3.438 \(\int \frac{x^{11}}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2}{21} \left (x^3+1\right )^{7/2}-\frac{2}{5} \left (x^3+1\right )^{5/2}+\frac{2}{3} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]

[Out]

(-2*Sqrt[1 + x^3])/3 + (2*(1 + x^3)^(3/2))/3 - (2*(1 + x^3)^(5/2))/5 + (2*(1 + x^3)^(7/2))/21

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Rubi [A]  time = 0.0178267, antiderivative size = 53, 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} \[ \frac{2}{21} \left (x^3+1\right )^{7/2}-\frac{2}{5} \left (x^3+1\right )^{5/2}+\frac{2}{3} \left (x^3+1\right )^{3/2}-\frac{2 \sqrt{x^3+1}}{3} \]

Antiderivative was successfully verified.

[In]

Int[x^11/Sqrt[1 + x^3],x]

[Out]

(-2*Sqrt[1 + x^3])/3 + (2*(1 + x^3)^(3/2))/3 - (2*(1 + x^3)^(5/2))/5 + (2*(1 + x^3)^(7/2))/21

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 \frac{x^{11}}{\sqrt{1+x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1+x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+3 \sqrt{1+x}-3 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,x^3\right )\\ &=-\frac{2}{3} \sqrt{1+x^3}+\frac{2}{3} \left (1+x^3\right )^{3/2}-\frac{2}{5} \left (1+x^3\right )^{5/2}+\frac{2}{21} \left (1+x^3\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0094386, size = 30, normalized size = 0.57 \[ \frac{2}{105} \sqrt{x^3+1} \left (5 x^9-6 x^6+8 x^3-16\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3]*(-16 + 8*x^3 - 6*x^6 + 5*x^9))/105

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Maple [A]  time = 0.005, size = 38, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 5\,{x}^{9}-6\,{x}^{6}+8\,{x}^{3}-16 \right ) }{105}{\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(x^3+1)^(1/2),x)

[Out]

2/105*(1+x)*(x^2-x+1)*(5*x^9-6*x^6+8*x^3-16)/(x^3+1)^(1/2)

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Maxima [A]  time = 0.953048, size = 50, normalized size = 0.94 \begin{align*} \frac{2}{21} \,{\left (x^{3} + 1\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

2/21*(x^3 + 1)^(7/2) - 2/5*(x^3 + 1)^(5/2) + 2/3*(x^3 + 1)^(3/2) - 2/3*sqrt(x^3 + 1)

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Fricas [A]  time = 1.70776, size = 68, normalized size = 1.28 \begin{align*} \frac{2}{105} \,{\left (5 \, x^{9} - 6 \, x^{6} + 8 \, x^{3} - 16\right )} \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

2/105*(5*x^9 - 6*x^6 + 8*x^3 - 16)*sqrt(x^3 + 1)

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Sympy [A]  time = 1.78113, size = 56, normalized size = 1.06 \begin{align*} \frac{2 x^{9} \sqrt{x^{3} + 1}}{21} - \frac{4 x^{6} \sqrt{x^{3} + 1}}{35} + \frac{16 x^{3} \sqrt{x^{3} + 1}}{105} - \frac{32 \sqrt{x^{3} + 1}}{105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(x**3+1)**(1/2),x)

[Out]

2*x**9*sqrt(x**3 + 1)/21 - 4*x**6*sqrt(x**3 + 1)/35 + 16*x**3*sqrt(x**3 + 1)/105 - 32*sqrt(x**3 + 1)/105

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Giac [A]  time = 1.14007, size = 50, normalized size = 0.94 \begin{align*} \frac{2}{21} \,{\left (x^{3} + 1\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

2/21*(x^3 + 1)^(7/2) - 2/5*(x^3 + 1)^(5/2) + 2/3*(x^3 + 1)^(3/2) - 2/3*sqrt(x^3 + 1)