3.1314 \(\int \frac{x^{13/2}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{5} x^{5/2} \sqrt{x^5+1}-\frac{1}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0102985, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {321, 329, 275, 215} \[ \frac{1}{5} x^{5/2} \sqrt{x^5+1}-\frac{1}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^(13/2)/Sqrt[1 + x^5],x]

[Out]

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

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

Mathematica [A]  time = 0.0056888, size = 29, normalized size = 1. \[ \frac{1}{5} x^{5/2} \sqrt{x^5+1}-\frac{1}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(13/2)/Sqrt[1 + x^5],x]

[Out]

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

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Maple [A]  time = 0.03, size = 39, normalized size = 1.3 \begin{align*}{\frac{1}{5}{x}^{{\frac{5}{2}}}\sqrt{{x}^{5}+1}}-{\frac{1}{5}{\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^(13/2)/(x^5+1)^(1/2),x)

[Out]

1/5*x^(5/2)*(x^5+1)^(1/2)-1/5*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 = 0.980035, size = 78, normalized size = 2.69 \begin{align*} \frac{\sqrt{x^{5} + 1}}{5 \, x^{\frac{5}{2}}{\left (\frac{x^{5} + 1}{x^{5}} - 1\right )}} - \frac{1}{10} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} + 1\right ) + \frac{1}{10} \, \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^(13/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

1/5*sqrt(x^5 + 1)/(x^(5/2)*((x^5 + 1)/x^5 - 1)) - 1/10*log(sqrt(x^5 + 1)/x^(5/2) + 1) + 1/10*log(sqrt(x^5 + 1)
/x^(5/2) - 1)

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Fricas [A]  time = 2.05121, size = 103, normalized size = 3.55 \begin{align*} \frac{1}{5} \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + \frac{1}{10} \, \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^(13/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

1/5*sqrt(x^5 + 1)*x^(5/2) + 1/10*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**(13/2)/(x**5+1)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.19357, size = 39, normalized size = 1.34 \begin{align*} \frac{1}{5} \, \sqrt{x^{5} + 1} x^{\frac{5}{2}} + \frac{1}{5} \, \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^(13/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

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