Integrand size = 26, antiderivative size = 119 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=-\frac {4}{3} \sqrt {1+\sqrt {1+x}}+\sqrt {1+x} \left (\frac {2 (1+x)}{3}-\frac {4}{3} \sqrt {1+\sqrt {1+x}}\right )-\frac {2}{5} \left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right ) \] Output:
-4/3*(1+(1+x)^(1/2))^(1/2)+(1+x)^(1/2)*(2/3+2/3*x-4/3*(1+(1+x)^(1/2))^(1/2 ))-2/5*(-5+5^(1/2))*ln(-1+5^(1/2)-2*(1+(1+x)^(1/2))^(1/2))+2/5*(5+5^(1/2)) *ln(1+5^(1/2)+2*(1+(1+x)^(1/2))^(1/2))
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.80 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{15} \left (-10 \left (1+\sqrt {1+x}\right )^{3/2}+5 \left (1+(1+x)^{3/2}\right )-3 \left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+3 \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )\right ) \] Input:
Integrate[(x*Sqrt[1 + x])/(x + Sqrt[1 + Sqrt[1 + x]]),x]
Output:
(2*(-10*(1 + Sqrt[1 + x])^(3/2) + 5*(1 + (1 + x)^(3/2)) - 3*(-5 + Sqrt[5]) *Log[-1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]] + 3*(5 + Sqrt[5])*Log[1 + Sqr t[5] + 2*Sqrt[1 + Sqrt[1 + x]]]))/15
Time = 0.87 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {7267, 7267, 25, 2159, 2009}
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.
\(\displaystyle \int \frac {x \sqrt {x+1}}{x+\sqrt {\sqrt {x+1}+1}} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int -\frac {x (x+1)}{-x-\sqrt {\sqrt {x+1}+1}}d\sqrt {x+1}\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 4 \int -\frac {(1-x) (x+1) \left (1-\sqrt {\sqrt {x+1}+1}\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )^2}{-x-\sqrt {\sqrt {x+1}+1}}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -4 \int \frac {(1-x) (x+1) \left (1-\sqrt {\sqrt {x+1}+1}\right ) \left (\sqrt {\sqrt {x+1}+1}+1\right )^2}{-x-\sqrt {\sqrt {x+1}+1}}d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle -4 \int \left (-(x+1)^{5/2}+2 (x+1)^{3/2}+x-\sqrt {\sqrt {x+1}+1}+\frac {\sqrt {\sqrt {x+1}+1}}{-x-\sqrt {\sqrt {x+1}+1}}+1\right )d\sqrt {\sqrt {x+1}+1}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (\frac {1}{6} (x+1)^3-\frac {1}{2} (x+1)^2-\frac {1}{3} (x+1)^{3/2}+\frac {x+1}{2}+\frac {1}{10} \left (5-\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right )\right )\) |
Input:
Int[(x*Sqrt[1 + x])/(x + Sqrt[1 + Sqrt[1 + x]]),x]
Output:
4*((1 + x)/2 - (1 + x)^(3/2)/3 - (1 + x)^2/2 + (1 + x)^3/6 + ((5 - Sqrt[5] )*Log[1 - Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/10 + ((5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/10)
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+2+2 \sqrt {1+x}+2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\) | \(85\) |
default | \(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+2+2 \sqrt {1+x}+2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\) | \(85\) |
Input:
int(x*(1+x)^(1/2)/(x+(1+(1+x)^(1/2))^(1/2)),x,method=_RETURNVERBOSE)
Output:
2/3*(1+(1+x)^(1/2))^3-2*(1+(1+x)^(1/2))^2-4/3*(1+(1+x)^(1/2))^(3/2)+2+2*(1 +x)^(1/2)+2*ln((1+x)^(1/2)+(1+(1+x)^(1/2))^(1/2))+4/5*5^(1/2)*arctanh(1/5* (2*(1+(1+x)^(1/2))^(1/2)+1)*5^(1/2))
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} + \frac {2}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} - {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) \] Input:
integrate(x*(1+x)^(1/2)/(x+(1+(1+x)^(1/2))^(1/2)),x, algorithm="fricas")
Output:
