Integrand size = 23, antiderivative size = 249 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {234639860780381988229271893709089+599643393101900122105957184226864 x+1129650644023239063399012556878492 x^2+1387443210853276135686753647625176 x^3+1378491402159939739304774333547654 x^4+1027228435068267876872457367335328 x^5+628530823497779895944692654279120 x^6+287852425478772976908575001303368 x^7+105092569428910116088968496479929 x^8+24945598143999361560228474400848 x^9+4150400378441530192773231110852 x^{10}+\sqrt {2} \left (165915436694475540805635728313976+424011909556064190889798701120701 x+798783630760582994655573305295524 x^2+981070502905588448116241178725596 x^3+974740618274645593869948214329596 x^4+726360192264417334196307996315058 x^5+444438407480045173855448430747052 x^6+203542402037035701060244675374692 x^7+74311668495500397843180232944316 x^8+17639201608376503319286232221121 x^9+2934776252235219007085702870688 x^{10}\right )}{12 \sqrt {2} \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2} \left (4276282378397179351700892708442+5009976441455984103472318426114 x+5009976441455984103472318426114 x^2+2075200189220765096386615555426 x^3+733694063058804751771425717672 x^4\right )+12 \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2} \left (6047576536066366651665626203827+7085176630676749199858933981540 x+7085176630676749199858933981540 x^2+2934776252235219007085702870688 x^3+1037600094610382548193307777713 x^4\right )}+\left (\frac {1}{2}+\frac {1}{4 \sqrt {2}}\right ) \log \left (\sqrt {2}+2 x-2 \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}\right ) \]
(234639860780381988229271893709089+599643393101900122105957184226864*x+112 9650644023239063399012556878492*x^2+1387443210853276135686753647625176*x^3 +1378491402159939739304774333547654*x^4+1027228435068267876872457367335328 *x^5+628530823497779895944692654279120*x^6+2878524254787729769085750013033 68*x^7+105092569428910116088968496479929*x^8+24945598143999361560228474400 848*x^9+4150400378441530192773231110852*x^10+2^(1/2)*(29347762522352190070 85702870688*x^10+17639201608376503319286232221121*x^9+74311668495500397843 180232944316*x^8+203542402037035701060244675374692*x^7+4444384074800451738 55448430747052*x^6+726360192264417334196307996315058*x^5+97474061827464559 3869948214329596*x^4+981070502905588448116241178725596*x^3+798783630760582 994655573305295524*x^2+424011909556064190889798701120701*x+165915436694475 540805635728313976))/(12*2^(1/2)*(1+2^(1/2)+x*2^(1/2)+x^2)^(3/2)*(73369406 3058804751771425717672*x^4+2075200189220765096386615555426*x^3+50099764414 55984103472318426114*x^2+5009976441455984103472318426114*x+427628237839717 9351700892708442)+12*(1+2^(1/2)+x*2^(1/2)+x^2)^(3/2)*(10376000946103825481 93307777713*x^4+2934776252235219007085702870688*x^3+7085176630676749199858 933981540*x^2+7085176630676749199858933981540*x+60475765360663666516656262 03827))+(1/2+1/8*2^(1/2))*ln(2^(1/2)+2*x-2*(1+2^(1/2)+x*2^(1/2)+x^2)^(1/2) )
Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.38 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {1}{12} \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \left (1+4 \sqrt {2}+\sqrt {2} x+4 x^2\right )+\frac {1}{8} \left (-4-\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {2}+2 x}{2 \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}}\right ) \]
(Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2]*(1 + 4*Sqrt[2] + Sqrt[2]*x + 4*x^2))/ 12 + ((-4 - Sqrt[2])*ArcTanh[(Sqrt[2] + 2*x)/(2*Sqrt[1 + Sqrt[2] + Sqrt[2] *x + x^2])])/8
Time = 0.25 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.46, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1160, 1087, 1090, 222}
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.
