3.389 \(\int \frac {\sqrt {\frac {a}{x}}}{\sqrt {1+x^3}} \, dx\)

Optimal. Leaf size=116 \[ \frac {x (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {\frac {a}{x}} F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^3+1}} \]

[Out]

1/3*x*(1+x)*((1+x*(1-3^(1/2)))^2/(1+x*(1+3^(1/2)))^2)^(1/2)/(1+x*(1-3^(1/2)))*(1+x*(1+3^(1/2)))*EllipticF((1-(
1+x*(1-3^(1/2)))^2/(1+x*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(a/x)^(1/2)*((x^2-x+1)/(1+x*(1+3^(1/2))
)^2)^(1/2)*3^(3/4)/(x^3+1)^(1/2)/(x*(1+x)/(1+x*(1+3^(1/2)))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {15, 329, 225} \[ \frac {x (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {\frac {a}{x}} F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^3+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x]/Sqrt[1 + x^3],x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {1+x^3}} \, dx &=\left (\sqrt {\frac {a}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x^3}} \, dx\\ &=\left (2 \sqrt {\frac {a}{x}} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )\\ &=\frac {\sqrt {\frac {a}{x}} x (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1+\left (1-\sqrt {3}\right ) x}{1+\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x (1+x)}{\left (1+\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {1+x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 27, normalized size = 0.23 \[ 2 x \sqrt {\frac {a}{x}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x]/Sqrt[1 + x^3],x]

[Out]

2*Sqrt[a/x]*x*Hypergeometric2F1[1/6, 1/2, 7/6, -x^3]

________________________________________________________________________________________

fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\frac {a}{x}}}{\sqrt {x^{3} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a/x)/sqrt(x^3 + 1), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a/x)/sqrt(x^3 + 1), x)

________________________________________________________________________________________

maple [C]  time = 0.18, size = 232, normalized size = 2.00 \[ \frac {4 \sqrt {\frac {a}{x}}\, \sqrt {x^{3}+1}\, \left (1+i \sqrt {3}\right ) \sqrt {\frac {\left (3+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x +1\right )}}\, \left (x +1\right )^{2} \sqrt {\frac {2 x +i \sqrt {3}-1}{\left (i \sqrt {3}-1\right ) \left (x +1\right )}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{\left (1+i \sqrt {3}\right ) \left (x +1\right )}}\, x \EllipticF \left (\sqrt {\frac {\left (3+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x +1\right )}}, \sqrt {\frac {\left (-3+i \sqrt {3}\right ) \left (1+i \sqrt {3}\right )}{\left (i \sqrt {3}-1\right ) \left (3+i \sqrt {3}\right )}}\right )}{\sqrt {\left (x^{3}+1\right ) x}\, \left (3+i \sqrt {3}\right ) \sqrt {-\left (x +1\right ) \left (2 x +i \sqrt {3}-1\right ) \left (-2 x +i \sqrt {3}+1\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(a/x)^(1/2)*x*(x^3+1)^(1/2)*(1+I*3^(1/2))*((3+I*3^(1/2))/(1+I*3^(1/2))/(x+1)*x)^(1/2)*(x+1)^2*((2*x+I*3^(1/2
)-1)/(I*3^(1/2)-1)/(x+1))^(1/2)*((-2*x+I*3^(1/2)+1)/(1+I*3^(1/2))/(x+1))^(1/2)*EllipticF(((3+I*3^(1/2))/(1+I*3
^(1/2))/(x+1)*x)^(1/2),((-3+I*3^(1/2))*(1+I*3^(1/2))/(I*3^(1/2)-1)/(3+I*3^(1/2)))^(1/2))/((x^3+1)*x)^(1/2)/(3+
I*3^(1/2))/(-x*(x+1)*(2*x+I*3^(1/2)-1)*(-2*x+I*3^(1/2)+1))^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a/x)/sqrt(x^3 + 1), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {x^3+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x)^(1/2)/(x^3 + 1)^(1/2),x)

[Out]

int((a/x)^(1/2)/(x^3 + 1)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x}}}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a/x)/sqrt((x + 1)*(x**2 - x + 1)), x)

________________________________________________________________________________________