3.1313 \(\int \frac{x^{23/2}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{10} \sqrt{x^5+1} x^{15/2}-\frac{3}{20} \sqrt{x^5+1} x^{5/2}+\frac{3}{20} \sinh ^{-1}\left (x^{5/2}\right ) \]

[Out]

(-3*x^(5/2)*Sqrt[1 + x^5])/20 + (x^(15/2)*Sqrt[1 + x^5])/10 + (3*ArcSinh[x^(5/2)])/20

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Rubi [A]  time = 0.014652, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {321, 329, 275, 215} \[ \frac{1}{10} \sqrt{x^5+1} x^{15/2}-\frac{3}{20} \sqrt{x^5+1} x^{5/2}+\frac{3}{20} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^(23/2)/Sqrt[1 + x^5],x]

[Out]

(-3*x^(5/2)*Sqrt[1 + x^5])/20 + (x^(15/2)*Sqrt[1 + x^5])/10 + (3*ArcSinh[x^(5/2)])/20

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 329

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

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 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^{23/2}}{\sqrt{1+x^5}} \, dx &=\frac{1}{10} x^{15/2} \sqrt{1+x^5}-\frac{3}{4} \int \frac{x^{13/2}}{\sqrt{1+x^5}} \, dx\\ &=-\frac{3}{20} x^{5/2} \sqrt{1+x^5}+\frac{1}{10} x^{15/2} \sqrt{1+x^5}+\frac{3}{8} \int \frac{x^{3/2}}{\sqrt{1+x^5}} \, dx\\ &=-\frac{3}{20} x^{5/2} \sqrt{1+x^5}+\frac{1}{10} x^{15/2} \sqrt{1+x^5}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1+x^{10}}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{3}{20} x^{5/2} \sqrt{1+x^5}+\frac{1}{10} x^{15/2} \sqrt{1+x^5}+\frac{3}{20} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^{5/2}\right )\\ &=-\frac{3}{20} x^{5/2} \sqrt{1+x^5}+\frac{1}{10} x^{15/2} \sqrt{1+x^5}+\frac{3}{20} \sinh ^{-1}\left (x^{5/2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0132086, size = 35, normalized size = 0.74 \[ \frac{1}{20} \left (\sqrt{x^5+1} \left (2 x^5-3\right ) x^{5/2}+3 \sinh ^{-1}\left (x^{5/2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(23/2)/Sqrt[1 + x^5],x]

[Out]

(x^(5/2)*Sqrt[1 + x^5]*(-3 + 2*x^5) + 3*ArcSinh[x^(5/2)])/20

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Maple [A]  time = 0.051, size = 46, normalized size = 1. \begin{align*}{\frac{2\,{x}^{5}-3}{20}{x}^{{\frac{5}{2}}}\sqrt{{x}^{5}+1}}+{\frac{3}{20}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{x \left ({x}^{5}+1 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(23/2)/(x^5+1)^(1/2),x)

[Out]

1/20*x^(5/2)*(2*x^5-3)*(x^5+1)^(1/2)+3/20*arcsinh(x^(5/2))*(x*(x^5+1))^(1/2)/x^(1/2)/(x^5+1)^(1/2)

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Maxima [B]  time = 0.97478, size = 116, normalized size = 2.47 \begin{align*} -\frac{\frac{5 \, \sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} - \frac{3 \,{\left (x^{5} + 1\right )}^{\frac{3}{2}}}{x^{\frac{15}{2}}}}{20 \,{\left (\frac{2 \,{\left (x^{5} + 1\right )}}{x^{5}} - \frac{{\left (x^{5} + 1\right )}^{2}}{x^{10}} - 1\right )}} + \frac{3}{40} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} + 1\right ) - \frac{3}{40} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

-1/20*(5*sqrt(x^5 + 1)/x^(5/2) - 3*(x^5 + 1)^(3/2)/x^(15/2))/(2*(x^5 + 1)/x^5 - (x^5 + 1)^2/x^10 - 1) + 3/40*l
og(sqrt(x^5 + 1)/x^(5/2) + 1) - 3/40*log(sqrt(x^5 + 1)/x^(5/2) - 1)

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Fricas [A]  time = 2.20182, size = 124, normalized size = 2.64 \begin{align*} \frac{1}{20} \,{\left (2 \, x^{7} - 3 \, x^{2}\right )} \sqrt{x^{5} + 1} \sqrt{x} + \frac{3}{40} \, \log \left (2 \, x^{5} + 2 \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

1/20*(2*x^7 - 3*x^2)*sqrt(x^5 + 1)*sqrt(x) + 3/40*log(2*x^5 + 2*sqrt(x^5 + 1)*x^(5/2) + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(23/2)/(x**5+1)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.2106, size = 49, normalized size = 1.04 \begin{align*} \frac{1}{20} \,{\left (2 \, x^{5} - 3\right )} \sqrt{x^{5} + 1} x^{\frac{5}{2}} - \frac{3}{20} \, \log \left (-x^{\frac{5}{2}} + \sqrt{x^{5} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

1/20*(2*x^5 - 3)*sqrt(x^5 + 1)*x^(5/2) - 3/20*log(-x^(5/2) + sqrt(x^5 + 1))