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

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 = 0.0143123, 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, Rules used = {12, 270} \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10} \]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int b x \left (e+f x^4\right )^2 \, dx &=b \int x \left (e+f x^4\right )^2 \, dx\\ &=b \int \left (e^2 x+2 e f x^5+f^2 x^9\right ) \, dx\\ &=\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}\\ \end{align*}

Mathematica [A]  time = 0.001375, 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 = 0.039, size = 27, normalized size = 0.8 \begin{align*} b \left ({\frac{{f}^{2}{x}^{10}}{10}}+{\frac{ef{x}^{6}}{3}}+{\frac{{e}^{2}{x}^{2}}{2}} \right ) \end{align*}

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 = 1.05494, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{30} \,{\left (3 \, f^{2} x^{10} + 10 \, e f x^{6} + 15 \, e^{2} x^{2}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x*(f*x^4+e)^2,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 = 1.06542, size = 66, normalized size = 2. \begin{align*} \frac{1}{10} x^{10} f^{2} b + \frac{1}{3} x^{6} f e b + \frac{1}{2} x^{2} e^{2} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b*x*(f*x^4+e)^2,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 = 0.061032, size = 29, normalized size = 0.88 \begin{align*} \frac{b e^{2} x^{2}}{2} + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} \end{align*}

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 [A]  time = 1.04414, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{30} \,{\left (3 \, f^{2} x^{10} + 10 \, f x^{6} e + 15 \, x^{2} e^{2}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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