3.140 \(\int d x^3 \left (e+f x^4\right )^2 \, dx\)

Optimal. Leaf size=17 \[ \frac{d \left (e+f x^4\right )^3}{12 f} \]

[Out]

(d*(e + f*x^4)^3)/(12*f)

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Rubi [A]  time = 0.0180704, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{d \left (e+f x^4\right )^3}{12 f} \]

Antiderivative was successfully verified.

[In]  Int[d*x^3*(e + f*x^4)^2,x]

[Out]

(d*(e + f*x^4)^3)/(12*f)

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Rubi in Sympy [A]  time = 2.56475, size = 12, normalized size = 0.71 \[ \frac{d \left (e + f x^{4}\right )^{3}}{12 f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(d*x**3*(f*x**4+e)**2,x)

[Out]

d*(e + f*x**4)**3/(12*f)

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Mathematica [A]  time = 0.00149336, size = 33, normalized size = 1.94 \[ \frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

[In]  Integrate[d*x^3*(e + f*x^4)^2,x]

[Out]

(d*e^2*x^4)/4 + (d*e*f*x^8)/4 + (d*f^2*x^12)/12

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Maple [A]  time = 0.001, size = 27, normalized size = 1.6 \[ d \left ({\frac{{f}^{2}{x}^{12}}{12}}+{\frac{ef{x}^{8}}{4}}+{\frac{{e}^{2}{x}^{4}}{4}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(d*x^3*(f*x^4+e)^2,x)

[Out]

d*(1/12*f^2*x^12+1/4*e*f*x^8+1/4*e^2*x^4)

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Maxima [A]  time = 1.41036, size = 20, normalized size = 1.18 \[ \frac{{\left (f x^{4} + e\right )}^{3} d}{12 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e)^2*d*x^3,x, algorithm="maxima")

[Out]

1/12*(f*x^4 + e)^3*d/f

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Fricas [A]  time = 0.203853, size = 1, normalized size = 0.06 \[ \frac{1}{12} x^{12} f^{2} d + \frac{1}{4} x^{8} f e d + \frac{1}{4} x^{4} e^{2} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e)^2*d*x^3,x, algorithm="fricas")

[Out]

1/12*x^12*f^2*d + 1/4*x^8*f*e*d + 1/4*x^4*e^2*d

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Sympy [A]  time = 0.049223, size = 29, normalized size = 1.71 \[ \frac{d e^{2} x^{4}}{4} + \frac{d e f x^{8}}{4} + \frac{d f^{2} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d*x**3*(f*x**4+e)**2,x)

[Out]

d*e**2*x**4/4 + d*e*f*x**8/4 + d*f**2*x**12/12

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GIAC/XCAS [A]  time = 0.209642, size = 22, normalized size = 1.29 \[ \frac{{\left (f x^{4} + e\right )}^{3} d}{12 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e)^2*d*x^3,x, algorithm="giac")

[Out]

1/12*(f*x^4 + e)^3*d/f