3.621 \(\int x^5 (a+b x^4)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^6}{6}+\frac{1}{5} a b x^{10}+\frac{b^2 x^{14}}{14} \]

[Out]

(a^2*x^6)/6 + (a*b*x^10)/5 + (b^2*x^14)/14

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Rubi [A]  time = 0.0100878, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^6}{6}+\frac{1}{5} a b x^{10}+\frac{b^2 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^4)^2,x]

[Out]

(a^2*x^6)/6 + (a*b*x^10)/5 + (b^2*x^14)/14

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 x^5 \left (a+b x^4\right )^2 \, dx &=\int \left (a^2 x^5+2 a b x^9+b^2 x^{13}\right ) \, dx\\ &=\frac{a^2 x^6}{6}+\frac{1}{5} a b x^{10}+\frac{b^2 x^{14}}{14}\\ \end{align*}

Mathematica [A]  time = 0.0007284, size = 30, normalized size = 1. \[ \frac{a^2 x^6}{6}+\frac{1}{5} a b x^{10}+\frac{b^2 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^4)^2,x]

[Out]

(a^2*x^6)/6 + (a*b*x^10)/5 + (b^2*x^14)/14

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{ab{x}^{10}}{5}}+{\frac{{b}^{2}{x}^{14}}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^4+a)^2,x)

[Out]

1/6*a^2*x^6+1/5*a*b*x^10+1/14*b^2*x^14

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Maxima [A]  time = 0.968967, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{14} \, b^{2} x^{14} + \frac{1}{5} \, a b x^{10} + \frac{1}{6} \, a^{2} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^2,x, algorithm="maxima")

[Out]

1/14*b^2*x^14 + 1/5*a*b*x^10 + 1/6*a^2*x^6

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Fricas [A]  time = 1.21131, size = 59, normalized size = 1.97 \begin{align*} \frac{1}{14} x^{14} b^{2} + \frac{1}{5} x^{10} b a + \frac{1}{6} x^{6} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^2,x, algorithm="fricas")

[Out]

1/14*x^14*b^2 + 1/5*x^10*b*a + 1/6*x^6*a^2

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Sympy [A]  time = 0.065103, size = 24, normalized size = 0.8 \begin{align*} \frac{a^{2} x^{6}}{6} + \frac{a b x^{10}}{5} + \frac{b^{2} x^{14}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**4+a)**2,x)

[Out]

a**2*x**6/6 + a*b*x**10/5 + b**2*x**14/14

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Giac [A]  time = 1.09294, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{14} \, b^{2} x^{14} + \frac{1}{5} \, a b x^{10} + \frac{1}{6} \, a^{2} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^2,x, algorithm="giac")

[Out]

1/14*b^2*x^14 + 1/5*a*b*x^10 + 1/6*a^2*x^6