Optimal. Leaf size=77 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{d \left (e+f x^4\right )^3}{12 f} \]
[Out]
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Rubi [A] time = 0.116891, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{d \left (e+f x^4\right )^3}{12 f} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2 + d*x^3)*(e + f*x^4)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} + \frac{d \left (e + f x^{4}\right )^{3}}{12 f} + e^{2} \int a\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c*x**2+a)*(f*x**4+e)**2,x)
[Out]
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Mathematica [A] time = 0.00620127, 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.
[In] Integrate[(a + c*x^2 + d*x^3)*(e + f*x^4)^2,x]
[Out]
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Maple [A] time = 0.001, size = 77, normalized size = 1. \[{\frac{d{f}^{2}{x}^{12}}{12}}+{\frac{c{f}^{2}{x}^{11}}{11}}+{\frac{a{f}^{2}{x}^{9}}{9}}+{\frac{def{x}^{8}}{4}}+{\frac{2\,cef{x}^{7}}{7}}+{\frac{2\,aef{x}^{5}}{5}}+{\frac{d{e}^{2}{x}^{4}}{4}}+{\frac{c{e}^{2}{x}^{3}}{3}}+a{e}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c*x^2+a)*(f*x^4+e)^2,x)
[Out]
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Maxima [A] time = 1.37371, size = 103, normalized size = 1.34 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{1}{4} \, d e f x^{8} + \frac{2}{7} \, c e f x^{7} + \frac{2}{5} \, a e f x^{5} + \frac{1}{4} \, d e^{2} x^{4} + \frac{1}{3} \, c e^{2} x^{3} + a e^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^4 + e)^2*(d*x^3 + c*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202007, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} f^{2} d + \frac{1}{11} x^{11} f^{2} c + \frac{1}{9} x^{9} f^{2} a + \frac{1}{4} x^{8} f e d + \frac{2}{7} x^{7} f e c + \frac{2}{5} x^{5} f e a + \frac{1}{4} x^{4} e^{2} d + \frac{1}{3} x^{3} e^{2} c + x e^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^4 + e)^2*(d*x^3 + c*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.066001, 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.
[In] integrate((d*x**3+c*x**2+a)*(f*x**4+e)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.207852, size = 103, normalized size = 1.34 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{1}{4} \, d f x^{8} e + \frac{2}{7} \, c f x^{7} e + \frac{2}{5} \, a f x^{5} e + \frac{1}{4} \, d x^{4} e^{2} + \frac{1}{3} \, c x^{3} e^{2} + a x e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^4 + e)^2*(d*x^3 + c*x^2 + a),x, algorithm="giac")
[Out]