3.1514 \(\int \frac{x^{19}}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{16} \sqrt{x^8+1} x^{12}-\frac{3}{32} \sqrt{x^8+1} x^4+\frac{3}{32} \sinh ^{-1}\left (x^4\right ) \]

[Out]

(-3*x^4*Sqrt[1 + x^8])/32 + (x^12*Sqrt[1 + x^8])/16 + (3*ArcSinh[x^4])/32

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Rubi [A]  time = 0.0162791, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 321, 215} \[ \frac{1}{16} \sqrt{x^8+1} x^{12}-\frac{3}{32} \sqrt{x^8+1} x^4+\frac{3}{32} \sinh ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^19/Sqrt[1 + x^8],x]

[Out]

(-3*x^4*Sqrt[1 + x^8])/32 + (x^12*Sqrt[1 + x^8])/16 + (3*ArcSinh[x^4])/32

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{x^{19}}{\sqrt{1+x^8}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{16} x^{12} \sqrt{1+x^8}-\frac{3}{16} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=-\frac{3}{32} x^4 \sqrt{1+x^8}+\frac{1}{16} x^{12} \sqrt{1+x^8}+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=-\frac{3}{32} x^4 \sqrt{1+x^8}+\frac{1}{16} x^{12} \sqrt{1+x^8}+\frac{3}{32} \sinh ^{-1}\left (x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0113924, size = 31, normalized size = 0.76 \[ \frac{1}{32} \left (\sqrt{x^8+1} \left (2 x^8-3\right ) x^4+3 \sinh ^{-1}\left (x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^19/Sqrt[1 + x^8],x]

[Out]

(x^4*Sqrt[1 + x^8]*(-3 + 2*x^8) + 3*ArcSinh[x^4])/32

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Maple [A]  time = 0.025, size = 27, normalized size = 0.7 \begin{align*}{\frac{{x}^{4} \left ( 2\,{x}^{8}-3 \right ) }{32}\sqrt{{x}^{8}+1}}+{\frac{3\,{\it Arcsinh} \left ({x}^{4} \right ) }{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(x^8+1)^(1/2),x)

[Out]

1/32*x^4*(2*x^8-3)*(x^8+1)^(1/2)+3/32*arcsinh(x^4)

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Maxima [B]  time = 0.956163, size = 116, normalized size = 2.83 \begin{align*} -\frac{\frac{5 \, \sqrt{x^{8} + 1}}{x^{4}} - \frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}}}{x^{12}}}{32 \,{\left (\frac{2 \,{\left (x^{8} + 1\right )}}{x^{8}} - \frac{{\left (x^{8} + 1\right )}^{2}}{x^{16}} - 1\right )}} + \frac{3}{64} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) - \frac{3}{64} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

-1/32*(5*sqrt(x^8 + 1)/x^4 - 3*(x^8 + 1)^(3/2)/x^12)/(2*(x^8 + 1)/x^8 - (x^8 + 1)^2/x^16 - 1) + 3/64*log(sqrt(
x^8 + 1)/x^4 + 1) - 3/64*log(sqrt(x^8 + 1)/x^4 - 1)

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Fricas [A]  time = 1.26612, size = 95, normalized size = 2.32 \begin{align*} \frac{1}{32} \,{\left (2 \, x^{12} - 3 \, x^{4}\right )} \sqrt{x^{8} + 1} - \frac{3}{32} \, \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

1/32*(2*x^12 - 3*x^4)*sqrt(x^8 + 1) - 3/32*log(-x^4 + sqrt(x^8 + 1))

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Sympy [A]  time = 5.99897, size = 49, normalized size = 1.2 \begin{align*} \frac{x^{20}}{16 \sqrt{x^{8} + 1}} - \frac{x^{12}}{32 \sqrt{x^{8} + 1}} - \frac{3 x^{4}}{32 \sqrt{x^{8} + 1}} + \frac{3 \operatorname{asinh}{\left (x^{4} \right )}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19/(x**8+1)**(1/2),x)

[Out]

x**20/(16*sqrt(x**8 + 1)) - x**12/(32*sqrt(x**8 + 1)) - 3*x**4/(32*sqrt(x**8 + 1)) + 3*asinh(x**4)/32

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Giac [A]  time = 1.26175, size = 59, normalized size = 1.44 \begin{align*} \frac{1}{32} \,{\left (2 \, x^{8} - 3\right )} \sqrt{x^{8} + 1} x^{4} + \frac{3}{64} \, \log \left (\sqrt{\frac{1}{x^{8}} + 1} + 1\right ) - \frac{3}{64} \, \log \left (\sqrt{\frac{1}{x^{8}} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

1/32*(2*x^8 - 3)*sqrt(x^8 + 1)*x^4 + 3/64*log(sqrt(1/x^8 + 1) + 1) - 3/64*log(sqrt(1/x^8 + 1) - 1)