Optimal. Leaf size=77 \[ a e^2 x+\frac {2}{5} a e f x^5+\frac {1}{9} a f^2 x^9+\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11}+\frac {d \left (e+f x^4\right )^3}{12 f} \]
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Rubi [A] time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1582, 1154} \[ a e^2 x+\frac {2}{5} a e f x^5+\frac {1}{9} a f^2 x^9+\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11}+\frac {d \left (e+f x^4\right )^3}{12 f} \]
Antiderivative was successfully verified.
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Rule 1154
Rule 1582
Rubi steps
\begin {align*} \int \left (a+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 \left (a+c x^2\right ) \left (e+f x^4\right )^2 \, dx\\ &=\frac {d \left (e+f x^4\right )^3}{12 f}+\int \left (a e^2+c e^2 x^2+2 a e f x^4+2 c e f x^6+a f^2 x^8+c f^2 x^{10}\right ) \, dx\\ &=a e^2 x+\frac {1}{3} c e^2 x^3+\frac {2}{5} a e f x^5+\frac {2}{7} c e f x^7+\frac {1}{9} a f^2 x^9+\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 = 0.01, size = 92, normalized size = 1.19 \[ a e^2 x+\frac {2}{5} a e f x^5+\frac {1}{9} a f^2 x^9+\frac {1}{3} c e^2 x^3+\frac {2}{7} c e f x^7+\frac {1}{11} c f^2 x^{11}+\frac {1}{4} d e^2 x^4+\frac {1}{4} d e f x^8+\frac {1}{12} d f^2 x^{12} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 76, normalized size = 0.99 \[ \frac {1}{12} x^{12} f^{2} d + \frac {1}{11} x^{11} f^{2} c + \frac {1}{9} x^{9} f^{2} a + \frac {1}{4} x^{8} f e d + \frac {2}{7} x^{7} f e c + \frac {2}{5} x^{5} f e a + \frac {1}{4} x^{4} e^{2} d + \frac {1}{3} x^{3} e^{2} c + x e^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 76, normalized size = 0.99 \[ \frac {1}{12} \, d f^{2} x^{12} + \frac {1}{11} \, c f^{2} x^{11} + \frac {1}{9} \, a f^{2} x^{9} + \frac {1}{4} \, d f x^{8} e + \frac {2}{7} \, c f x^{7} e + \frac {2}{5} \, a f x^{5} e + \frac {1}{4} \, d x^{4} e^{2} + \frac {1}{3} \, c x^{3} e^{2} + a x e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 77, normalized size = 1.00 \[ \frac {1}{12} d \,f^{2} x^{12}+\frac {1}{11} c \,f^{2} x^{11}+\frac {1}{9} a \,f^{2} x^{9}+\frac {1}{4} d e f \,x^{8}+\frac {2}{7} c e f \,x^{7}+\frac {2}{5} a e f \,x^{5}+\frac {1}{4} d \,e^{2} x^{4}+\frac {1}{3} c \,e^{2} x^{3}+a \,e^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 76, normalized size = 0.99 \[ \frac {1}{12} \, d f^{2} x^{12} + \frac {1}{11} \, c f^{2} x^{11} + \frac {1}{9} \, a f^{2} x^{9} + \frac {1}{4} \, d e f x^{8} + \frac {2}{7} \, c e f x^{7} + \frac {2}{5} \, a e f x^{5} + \frac {1}{4} \, d e^{2} x^{4} + \frac {1}{3} \, c e^{2} x^{3} + a e^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 76, normalized size = 0.99 \[ \frac {d\,e^2\,x^4}{4}+\frac {c\,e^2\,x^3}{3}+a\,e^2\,x+\frac {d\,e\,f\,x^8}{4}+\frac {2\,c\,e\,f\,x^7}{7}+\frac {2\,a\,e\,f\,x^5}{5}+\frac {d\,f^2\,x^{12}}{12}+\frac {c\,f^2\,x^{11}}{11}+\frac {a\,f^2\,x^9}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 90, normalized size = 1.17 \[ a e^{2} x + \frac {2 a e f x^{5}}{5} + \frac {a f^{2} x^{9}}{9} + \frac {c e^{2} x^{3}}{3} + \frac {2 c e f x^{7}}{7} + \frac {c f^{2} x^{11}}{11} + \frac {d e^{2} x^{4}}{4} + \frac {d e f x^{8}}{4} + \frac {d f^{2} x^{12}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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