Integrand size = 32, antiderivative size = 140 \[ \int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {\sqrt {x+x^2} \left (-1575+840 x-640 x^2-3072 x^3\right ) \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{10752 x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (\frac {525-120 x+3968 x^2+3072 x^3}{10752}+\frac {75 \sqrt {-x+\sqrt {x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{512 \sqrt {2} x}\right ) \]
1/10752*(x^2+x)^(1/2)*(-3072*x^3-640*x^2+840*x-1575)*(x*(x+(x^2+x)^(1/2))) ^(1/2)/x+(x*(x+(x^2+x)^(1/2)))^(1/2)*(2/7*x^3+31/84*x^2-5/448*x+25/512+75/ 1024*2^(1/2)*(-x+(x^2+x)^(1/2))^(1/2)*arctanh(2^(1/2)*(-x+(x^2+x)^(1/2))^( 1/2))/x)
Time = 3.03 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.14 \[ \int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {(1+x) \sqrt {x \left (x+\sqrt {x (1+x)}\right )} \left (-6144 x^5+1050 x \left (-3+\sqrt {x (1+x)}\right )-30 x^2 \left (49+8 \sqrt {x (1+x)}\right )+256 x^4 \left (-29+24 \sqrt {x (1+x)}\right )+16 x^3 \left (25+496 \sqrt {x (1+x)}\right )+1575 \sqrt {2} \sqrt {x (1+x)} \sqrt {-x+\sqrt {x (1+x)}} \text {arctanh}\left (\sqrt {-2 x+2 \sqrt {x (1+x)}}\right )\right )}{21504 (x (1+x))^{3/2}} \]
((1 + x)*Sqrt[x*(x + Sqrt[x*(1 + x)])]*(-6144*x^5 + 1050*x*(-3 + Sqrt[x*(1 + x)]) - 30*x^2*(49 + 8*Sqrt[x*(1 + x)]) + 256*x^4*(-29 + 24*Sqrt[x*(1 + x)]) + 16*x^3*(25 + 496*Sqrt[x*(1 + x)]) + 1575*Sqrt[2]*Sqrt[x*(1 + x)]*Sq rt[-x + Sqrt[x*(1 + x)]]*ArcTanh[Sqrt[-2*x + 2*Sqrt[x*(1 + x)]]]))/(21504* (x*(1 + x))^(3/2))
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.
\(\displaystyle \int \frac {x^2 \sqrt {x^2+x}}{\sqrt {x^2+\sqrt {x^2+x} x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x^2+x} \int \frac {x^{5/2} \sqrt {x+1}}{\sqrt {x^2+\sqrt {x^2+x} x}}dx}{\sqrt {x} \sqrt {x+1}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {2 \sqrt {x^2+x} \int \frac {x^3 \sqrt {x+1}}{\sqrt {x^2+\sqrt {x^2+x} x}}d\sqrt {x}}{\sqrt {x} \sqrt {x+1}}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {2 \sqrt {x^2+x} \int \frac {x^3 \sqrt {x+1}}{\sqrt {x^2+\sqrt {x^2+x} x}}d\sqrt {x}}{\sqrt {x} \sqrt {x+1}}\) |
3.20.86.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
\[\int \frac {x^{2} \sqrt {x^{2}+x}}{\sqrt {x^{2}+x \sqrt {x^{2}+x}}}d x\]
Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {1575 \, \sqrt {2} x \log \left (\frac {4 \, x^{2} + 2 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + x}\right )} + 4 \, \sqrt {x^{2} + x} x + x}{x}\right ) + 4 \, {\left (3072 \, x^{4} + 3968 \, x^{3} - 120 \, x^{2} - {\left (3072 \, x^{3} + 640 \, x^{2} - 840 \, x + 1575\right )} \sqrt {x^{2} + x} + 525 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} + x} x}}{43008 \, x} \]
1/43008*(1575*sqrt(2)*x*log((4*x^2 + 2*sqrt(x^2 + sqrt(x^2 + x)*x)*(sqrt(2 )*x + sqrt(2)*sqrt(x^2 + x)) + 4*sqrt(x^2 + x)*x + x)/x) + 4*(3072*x^4 + 3 968*x^3 - 120*x^2 - (3072*x^3 + 640*x^2 - 840*x + 1575)*sqrt(x^2 + x) + 52 5*x)*sqrt(x^2 + sqrt(x^2 + x)*x))/x
\[ \int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {x^{2} \sqrt {x \left (x + 1\right )}}{\sqrt {x \left (x + \sqrt {x^{2} + x}\right )}}\, dx \]
\[ \int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + x} x^{2}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}} \,d x } \]
\[ \int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + x} x^{2}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}} \,d x } \]
Timed out. \[ \int \frac {x^2 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {x^2\,\sqrt {x^2+x}}{\sqrt {x^2+x\,\sqrt {x^2+x}}} \,d x \]