3.985 \(\int x^7 \sqrt [3]{1+x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac{3}{28} \left (x^4+1\right )^{7/3}-\frac{3}{16} \left (x^4+1\right )^{4/3} \]

[Out]

(-3*(1 + x^4)^(4/3))/16 + (3*(1 + x^4)^(7/3))/28

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Rubi [A]  time = 0.0102084, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3}{28} \left (x^4+1\right )^{7/3}-\frac{3}{16} \left (x^4+1\right )^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(1 + x^4)^(1/3),x]

[Out]

(-3*(1 + x^4)^(4/3))/16 + (3*(1 + x^4)^(7/3))/28

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^7 \sqrt [3]{1+x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x \sqrt [3]{1+x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\sqrt [3]{1+x}+(1+x)^{4/3}\right ) \, dx,x,x^4\right )\\ &=-\frac{3}{16} \left (1+x^4\right )^{4/3}+\frac{3}{28} \left (1+x^4\right )^{7/3}\\ \end{align*}

Mathematica [A]  time = 0.0062148, size = 20, normalized size = 0.74 \[ \frac{3}{112} \left (x^4+1\right )^{4/3} \left (4 x^4-3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(1 + x^4)^(1/3),x]

[Out]

(3*(1 + x^4)^(4/3)*(-3 + 4*x^4))/112

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Maple [A]  time = 0.003, size = 17, normalized size = 0.6 \begin{align*}{\frac{12\,{x}^{4}-9}{112} \left ({x}^{4}+1 \right ) ^{{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(x^4+1)^(1/3),x)

[Out]

3/112*(x^4+1)^(4/3)*(4*x^4-3)

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Maxima [A]  time = 0.956257, size = 26, normalized size = 0.96 \begin{align*} \frac{3}{28} \,{\left (x^{4} + 1\right )}^{\frac{7}{3}} - \frac{3}{16} \,{\left (x^{4} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(x^4+1)^(1/3),x, algorithm="maxima")

[Out]

3/28*(x^4 + 1)^(7/3) - 3/16*(x^4 + 1)^(4/3)

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Fricas [A]  time = 1.41467, size = 55, normalized size = 2.04 \begin{align*} \frac{3}{112} \,{\left (4 \, x^{8} + x^{4} - 3\right )}{\left (x^{4} + 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(x^4+1)^(1/3),x, algorithm="fricas")

[Out]

3/112*(4*x^8 + x^4 - 3)*(x^4 + 1)^(1/3)

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Sympy [A]  time = 0.972573, size = 41, normalized size = 1.52 \begin{align*} \frac{3 x^{8} \sqrt [3]{x^{4} + 1}}{28} + \frac{3 x^{4} \sqrt [3]{x^{4} + 1}}{112} - \frac{9 \sqrt [3]{x^{4} + 1}}{112} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(x**4+1)**(1/3),x)

[Out]

3*x**8*(x**4 + 1)**(1/3)/28 + 3*x**4*(x**4 + 1)**(1/3)/112 - 9*(x**4 + 1)**(1/3)/112

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Giac [A]  time = 1.13798, size = 26, normalized size = 0.96 \begin{align*} \frac{3}{28} \,{\left (x^{4} + 1\right )}^{\frac{7}{3}} - \frac{3}{16} \,{\left (x^{4} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(x^4+1)^(1/3),x, algorithm="giac")

[Out]

3/28*(x^4 + 1)^(7/3) - 3/16*(x^4 + 1)^(4/3)