3.650 \(\int \frac{1}{\sqrt{x} \sqrt{1+a x^2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{\left (\sqrt{a} x+1\right ) \sqrt{\frac{a x^2+1}{\left (\sqrt{a} x+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{a} \sqrt{x}\right ),\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt{a x^2+1}} \]

[Out]

((1 + Sqrt[a]*x)*Sqrt[(1 + a*x^2)/(1 + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[a^(1/4)*Sqrt[x]], 1/2])/(a^(1/4)*Sqrt[
1 + a*x^2])

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Rubi [A]  time = 0.0377915, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {329, 220} \[ \frac{\left (\sqrt{a} x+1\right ) \sqrt{\frac{a x^2+1}{\left (\sqrt{a} x+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{a} \sqrt{x}\right )|\frac{1}{2}\right )}{\sqrt [4]{a} \sqrt{a x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[1 + a*x^2]),x]

[Out]

((1 + Sqrt[a]*x)*Sqrt[(1 + a*x^2)/(1 + Sqrt[a]*x)^2]*EllipticF[2*ArcTan[a^(1/4)*Sqrt[x]], 1/2])/(a^(1/4)*Sqrt[
1 + a*x^2])

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 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

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

Mathematica [C]  time = 0.0061015, size = 23, normalized size = 0.34 \[ 2 \sqrt{x} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-a x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*Sqrt[1 + a*x^2]),x]

[Out]

2*Sqrt[x]*Hypergeometric2F1[1/4, 1/2, 5/4, -(a*x^2)]

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Maple [A]  time = 0.041, size = 73, normalized size = 1.1 \begin{align*} -{\sqrt{2}\sqrt{-x\sqrt{-a}+1}\sqrt{x\sqrt{-a}+1}\sqrt{x\sqrt{-a}}{\it EllipticF} \left ( \sqrt{-x\sqrt{-a}+1},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{a{x}^{2}+1}}}{\frac{1}{\sqrt{-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(a*x^2+1)^(1/2),x)

[Out]

-1/x^(1/2)/(a*x^2+1)^(1/2)*(-x*(-a)^(1/2)+1)^(1/2)*2^(1/2)*(x*(-a)^(1/2)+1)^(1/2)*(x*(-a)^(1/2))^(1/2)*Ellipti
cF((-x*(-a)^(1/2)+1)^(1/2),1/2*2^(1/2))/(-a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x^{2} + 1} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a*x^2 + 1)*sqrt(x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a x^{2} + 1} \sqrt{x}}{a x^{3} + x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*x^2 + 1)*sqrt(x)/(a*x^3 + x), x)

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Sympy [C]  time = 0.640454, size = 32, normalized size = 0.48 \begin{align*} \frac{\sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{a x^{2} e^{i \pi }} \right )}}{2 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/(a*x**2+1)**(1/2),x)

[Out]

sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), a*x**2*exp_polar(I*pi))/(2*gamma(5/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a x^{2} + 1} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a*x^2 + 1)*sqrt(x)), x)