Optimal. Leaf size=50 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{d \left (e+f x^4\right )^3}{12 f} \]
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Rubi [A] time = 0.0198986, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1582, 12, 270} \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{d \left (e+f x^4\right )^3}{12 f} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 12
Rule 270
Rubi steps
\begin{align*} \int \left (b x+d x^3\right ) \left (e+f x^4\right )^2 \, dx &=\frac{d \left (e+f x^4\right )^3}{12 f}+\int b x \left (e+f x^4\right )^2 \, dx\\ &=\frac{d \left (e+f x^4\right )^3}{12 f}+b \int x \left (e+f x^4\right )^2 \, dx\\ &=\frac{d \left (e+f x^4\right )^3}{12 f}+b \int \left (e^2 x+2 e f x^5+f^2 x^9\right ) \, dx\\ &=\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{d \left (e+f x^4\right )^3}{12 f}\\ \end{align*}
Mathematica [A] time = 0.002989, size = 65, normalized size = 1.3 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 54, normalized size = 1.1 \begin{align*}{\frac{d{f}^{2}{x}^{12}}{12}}+{\frac{b{f}^{2}{x}^{10}}{10}}+{\frac{def{x}^{8}}{4}}+{\frac{bef{x}^{6}}{3}}+{\frac{d{e}^{2}{x}^{4}}{4}}+{\frac{b{e}^{2}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00185, size = 72, normalized size = 1.44 \begin{align*} \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{4} \, d e f x^{8} + \frac{1}{3} \, b e f x^{6} + \frac{1}{4} \, d e^{2} x^{4} + \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 = 1.02542, size = 134, normalized size = 2.68 \begin{align*} \frac{1}{12} x^{12} f^{2} d + \frac{1}{10} x^{10} f^{2} b + \frac{1}{4} x^{8} f e d + \frac{1}{3} x^{6} f e b + \frac{1}{4} x^{4} e^{2} d + \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 = 0.065855, size = 60, normalized size = 1.2 \begin{align*} \frac{b e^{2} x^{2}}{2} + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} + \frac{d e^{2} x^{4}}{4} + \frac{d e f x^{8}}{4} + \frac{d f^{2} x^{12}}{12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05355, size = 72, normalized size = 1.44 \begin{align*} \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{4} \, d f x^{8} e + \frac{1}{3} \, b f x^{6} e + \frac{1}{4} \, d x^{4} 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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