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

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

[Out]

(cos(2*arctan(a^(1/4)*x^(1/2)))^2)^(1/2)/cos(2*arctan(a^(1/4)*x^(1/2)))*EllipticF(sin(2*arctan(a^(1/4)*x^(1/2)
)),1/2*2^(1/2))*(1+x*a^(1/2))*((a*x^2+1)/(1+x*a^(1/2))^2)^(1/2)/a^(1/4)/(a*x^2+1)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [C]  time = 0.01, 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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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a x^{2} + 1} \sqrt {x}}{a x^{3} + x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a x^{2} + 1} \sqrt {x}}\,{d x} \]

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)

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maple [A]  time = 0.05, size = 73, normalized size = 1.09 \[ -\frac {\sqrt {-\sqrt {-a}\, x +1}\, \sqrt {2}\, \sqrt {\sqrt {-a}\, x +1}\, \sqrt {\sqrt {-a}\, x}\, \EllipticF \left (\sqrt {-\sqrt {-a}\, x +1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {a \,x^{2}+1}\, \sqrt {-a}\, \sqrt {x}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a x^{2} + 1} \sqrt {x}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {x}\,\sqrt {a\,x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a*x^2 + 1)^(1/2)),x)

[Out]

int(1/(x^(1/2)*(a*x^2 + 1)^(1/2)), x)

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sympy [C]  time = 0.75, size = 32, normalized size = 0.48 \[ \frac {\sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {a x^{2} e^{i \pi }} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} \]

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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