\(\int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx\) [841]

Optimal result

Integrand size = 32, antiderivative size = 63 \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {4 (-3+4 x) \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{15 x^2}+\frac {16 \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{15 x^2} \]

[Out]

4/15*(-3+4*x)*(x*(x+(x^2+x)^(1/2)))^(1/2)/x^2+16/15*(x^2+x)^(1/2)*(x*(x+(x^2+x)^(1/2)))^(1/2)/x^2

Rubi [F]

\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx \]

[In]

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

[Out]

(2*Sqrt[x]*Sqrt[1 + x]*Defer[Subst][Defer[Int][1/(x^2*Sqrt[1 + x^2]*Sqrt[x^4 + x^2*Sqrt[x^2 + x^4]]), x], x, S
qrt[x]])/Sqrt[x + x^2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x}\right ) \int \frac {1}{x^{3/2} \sqrt {1+x} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx}{\sqrt {x+x^2}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^2} \sqrt {x^4+x^2 \sqrt {x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {4 \left (-3+8 x^2+\sqrt {x (1+x)}+x \left (5+8 \sqrt {x (1+x)}\right )\right )}{15 \sqrt {x (1+x)} \sqrt {x \left (x+\sqrt {x (1+x)}\right )}} \]

[In]

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

[Out]

(4*(-3 + 8*x^2 + Sqrt[x*(1 + x)] + x*(5 + 8*Sqrt[x*(1 + x)])))/(15*Sqrt[x*(1 + x)]*Sqrt[x*(x + Sqrt[x*(1 + x)]
)])

Maple [F]

\[\int \frac {1}{x \sqrt {x^{2}+x}\, \sqrt {x^{2}+x \sqrt {x^{2}+x}}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {4 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (4 \, x + 4 \, \sqrt {x^{2} + x} - 3\right )}}{15 \, x^{2}} \]

[In]

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

[Out]

4/15*sqrt(x^2 + sqrt(x^2 + x)*x)*(4*x + 4*sqrt(x^2 + x) - 3)/x^2

Sympy [F]

\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {1}{x \sqrt {x \left (x + 1\right )} \sqrt {x \left (x + \sqrt {x^{2} + x}\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {1}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} \sqrt {x^{2} + x} x} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {1}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} \sqrt {x^{2} + x} x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {1}{x\,\sqrt {x^2+x\,\sqrt {x^2+x}}\,\sqrt {x^2+x}} \,d x \]

[In]

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

[Out]

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