Optimal. Leaf size=65 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]
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Rubi [A] time = 0.0975165, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1593, 1620} \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1620
Rubi steps
\begin{align*} \int \left (b x+c x^2\right ) \left (e+f x^4\right )^2 \, dx &=\int x (b+c x) \left (e+f x^4\right )^2 \, dx\\ &=\int \left (b e^2 x+c e^2 x^2+2 b e f x^5+2 c e f x^6+b f^2 x^9+c f^2 x^{10}\right ) \, dx\\ &=\frac{1}{2} b e^2 x^2+\frac{1}{3} c e^2 x^3+\frac{1}{3} b e f x^6+\frac{2}{7} c e f x^7+\frac{1}{10} b f^2 x^{10}+\frac{1}{11} c f^2 x^{11}\\ \end{align*}
Mathematica [A] time = 0.0025941, size = 65, normalized size = 1. \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 54, normalized size = 0.8 \begin{align*}{\frac{b{e}^{2}{x}^{2}}{2}}+{\frac{c{e}^{2}{x}^{3}}{3}}+{\frac{bef{x}^{6}}{3}}+{\frac{2\,cef{x}^{7}}{7}}+{\frac{b{f}^{2}{x}^{10}}{10}}+{\frac{c{f}^{2}{x}^{11}}{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01028, size = 72, normalized size = 1.11 \begin{align*} \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{10} \, b f^{2} x^{10} + \frac{2}{7} \, c e f x^{7} + \frac{1}{3} \, b e f x^{6} + \frac{1}{3} \, c e^{2} x^{3} + \frac{1}{2} \, b e^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08661, size = 134, normalized size = 2.06 \begin{align*} \frac{1}{11} x^{11} f^{2} c + \frac{1}{10} x^{10} f^{2} b + \frac{2}{7} x^{7} f e c + \frac{1}{3} x^{6} f e b + \frac{1}{3} x^{3} e^{2} c + \frac{1}{2} x^{2} e^{2} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.068121, size = 61, normalized size = 0.94 \begin{align*} \frac{b e^{2} x^{2}}{2} + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0566, size = 72, normalized size = 1.11 \begin{align*} \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{10} \, b f^{2} x^{10} + \frac{2}{7} \, c f x^{7} e + \frac{1}{3} \, b f x^{6} e + \frac{1}{3} \, c x^{3} e^{2} + \frac{1}{2} \, b x^{2} e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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