3.299 \(\int x^6 \sqrt [3]{1+x^7} \, dx\)

Optimal. Leaf size=13 \[ \frac{3}{28} \left (x^7+1\right )^{4/3} \]

[Out]

(3*(1 + x^7)^(4/3))/28

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Rubi [A]  time = 0.0027798, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{3}{28} \left (x^7+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(1 + x^7)^(1/3),x]

[Out]

(3*(1 + x^7)^(4/3))/28

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^6 \sqrt [3]{1+x^7} \, dx &=\frac{3}{28} \left (1+x^7\right )^{4/3}\\ \end{align*}

Mathematica [A]  time = 0.0029953, size = 13, normalized size = 1. \[ \frac{3}{28} \left (x^7+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(1 + x^7)^(1/3),x]

[Out]

(3*(1 + x^7)^(4/3))/28

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Maple [B]  time = 0.006, size = 37, normalized size = 2.9 \begin{align*}{\frac{ \left ( 3+3\,x \right ) \left ({x}^{6}-{x}^{5}+{x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) }{28}\sqrt [3]{{x}^{7}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^7+1)^(1/3),x)

[Out]

3/28*(1+x)*(x^6-x^5+x^4-x^3+x^2-x+1)*(x^7+1)^(1/3)

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Maxima [A]  time = 0.937423, size = 12, normalized size = 0.92 \begin{align*} \frac{3}{28} \,{\left (x^{7} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^7+1)^(1/3),x, algorithm="maxima")

[Out]

3/28*(x^7 + 1)^(4/3)

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Fricas [A]  time = 1.73788, size = 30, normalized size = 2.31 \begin{align*} \frac{3}{28} \,{\left (x^{7} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^7+1)^(1/3),x, algorithm="fricas")

[Out]

3/28*(x^7 + 1)^(4/3)

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Sympy [B]  time = 0.560205, size = 26, normalized size = 2. \begin{align*} \frac{3 x^{7} \sqrt [3]{x^{7} + 1}}{28} + \frac{3 \sqrt [3]{x^{7} + 1}}{28} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(x**7+1)**(1/3),x)

[Out]

3*x**7*(x**7 + 1)**(1/3)/28 + 3*(x**7 + 1)**(1/3)/28

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Giac [A]  time = 1.06003, size = 12, normalized size = 0.92 \begin{align*} \frac{3}{28} \,{\left (x^{7} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^7+1)^(1/3),x, algorithm="giac")

[Out]

3/28*(x^7 + 1)^(4/3)