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

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {3}{28} \left (x^7+1\right )^{4/3} \]

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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IntegrateAlgebraic [A]  time = 0.02, size = 13, normalized size = 1.00 \[ \frac {3}{28} \left (x^7+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6*(1 + x^7)^(1/3),x]

[Out]

(3*(1 + x^7)^(4/3))/28

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fricas [A]  time = 0.65, size = 9, normalized size = 0.69 \[ \frac {3}{28} \, {\left (x^{7} + 1\right )}^{\frac {4}{3}} \]

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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giac [A]  time = 0.63, size = 9, normalized size = 0.69 \[ \frac {3}{28} \, {\left (x^{7} + 1\right )}^{\frac {4}{3}} \]

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)

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maple [A]  time = 0.28, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {3 \left (x^{7}+1\right )^{\frac {4}{3}}}{28}\)	\(10\)
default	\(\frac {3 \left (x^{7}+1\right )^{\frac {4}{3}}}{28}\)	\(10\)
risch	\(\frac {3 \left (x^{7}+1\right )^{\frac {4}{3}}}{28}\)	\(10\)
trager	\(\left (\frac {3}{28}+\frac {3 x^{7}}{28}\right ) \left (x^{7}+1\right )^{\frac {1}{3}}\)	\(16\)
meijerg	\(\frac {x^{7} \hypergeom \left (\left [-\frac {1}{3}, 1\right ], \relax [2], -x^{7}\right )}{7}\)	\(17\)
gosper	\(\frac {3 \left (1+x \right ) \left (x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x +1\right ) \left (x^{7}+1\right )^{\frac {1}{3}}}{28}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/28*(x^7+1)^(4/3)

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maxima [A]  time = 0.48, size = 9, normalized size = 0.69 \[ \frac {3}{28} \, {\left (x^{7} + 1\right )}^{\frac {4}{3}} \]

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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mupad [B]  time = 0.29, size = 9, normalized size = 0.69 \[ \frac {3\,{\left (x^7+1\right )}^{4/3}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^7 + 1)^(1/3),x)

[Out]

(3*(x^7 + 1)^(4/3))/28

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sympy [B]  time = 0.54, size = 26, normalized size = 2.00 \[ \frac {3 x^{7} \sqrt [3]{x^{7} + 1}}{28} + \frac {3 \sqrt [3]{x^{7} + 1}}{28} \]

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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