Optimal. Leaf size=50 \[ \frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11}+\frac {d \left (e+f x^4\right )^3}{12 f} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1582, 12, 270} \[ \frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11}+\frac {d \left (e+f x^4\right )^3}{12 f} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 1582
Rubi steps
\begin {align*} \int \left (c x^2+d x^3\right ) \left (e+f x^4\right )^2 \, dx &=\frac {d \left (e+f x^4\right )^3}{12 f}+\int c x^2 \left (e+f x^4\right )^2 \, dx\\ &=\frac {d \left (e+f x^4\right )^3}{12 f}+c \int x^2 \left (e+f x^4\right )^2 \, dx\\ &=\frac {d \left (e+f x^4\right )^3}{12 f}+c \int \left (e^2 x^2+2 e f x^6+f^2 x^{10}\right ) \, dx\\ &=\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11}+\frac {d \left (e+f x^4\right )^3}{12 f}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 65, normalized size = 1.30 \[ \frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11}+\frac {1}{4} d e^2 x^4+\frac {1}{4} d e f x^8+\frac {1}{12} d f^2 x^{12} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 53, normalized size = 1.06 \[ \frac {1}{12} x^{12} f^{2} d + \frac {1}{11} x^{11} f^{2} c + \frac {1}{4} x^{8} f e d + \frac {2}{7} x^{7} f e c + \frac {1}{4} x^{4} e^{2} d + \frac {1}{3} x^{3} e^{2} c \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 1.06 \[ \frac {1}{12} \, d f^{2} x^{12} + \frac {1}{11} \, c f^{2} x^{11} + \frac {1}{4} \, d f x^{8} e + \frac {2}{7} \, c f x^{7} e + \frac {1}{4} \, d x^{4} e^{2} + \frac {1}{3} \, c x^{3} e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 54, normalized size = 1.08 \[ \frac {1}{12} d \,f^{2} x^{12}+\frac {1}{11} c \,f^{2} x^{11}+\frac {1}{4} d e f \,x^{8}+\frac {2}{7} c e f \,x^{7}+\frac {1}{4} d \,e^{2} x^{4}+\frac {1}{3} c \,e^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 53, normalized size = 1.06 \[ \frac {1}{12} \, d f^{2} x^{12} + \frac {1}{11} \, c f^{2} x^{11} + \frac {1}{4} \, d e f x^{8} + \frac {2}{7} \, c e f x^{7} + \frac {1}{4} \, d e^{2} x^{4} + \frac {1}{3} \, c e^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 53, normalized size = 1.06 \[ \frac {d\,e^2\,x^4}{4}+\frac {c\,e^2\,x^3}{3}+\frac {d\,e\,f\,x^8}{4}+\frac {2\,c\,e\,f\,x^7}{7}+\frac {d\,f^2\,x^{12}}{12}+\frac {c\,f^2\,x^{11}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 61, normalized size = 1.22 \[ \frac {c e^{2} x^{3}}{3} + \frac {2 c e f x^{7}}{7} + \frac {c f^{2} x^{11}}{11} + \frac {d e^{2} x^{4}}{4} + \frac {d e f x^{8}}{4} + \frac {d f^{2} x^{12}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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