Integrand size = 24, antiderivative size = 93 \[ \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {\left (563+328 x+128 x^2\right ) \sqrt {x+\sqrt {1+x}}}{3840}+\frac {\sqrt {1+x} \left (975-872 x+640 x^2\right ) \sqrt {x+\sqrt {1+x}}}{1920}+\frac {385}{512} \log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right ) \]
1/3840*(128*x^2+328*x+563)*(x+(1+x)^(1/2))^(1/2)+1/1920*(1+x)^(1/2)*(640*x ^2-872*x+975)*(x+(1+x)^(1/2))^(1/2)+385/512*ln(1+2*(1+x)^(1/2)-2*(x+(1+x)^ (1/2))^(1/2))
Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {\sqrt {x+\sqrt {1+x}} \left (363+4974 \sqrt {1+x}+72 (1+x)-4304 (1+x)^{3/2}+128 (1+x)^2+1280 (1+x)^{5/2}\right )}{3840}+\frac {385}{512} \log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right ) \]
(Sqrt[x + Sqrt[1 + x]]*(363 + 4974*Sqrt[1 + x] + 72*(1 + x) - 4304*(1 + x) ^(3/2) + 128*(1 + x)^2 + 1280*(1 + x)^(5/2)))/3840 + (385*Log[-1 - 2*Sqrt[ 1 + x] + 2*Sqrt[x + Sqrt[1 + x]]])/512
Time = 0.64 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.80, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {7267, 2192, 27, 2192, 27, 2192, 27, 1160, 1087, 1092, 219}
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.
| \(\displaystyle \int \frac {x^2 \sqrt {x+\sqrt {x+1}}}{\sqrt {x+1}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int x^2 \sqrt {x+\sqrt {x+1}}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle 2 \left (\frac {1}{6} \int \frac {3}{2} \sqrt {x+\sqrt {x+1}} \left (-3 (x+1)^{3/2}-6 (x+1)+4\right )d\sqrt {x+1}+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {1}{4} \int \sqrt {x+\sqrt {x+1}} \left (-3 (x+1)^{3/2}-6 (x+1)+4\right )d\sqrt {x+1}+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{5} \int \frac {1}{2} \sqrt {x+\sqrt {x+1}} \left (-39 (x+1)-12 \sqrt {x+1}+40\right )d\sqrt {x+1}-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{10} \int \sqrt {x+\sqrt {x+1}} \left (-39 (x+1)-12 \sqrt {x+1}+40\right )d\sqrt {x+1}-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {1}{4} \int \frac {11}{2} \sqrt {x+\sqrt {x+1}} \left (9 \sqrt {x+1}+22\right )d\sqrt {x+1}-\frac {39}{4} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {11}{8} \int \sqrt {x+\sqrt {x+1}} \left (9 \sqrt {x+1}+22\right )d\sqrt {x+1}-\frac {39}{4} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {11}{8} \left (\frac {35}{2} \int \sqrt {x+\sqrt {x+1}}d\sqrt {x+1}+3 \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {39}{4} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {11}{8} \left (\frac {35}{2} \left (\frac {1}{4} \sqrt {x+\sqrt {x+1}} \left (2 \sqrt {x+1}+1\right )-\frac {5}{8} \int \frac {1}{\sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\right )+3 \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {39}{4} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {11}{8} \left (\frac {35}{2} \left (\frac {1}{4} \sqrt {x+\sqrt {x+1}} \left (2 \sqrt {x+1}+1\right )-\frac {5}{4} \int \frac {1}{3-x}d\frac {2 \sqrt {x+1}+1}{\sqrt {x+\sqrt {x+1}}}\right )+3 \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {39}{4} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {11}{8} \left (\frac {35}{2} \left (\frac {1}{4} \sqrt {x+\sqrt {x+1}} \left (2 \sqrt {x+1}+1\right )-\frac {5}{8} \text {arctanh}\left (\frac {2 \sqrt {x+1}+1}{2 \sqrt {x+\sqrt {x+1}}}\right )\right )+3 \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {39}{4} \sqrt {x+1} \left (x+\sqrt {x+1}\right )^{3/2}\right )-\frac {3}{5} (x+1) \left (x+\sqrt {x+1}\right )^{3/2}\right )+\frac {1}{6} (x+1)^{3/2} \left (x+\sqrt {x+1}\right )^{3/2}\right )\) |