2/3*(x + 1)^(3/2) - 4/3*(sqrt(x + 1) + 1)^(3/2) + 2/5*sqrt(5)*log((2*x^2 - sqrt(5)*(3*x + 1) - (sqrt(5)*(x + 2) - 5*x)*sqrt(x + 1) + (sqrt(5)*(x + 2 ) + (sqrt(5)*(2*x - 1) - 5)*sqrt(x + 1) - 5*x)*sqrt(sqrt(x + 1) + 1) + 3*x + 3)/(x^2 - x - 1)) + 2*log(sqrt(x + 1) + sqrt(sqrt(x + 1) + 1))
Time = 5.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=2 \sqrt {x + 1} - \frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {3}{2}}}{3} + \frac {2 \left (\sqrt {x + 1} + 1\right )^{3}}{3} - 2 \left (\sqrt {x + 1} + 1\right )^{2} - \frac {2 \sqrt {5} \left (- \log {\left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2} + \frac {\sqrt {5}}{2} \right )} + \log {\left (\sqrt {\sqrt {x + 1} + 1} - \frac {\sqrt {5}}{2} + \frac {1}{2} \right )}\right )}{5} + 2 \log {\left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1} \right )} + 2 \] Input:
integrate(x*(1+x)**(1/2)/(x+(1+(1+x)**(1/2))**(1/2)),x)
Output:
2*sqrt(x + 1) - 4*(sqrt(x + 1) + 1)**(3/2)/3 + 2*(sqrt(x + 1) + 1)**3/3 - 2*(sqrt(x + 1) + 1)**2 - 2*sqrt(5)*(-log(sqrt(sqrt(x + 1) + 1) + 1/2 + sqr t(5)/2) + log(sqrt(sqrt(x + 1) + 1) - sqrt(5)/2 + 1/2))/5 + 2*log(sqrt(x + 1) + sqrt(sqrt(x + 1) + 1)) + 2
Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{3} - 2 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - \frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \] Input:
integrate(x*(1+x)^(1/2)/(x+(1+(1+x)^(1/2))^(1/2)),x, algorithm="maxima")
Output:
2/3*(sqrt(x + 1) + 1)^3 - 2*(sqrt(x + 1) + 1)^2 - 4/3*(sqrt(x + 1) + 1)^(3 /2) - 2/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(sqrt(x + 1) + 1) - 1)/(sqrt(5) + 2*sqrt(sqrt(x + 1) + 1) + 1)) + 2*sqrt(x + 1) + 2*log(sqrt(x + 1) + sqrt(s qrt(x + 1) + 1)) + 2
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=\frac {2}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{3} - 2 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - \frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \] Input:
integrate(x*(1+x)^(1/2)/(x+(1+(1+x)^(1/2))^(1/2)),x, algorithm="giac")
Output:
2/3*(sqrt(x + 1) + 1)^3 - 2*(sqrt(x + 1) + 1)^2 - 4/3*(sqrt(x + 1) + 1)^(3 /2) - 2/5*sqrt(5)*log(-(sqrt(5) - 2*sqrt(sqrt(x + 1) + 1) - 1)/(sqrt(5) + 2*sqrt(sqrt(x + 1) + 1) + 1)) + 2*sqrt(x + 1) + 2*log(sqrt(x + 1) + sqrt(s qrt(x + 1) + 1)) + 2
Timed out. \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=\int \frac {x\,\sqrt {x+1}}{x+\sqrt {\sqrt {x+1}+1}} \,d x \] Input:
int((x*(x + 1)^(1/2))/(x + ((x + 1)^(1/2) + 1)^(1/2)),x)
Output:
int((x*(x + 1)^(1/2))/(x + ((x + 1)^(1/2) + 1)^(1/2)), x)
Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx=-\frac {4 \sqrt {x +1}\, \sqrt {\sqrt {x +1}+1}}{3}-\frac {4 \sqrt {\sqrt {x +1}+1}}{3}+\frac {2 \sqrt {x +1}\, x}{3}+\frac {2 \sqrt {x +1}}{3}-\frac {2 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {\sqrt {x +1}+1}-\sqrt {5}+1\right )}{5}+\frac {2 \sqrt {5}\, \mathrm {log}\left (2 \sqrt {\sqrt {x +1}+1}+\sqrt {5}+1\right )}{5}+2 \,\mathrm {log}\left (2 \sqrt {\sqrt {x +1}+1}-\sqrt {5}+1\right )+2 \,\mathrm {log}\left (2 \sqrt {\sqrt {x +1}+1}+\sqrt {5}+1\right )+\frac {2}{3} \] Input:
int(x*(1+x)^(1/2)/(x+(1+(1+x)^(1/2))^(1/2)),x)
Output:
(2*( - 10*sqrt(x + 1)*sqrt(sqrt(x + 1) + 1) - 10*sqrt(sqrt(x + 1) + 1) + 5 *sqrt(x + 1)*x + 5*sqrt(x + 1) - 3*sqrt(5)*log(2*sqrt(sqrt(x + 1) + 1) - s qrt(5) + 1) + 3*sqrt(5)*log(2*sqrt(sqrt(x + 1) + 1) + sqrt(5) + 1) + 15*lo g(2*sqrt(sqrt(x + 1) + 1) - sqrt(5) + 1) + 15*log(2*sqrt(sqrt(x + 1) + 1) + sqrt(5) + 1) + 5))/15