3.1515 \(\int \frac{x^{15}}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{12} \left (x^8+1\right )^{3/2}-\frac{\sqrt{x^8+1}}{4} \]

[Out]

-Sqrt[1 + x^8]/4 + (1 + x^8)^(3/2)/12

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Rubi [A]  time = 0.0104585, 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} \[ \frac{1}{12} \left (x^8+1\right )^{3/2}-\frac{\sqrt{x^8+1}}{4} \]

Antiderivative was successfully verified.

[In]

Int[x^15/Sqrt[1 + x^8],x]

[Out]

-Sqrt[1 + x^8]/4 + (1 + x^8)^(3/2)/12

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^{15}}{\sqrt{1+x^8}} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x}} \, dx,x,x^8\right )\\ &=\frac{1}{8} \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+\sqrt{1+x}\right ) \, dx,x,x^8\right )\\ &=-\frac{1}{4} \sqrt{1+x^8}+\frac{1}{12} \left (1+x^8\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0051065, size = 18, normalized size = 0.67 \[ \frac{1}{12} \left (x^8-2\right ) \sqrt{x^8+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15/Sqrt[1 + x^8],x]

[Out]

((-2 + x^8)*Sqrt[1 + x^8])/12

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Maple [A]  time = 0.004, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{8}-2}{12}\sqrt{{x}^{8}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/(x^8+1)^(1/2),x)

[Out]

1/12*(x^8+1)^(1/2)*(x^8-2)

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Maxima [A]  time = 1.47021, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{12} \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - \frac{1}{4} \, \sqrt{x^{8} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

1/12*(x^8 + 1)^(3/2) - 1/4*sqrt(x^8 + 1)

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Fricas [A]  time = 1.2526, size = 41, normalized size = 1.52 \begin{align*} \frac{1}{12} \, \sqrt{x^{8} + 1}{\left (x^{8} - 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(x^8 + 1)*(x^8 - 2)

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Sympy [A]  time = 2.90231, size = 22, normalized size = 0.81 \begin{align*} \frac{x^{8} \sqrt{x^{8} + 1}}{12} - \frac{\sqrt{x^{8} + 1}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15/(x**8+1)**(1/2),x)

[Out]

x**8*sqrt(x**8 + 1)/12 - sqrt(x**8 + 1)/6

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Giac [A]  time = 1.18002, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{12} \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - \frac{1}{4} \, \sqrt{x^{8} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

1/12*(x^8 + 1)^(3/2) - 1/4*sqrt(x^8 + 1)