Integrand size = 32, antiderivative size = 63 \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {4 (-3+4 x) \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{15 x^2}+\frac {16 \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{15 x^2} \] Output:
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
Time = 2.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {4 \left (-3+8 x^2+\sqrt {x (1+x)}+x \left (5+8 \sqrt {x (1+x)}\right )\right )}{15 \sqrt {x (1+x)} \sqrt {x \left (x+\sqrt {x (1+x)}\right )}} \] Input:
Integrate[1/(x*Sqrt[x + x^2]*Sqrt[x^2 + x*Sqrt[x + x^2]]),x]
Output:
(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)])])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x \sqrt {x^2+x} \sqrt {x^2+\sqrt {x^2+x} x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt {x+1} \int \frac {1}{x^{3/2} \sqrt {x+1} \sqrt {x^2+\sqrt {x^2+x} x}}dx}{\sqrt {x^2+x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x+1} \int \frac {1}{x \sqrt {x+1} \sqrt {x^2+\sqrt {x^2+x} x}}d\sqrt {x}}{\sqrt {x^2+x}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {2 \sqrt {x} \sqrt {x+1} \int \frac {1}{x \sqrt {x+1} \sqrt {x^2+\sqrt {x^2+x} x}}d\sqrt {x}}{\sqrt {x^2+x}}\) |
Input:
Int[1/(x*Sqrt[x + x^2]*Sqrt[x^2 + x*Sqrt[x + x^2]]),x]
Output:
$Aborted
\[\int \frac {1}{x \sqrt {x^{2}+x}\, \sqrt {x^{2}+x \sqrt {x^{2}+x}}}d x\]
Input:
int(1/x/(x^2+x)^(1/2)/(x^2+x*(x^2+x)^(1/2))^(1/2),x)
Output:
int(1/x/(x^2+x)^(1/2)/(x^2+x*(x^2+x)^(1/2))^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {4 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (4 \, x + 4 \, \sqrt {x^{2} + x} - 3\right )}}{15 \, x^{2}} \] Input:
integrate(1/x/(x^2+x)^(1/2)/(x^2+x*(x^2+x)^(1/2))^(1/2),x, algorithm="fric as")
Output:
4/15*sqrt(x^2 + sqrt(x^2 + x)*x)*(4*x + 4*sqrt(x^2 + x) - 3)/x^2
\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {1}{x \sqrt {x \left (x + 1\right )} \sqrt {x \left (x + \sqrt {x^{2} + x}\right )}}\, dx \] Input:
integrate(1/x/(x**2+x)**(1/2)/(x**2+x*(x**2+x)**(1/2))**(1/2),x)
Output:
Integral(1/(x*sqrt(x*(x + 1))*sqrt(x*(x + sqrt(x**2 + x)))), x)
\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {1}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} \sqrt {x^{2} + x} x} \,d x } \] Input:
integrate(1/x/(x^2+x)^(1/2)/(x^2+x*(x^2+x)^(1/2))^(1/2),x, algorithm="maxi ma")
Output:
integrate(1/(sqrt(x^2 + sqrt(x^2 + x)*x)*sqrt(x^2 + x)*x), x)
\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {1}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} \sqrt {x^{2} + x} x} \,d x } \] Input:
integrate(1/x/(x^2+x)^(1/2)/(x^2+x*(x^2+x)^(1/2))^(1/2),x, algorithm="giac ")
Output:
integrate(1/(sqrt(x^2 + sqrt(x^2 + x)*x)*sqrt(x^2 + x)*x), x)
Timed out. \[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {1}{x\,\sqrt {x^2+x\,\sqrt {x^2+x}}\,\sqrt {x^2+x}} \,d x \] Input:
int(1/(x*(x^2 + x*(x + x^2)^(1/2))^(1/2)*(x + x^2)^(1/2)),x)
Output:
int(1/(x*(x^2 + x*(x + x^2)^(1/2))^(1/2)*(x + x^2)^(1/2)), x)
\[ \int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx=-\left (\int \frac {\sqrt {x +1}\, \sqrt {\sqrt {x}\, \sqrt {x +1}+x}}{x^{3}+x^{2}}d x \right )+\int \frac {\sqrt {x}\, \sqrt {\sqrt {x}\, \sqrt {x +1}+x}}{x^{4}+x^{3}}d x +\int \frac {\sqrt {x}\, \sqrt {\sqrt {x}\, \sqrt {x +1}+x}}{x^{3}+x^{2}}d x \] Input:
int(1/x/(x^2+x)^(1/2)/(x^2+x*(x^2+x)^(1/2))^(1/2),x)
Output:
- int((sqrt(x + 1)*sqrt(sqrt(x)*sqrt(x + 1) + x))/(x**3 + x**2),x) + int( (sqrt(x)*sqrt(sqrt(x)*sqrt(x + 1) + x))/(x**4 + x**3),x) + int((sqrt(x)*sq rt(sqrt(x)*sqrt(x + 1) + x))/(x**3 + x**2),x)