3.565 \(\int \frac{1}{\sqrt{x}+4 x^{3/2}} \, dx\)

Optimal. Leaf size=8 \[ \tan ^{-1}\left (2 \sqrt{x}\right ) \]

[Out]

ArcTan[2*Sqrt[x]]

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Rubi [A]  time = 0.0057815, antiderivative size = 8, 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 = {1593, 63, 203} \[ \tan ^{-1}\left (2 \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[x] + 4*x^(3/2))^(-1),x]

[Out]

ArcTan[2*Sqrt[x]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

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

Rubi steps

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

Mathematica [A]  time = 0.0024078, size = 8, normalized size = 1. \[ \tan ^{-1}\left (2 \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[x] + 4*x^(3/2))^(-1),x]

[Out]

ArcTan[2*Sqrt[x]]

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Maple [A]  time = 0.004, size = 7, normalized size = 0.9 \begin{align*} \arctan \left ( 2\,\sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^(3/2)+x^(1/2)),x)

[Out]

arctan(2*x^(1/2))

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Maxima [A]  time = 1.84293, size = 8, normalized size = 1. \begin{align*} \arctan \left (2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^(3/2)+x^(1/2)),x, algorithm="maxima")

[Out]

arctan(2*sqrt(x))

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Fricas [A]  time = 1.23929, size = 26, normalized size = 3.25 \begin{align*} \arctan \left (2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^(3/2)+x^(1/2)),x, algorithm="fricas")

[Out]

arctan(2*sqrt(x))

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Sympy [A]  time = 0.202828, size = 7, normalized size = 0.88 \begin{align*} \operatorname{atan}{\left (2 \sqrt{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x**(3/2)+x**(1/2)),x)

[Out]

atan(2*sqrt(x))

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Giac [A]  time = 1.09106, size = 8, normalized size = 1. \begin{align*} \arctan \left (2 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^(3/2)+x^(1/2)),x, algorithm="giac")

[Out]

arctan(2*sqrt(x))