3.50 \(\int x^4 (1+x^5)^5 \, dx\)

Optimal. Leaf size=11 \[ \frac{1}{30} \left (x^5+1\right )^6 \]

[Out]

(1 + x^5)^6/30

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Rubi [A]  time = 0.0018053, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {261} \[ \frac{1}{30} \left (x^5+1\right )^6 \]

Antiderivative was successfully verified.

[In]

Int[x^4*(1 + x^5)^5,x]

[Out]

(1 + x^5)^6/30

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^4 \left (1+x^5\right )^5 \, dx &=\frac{1}{30} \left (1+x^5\right )^6\\ \end{align*}

Mathematica [B]  time = 0.0012812, size = 43, normalized size = 3.91 \[ \frac{x^{30}}{30}+\frac{x^{25}}{5}+\frac{x^{20}}{2}+\frac{2 x^{15}}{3}+\frac{x^{10}}{2}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(1 + x^5)^5,x]

[Out]

x^5/5 + x^10/2 + (2*x^15)/3 + x^20/2 + x^25/5 + x^30/30

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Maple [B]  time = 0.001, size = 32, normalized size = 2.9 \begin{align*}{\frac{{x}^{30}}{30}}+{\frac{{x}^{25}}{5}}+{\frac{{x}^{20}}{2}}+{\frac{2\,{x}^{15}}{3}}+{\frac{{x}^{10}}{2}}+{\frac{{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(x^5+1)^5,x)

[Out]

1/30*x^30+1/5*x^25+1/2*x^20+2/3*x^15+1/2*x^10+1/5*x^5

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Maxima [A]  time = 0.94927, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{30} \,{\left (x^{5} + 1\right )}^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(x^5+1)^5,x, algorithm="maxima")

[Out]

1/30*(x^5 + 1)^6

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Fricas [B]  time = 0.357088, size = 88, normalized size = 8. \begin{align*} \frac{1}{30} x^{30} + \frac{1}{5} x^{25} + \frac{1}{2} x^{20} + \frac{2}{3} x^{15} + \frac{1}{2} x^{10} + \frac{1}{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(x^5+1)^5,x, algorithm="fricas")

[Out]

1/30*x^30 + 1/5*x^25 + 1/2*x^20 + 2/3*x^15 + 1/2*x^10 + 1/5*x^5

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Sympy [B]  time = 0.05131, size = 31, normalized size = 2.82 \begin{align*} \frac{x^{30}}{30} + \frac{x^{25}}{5} + \frac{x^{20}}{2} + \frac{2 x^{15}}{3} + \frac{x^{10}}{2} + \frac{x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(x**5+1)**5,x)

[Out]

x**30/30 + x**25/5 + x**20/2 + 2*x**15/3 + x**10/2 + x**5/5

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Giac [A]  time = 1.09281, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{30} \,{\left (x^{5} + 1\right )}^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(x^5+1)^5,x, algorithm="giac")

[Out]

1/30*(x^5 + 1)^6