3.134 \(\int b x \left (e+f x^4\right )^2 \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10} \]

[Out]

(b*e^2*x^2)/2 + (b*e*f*x^6)/3 + (b*f^2*x^10)/10

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Rubi [A]  time = 0.0429488, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10} \]

Antiderivative was successfully verified.

[In]  Int[b*x*(e + f*x^4)^2,x]

[Out]

(b*e^2*x^2)/2 + (b*e*f*x^6)/3 + (b*f^2*x^10)/10

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ b e^{2} \int x\, dx + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*e**2*Integral(x, x) + b*e*f*x**6/3 + b*f**2*x**10/10

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Mathematica [A]  time = 0.00245683, size = 32, normalized size = 0.97 \[ b \left (\frac{e^2 x^2}{2}+\frac{1}{3} e f x^6+\frac{f^2 x^{10}}{10}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[b*x*(e + f*x^4)^2,x]

[Out]

b*((e^2*x^2)/2 + (e*f*x^6)/3 + (f^2*x^10)/10)

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Maple [A]  time = 0.001, size = 27, normalized size = 0.8 \[ b \left ({\frac{{f}^{2}{x}^{10}}{10}}+{\frac{ef{x}^{6}}{3}}+{\frac{{e}^{2}{x}^{2}}{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(b*x*(f*x^4+e)^2,x)

[Out]

b*(1/10*f^2*x^10+1/3*e*f*x^6+1/2*e^2*x^2)

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Maxima [A]  time = 7.47718, size = 36, normalized size = 1.09 \[ \frac{1}{30} \,{\left (3 \, f^{2} x^{10} + 10 \, e f x^{6} + 15 \, e^{2} x^{2}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(3*f^2*x^10 + 10*e*f*x^6 + 15*e^2*x^2)*b

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Fricas [A]  time = 0.195803, size = 1, normalized size = 0.03 \[ \frac{1}{10} x^{10} f^{2} b + \frac{1}{3} x^{6} f e b + \frac{1}{2} x^{2} e^{2} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*f^2*b + 1/3*x^6*f*e*b + 1/2*x^2*e^2*b

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Sympy [A]  time = 0.048491, size = 29, normalized size = 0.88 \[ \frac{b e^{2} x^{2}}{2} + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(b*x*(f*x**4+e)**2,x)

[Out]

b*e**2*x**2/2 + b*e*f*x**6/3 + b*f**2*x**10/10

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GIAC/XCAS [A]  time = 0.207203, size = 36, normalized size = 1.09 \[ \frac{1}{30} \,{\left (3 \, f^{2} x^{10} + 10 \, f x^{6} e + 15 \, x^{2} e^{2}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(3*f^2*x^10 + 10*f*x^6*e + 15*x^2*e^2)*b