dx &=\int x^4 \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^4+3 b^2 c x^6+3 b c^2 x^8+c^3 x^{10}\right ) \, dx\\ &=\frac{b^3 x^5}{5}+\frac{3}{7} b^2 c x^7+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11}\\ \end{align*}
Mathematica [A] time = 0.0019193, size = 43, normalized size = 1. \[ \frac{3}{7} b^2 c x^7+\frac{b^3 x^5}{5}+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 36, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}{x}^{5}}{5}}+{\frac{3\,{b}^{2}c{x}^{7}}{7}}+{\frac{b{c}^{2}{x}^{9}}{3}}+{\frac{{c}^{3}{x}^{11}}{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01646, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{11} \, c^{3} x^{11} + \frac{1}{3} \, b c^{2} x^{9} + \frac{3}{7} \, b^{2} c x^{7} + \frac{1}{5} \, b^{3} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18463, size = 82, normalized size = 1.91 \begin{align*} \frac{1}{11} \, c^{3} x^{11} + \frac{1}{3} \, b c^{2} x^{9} + \frac{3}{7} \, b^{2} c x^{7} + \frac{1}{5} \, b^{3} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072205, size = 37, normalized size = 0.86 \begin{align*} \frac{b^{3} x^{5}}{5} + \frac{3 b^{2} c x^{7}}{7} + \frac{b c^{2} x^{9}}{3} + \frac{c^{3} x^{11}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.237, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{11} \, c^{3} x^{11} + \frac{1}{3} \, b c^{2} x^{9} + \frac{3}{7} \, b^{2} c x^{7} + \frac{1}{5} \, b^{3} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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