3.1315 \(\int \frac{x^{3/2}}{\sqrt{1+x^5}} \, dx\)

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

[Out]

(2*ArcSinh[x^(5/2)])/5

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Rubi [A]  time = 0.0067993, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {329, 275, 215} \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/Sqrt[1 + x^5],x]

[Out]

(2*ArcSinh[x^(5/2)])/5

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

Mathematica [A]  time = 0.0025837, size = 10, normalized size = 1. \[ \frac{2}{5} \sinh ^{-1}\left (x^{5/2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/Sqrt[1 + x^5],x]

[Out]

(2*ArcSinh[x^(5/2)])/5

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Maple [A]  time = 0.025, size = 7, normalized size = 0.7 \begin{align*}{\frac{2}{5}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(x^5+1)^(1/2),x)

[Out]

2/5*arcsinh(x^(5/2))

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Maxima [B]  time = 0.954905, size = 45, normalized size = 4.5 \begin{align*} \frac{1}{5} \, \log \left (\frac{\sqrt{x^{5} + 1}}{x^{\frac{5}{2}}} + 1\right ) - \frac{1}{5} \, \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^(3/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.18181, size = 62, normalized size = 6.2 \begin{align*} \frac{1}{5} \, \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^(3/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.46149, size = 8, normalized size = 0.8 \begin{align*} \frac{2 \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)/(x**5+1)**(1/2),x)

[Out]

2*asinh(x**(5/2))/5

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Giac [B]  time = 1.22061, size = 34, normalized size = 3.4 \begin{align*} \frac{1}{5} \, \log \left (\sqrt{\frac{1}{x^{5}} + 1} + 1\right ) - \frac{1}{5} \, \log \left (\sqrt{\frac{1}{x^{5}} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

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