3.133 \(\int a \left (e+f x^4\right )^2 \, dx\)

Optimal. Leaf size=28 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9 \]

[Out]

a*e^2*x + (2*a*e*f*x^5)/5 + (a*f^2*x^9)/9

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Rubi [A]  time = 0.0330414, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9 \]

Antiderivative was successfully verified.

[In]  Int[a*(e + f*x^4)^2,x]

[Out]

a*e^2*x + (2*a*e*f*x^5)/5 + (a*f^2*x^9)/9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} + a \int e^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*e*f*x**5/5 + a*f**2*x**9/9 + a*Integral(e**2, x)

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Mathematica [A]  time = 0.00308368, size = 27, normalized size = 0.96 \[ a \left (e^2 x+\frac{2}{5} e f x^5+\frac{f^2 x^9}{9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[a*(e + f*x^4)^2,x]

[Out]

a*(e^2*x + (2*e*f*x^5)/5 + (f^2*x^9)/9)

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Maple [A]  time = 0.002, size = 24, normalized size = 0.9 \[ a \left ({\frac{{f}^{2}{x}^{9}}{9}}+{\frac{2\,ef{x}^{5}}{5}}+{e}^{2}x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a*(f*x^4+e)^2,x)

[Out]

a*(1/9*f^2*x^9+2/5*e*f*x^5+e^2*x)

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Maxima [A]  time = 1.36085, size = 34, normalized size = 1.21 \[ \frac{1}{45} \,{\left (5 \, f^{2} x^{9} + 18 \, e f x^{5} + 45 \, e^{2} x\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(5*f^2*x^9 + 18*e*f*x^5 + 45*e^2*x)*a

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Fricas [A]  time = 0.198251, size = 1, normalized size = 0.04 \[ \frac{1}{9} x^{9} f^{2} a + \frac{2}{5} x^{5} f e a + x e^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^9*f^2*a + 2/5*x^5*f*e*a + x*e^2*a

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Sympy [A]  time = 0.043121, size = 27, normalized size = 0.96 \[ a e^{2} x + \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a*(f*x**4+e)**2,x)

[Out]

a*e**2*x + 2*a*e*f*x**5/5 + a*f**2*x**9/9

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GIAC/XCAS [A]  time = 0.20868, size = 34, normalized size = 1.21 \[ \frac{1}{45} \,{\left (5 \, f^{2} x^{9} + 18 \, f x^{5} e + 45 \, x e^{2}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(5*f^2*x^9 + 18*f*x^5*e + 45*x*e^2)*a