2*(((1 + x)^(3/2)*(x + Sqrt[1 + x])^(3/2))/6 + ((-3*(1 + x)*(x + Sqrt[1 + x])^(3/2))/5 + ((-39*Sqrt[1 + x]*(x + Sqrt[1 + x])^(3/2))/4 + (11*(3*(x + Sqrt[1 + x])^(3/2) + (35*((Sqrt[x + Sqrt[1 + x]]*(1 + 2*Sqrt[1 + x]))/4 - (5*ArcTanh[(1 + 2*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/8))/2))/8)/10)/ 4)
3.13.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
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 = 0.84 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {77 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{256}-\frac {385 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{512}+\frac {\left (1+x \right )^{\frac {3}{2}} \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {3 \left (1+x \right ) \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{10}-\frac {39 \sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{80}+\frac {33 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{160}\) | \(98\) |
| default | \(\frac {77 \left (2 \sqrt {1+x}+1\right ) \sqrt {x +\sqrt {1+x}}}{256}-\frac {385 \ln \left (\frac {1}{2}+\sqrt {1+x}+\sqrt {x +\sqrt {1+x}}\right )}{512}+\frac {\left (1+x \right )^{\frac {3}{2}} \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {3 \left (1+x \right ) \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{10}-\frac {39 \sqrt {1+x}\, \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{80}+\frac {33 \left (x +\sqrt {1+x}\right )^{\frac {3}{2}}}{160}\) | \(98\) |
77/256*(2*(1+x)^(1/2)+1)*(x+(1+x)^(1/2))^(1/2)-385/512*ln(1/2+(1+x)^(1/2)+ (x+(1+x)^(1/2))^(1/2))+1/3*(1+x)^(3/2)*(x+(1+x)^(1/2))^(3/2)-3/10*(1+x)*(x +(1+x)^(1/2))^(3/2)-39/80*(1+x)^(1/2)*(x+(1+x)^(1/2))^(3/2)+33/160*(x+(1+x )^(1/2))^(3/2)
Time = 0.70 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.80 \[ \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{3840} \, {\left (128 \, x^{2} + 2 \, {\left (640 \, x^{2} - 872 \, x + 975\right )} \sqrt {x + 1} + 328 \, x + 563\right )} \sqrt {x + \sqrt {x + 1}} + \frac {385}{1024} \, \log \left (4 \, \sqrt {x + \sqrt {x + 1}} {\left (2 \, \sqrt {x + 1} + 1\right )} - 8 \, x - 8 \, \sqrt {x + 1} - 5\right ) \]
1/3840*(128*x^2 + 2*(640*x^2 - 872*x + 975)*sqrt(x + 1) + 328*x + 563)*sqr t(x + sqrt(x + 1)) + 385/1024*log(4*sqrt(x + sqrt(x + 1))*(2*sqrt(x + 1) + 1) - 8*x - 8*sqrt(x + 1) - 5)
Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=2 \sqrt {x + \sqrt {x + 1}} \cdot \left (\frac {3 x}{320} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{6} - \frac {269 \left (x + 1\right )^{\frac {3}{2}}}{480} + \frac {829 \sqrt {x + 1}}{1280} + \frac {\left (x + 1\right )^{2}}{60} + \frac {29}{512}\right ) - \frac {385 \log {\left (2 \sqrt {x + 1} + 2 \sqrt {x + \sqrt {x + 1}} + 1 \right )}}{512} \]
2*sqrt(x + sqrt(x + 1))*(3*x/320 + (x + 1)**(5/2)/6 - 269*(x + 1)**(3/2)/4 80 + 829*sqrt(x + 1)/1280 + (x + 1)**2/60 + 29/512) - 385*log(2*sqrt(x + 1 ) + 2*sqrt(x + sqrt(x + 1)) + 1)/512
\[ \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {x + \sqrt {x + 1}} x^{2}}{\sqrt {x + 1}} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, \sqrt {x + 1} {\left (10 \, \sqrt {x + 1} + 1\right )} - 269\right )} \sqrt {x + 1} + 9\right )} \sqrt {x + 1} + 2487\right )} \sqrt {x + 1} + 363\right )} \sqrt {x + \sqrt {x + 1}} + \frac {385}{512} \, \log \left (-2 \, \sqrt {x + \sqrt {x + 1}} + 2 \, \sqrt {x + 1} + 1\right ) \]
1/3840*(2*(4*(2*(8*sqrt(x + 1)*(10*sqrt(x + 1) + 1) - 269)*sqrt(x + 1) + 9 )*sqrt(x + 1) + 2487)*sqrt(x + 1) + 363)*sqrt(x + sqrt(x + 1)) + 385/512*l og(-2*sqrt(x + sqrt(x + 1)) + 2*sqrt(x + 1) + 1)
Timed out. \[ \int \frac {x^2 \sqrt {x+\sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {x^2\,\sqrt {x+\sqrt {x+1}}}{\sqrt {x+1}} \,d x \]