\(\displaystyle \int x \sqrt {x^2+\sqrt {2} x+\sqrt {2}+1} \, dx\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{3} \left (x^2+\sqrt {2} x+\sqrt {2}+1\right )^{3/2}-\frac {\int \sqrt {x^2+\sqrt {2} x+\sqrt {2}+1}dx}{\sqrt {2}}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{3} \left (x^2+\sqrt {2} x+\sqrt {2}+1\right )^{3/2}-\frac {\frac {1}{4} \left (1+2 \sqrt {2}\right ) \int \frac {1}{\sqrt {x^2+\sqrt {2} x+\sqrt {2}+1}}dx+\frac {\sqrt {x^2+\sqrt {2} x+\sqrt {2}+1} \left (\sqrt {2} x+1\right )}{2 \sqrt {2}}}{\sqrt {2}}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{3} \left (x^2+\sqrt {2} x+\sqrt {2}+1\right )^{3/2}-\frac {\frac {1}{4} \sqrt {\frac {1}{2} \left (1+2 \sqrt {2}\right )} \int \frac {1}{\sqrt {\frac {\left (2 x+\sqrt {2}\right )^2}{2 \left (1+2 \sqrt {2}\right )}+1}}d\left (2 x+\sqrt {2}\right )+\frac {\sqrt {x^2+\sqrt {2} x+\sqrt {2}+1} \left (\sqrt {2} x+1\right )}{2 \sqrt {2}}}{\sqrt {2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{3} \left (x^2+\sqrt {2} x+\sqrt {2}+1\right )^{3/2}-\frac {\frac {1}{4} \left (1+2 \sqrt {2}\right ) \text {arcsinh}\left (\sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (2 x+\sqrt {2}\right )\right )+\frac {\sqrt {x^2+\sqrt {2} x+\sqrt {2}+1} \left (\sqrt {2} x+1\right )}{2 \sqrt {2}}}{\sqrt {2}}\) |
(1 + Sqrt[2] + Sqrt[2]*x + x^2)^(3/2)/3 - (((1 + Sqrt[2]*x)*Sqrt[1 + Sqrt[ 2] + Sqrt[2]*x + x^2])/(2*Sqrt[2]) + ((1 + 2*Sqrt[2])*ArcSinh[Sqrt[(-1 + 2 *Sqrt[2])/14]*(Sqrt[2] + 2*x)])/4)/Sqrt[2]
3.28.16.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
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]
Time = 0.75 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.24
method | result | size |
risch | \(\frac {\left (x \sqrt {2}+4 x^{2}+4 \sqrt {2}+1\right ) \sqrt {1+\sqrt {2}+x \sqrt {2}+x^{2}}}{12}+\left (-\frac {1}{2}-\frac {\sqrt {2}}{8}\right ) \operatorname {arcsinh}\left (\frac {x +\frac {\sqrt {2}}{2}}{\sqrt {\frac {1}{2}+\sqrt {2}}}\right )\) | \(60\) |
default | \(\frac {\left (1+\sqrt {2}+x \sqrt {2}+x^{2}\right )^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\frac {\left (\sqrt {2}+2 x \right ) \sqrt {1+\sqrt {2}+x \sqrt {2}+x^{2}}}{4}+\frac {\left (2+4 \sqrt {2}\right ) \operatorname {arcsinh}\left (\frac {x +\frac {\sqrt {2}}{2}}{\sqrt {\frac {1}{2}+\sqrt {2}}}\right )}{8}\right )}{2}\) | \(74\) |
trager | \(\left (\frac {x \sqrt {2}}{12}+\frac {x^{2}}{3}+\frac {\sqrt {2}}{3}+\frac {1}{12}\right ) \sqrt {1+\sqrt {2}+x \sqrt {2}+x^{2}}+\frac {\left (3 \sqrt {2}-2\right ) \left (1+\sqrt {2}\right ) \ln \left (-2 x +2 \sqrt {1+\sqrt {2}+x \sqrt {2}+x^{2}}-\sqrt {2}\right )}{8}\) | \(77\) |
1/12*(x*2^(1/2)+4*x^2+4*2^(1/2)+1)*(1+2^(1/2)+x*2^(1/2)+x^2)^(1/2)+(-1/2-1 /8*2^(1/2))*arcsinh(1/(1/2+2^(1/2))^(1/2)*(x+1/2*2^(1/2)))
Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {1}{8} \, {\left (\sqrt {2} + 4\right )} \log \left (-2 \, x - \sqrt {2} + 2 \, \sqrt {x^{2} + \sqrt {2} {\left (x + 1\right )} + 1}\right ) + \frac {1}{12} \, {\left (4 \, x^{2} + \sqrt {2} {\left (x + 4\right )} + 1\right )} \sqrt {x^{2} + \sqrt {2} {\left (x + 1\right )} + 1} \]
1/8*(sqrt(2) + 4)*log(-2*x - sqrt(2) + 2*sqrt(x^2 + sqrt(2)*(x + 1) + 1)) + 1/12*(4*x^2 + sqrt(2)*(x + 4) + 1)*sqrt(x^2 + sqrt(2)*(x + 1) + 1)
Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.39 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\left (\frac {x^{2}}{3} + \frac {\sqrt {2} x}{12} + \frac {1}{12} + \frac {\sqrt {2}}{3}\right ) \sqrt {x^{2} + \sqrt {2} x + 1 + \sqrt {2}} + \left (- \frac {\sqrt {2} \cdot \left (\frac {1}{12} + \frac {\sqrt {2}}{3}\right )}{2} - \frac {\sqrt {2} \cdot \left (1 + \sqrt {2}\right )}{12}\right ) \operatorname {asinh}{\left (\frac {x + \frac {\sqrt {2}}{2}}{\sqrt {\frac {1}{2} + \sqrt {2}}} \right )} \]
(x**2/3 + sqrt(2)*x/12 + 1/12 + sqrt(2)/3)*sqrt(x**2 + sqrt(2)*x + 1 + sqr t(2)) + (-sqrt(2)*(1/12 + sqrt(2)/3)/2 - sqrt(2)*(1 + sqrt(2))/12)*asinh(( x + sqrt(2)/2)/sqrt(1/2 + sqrt(2)))
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.43 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=-\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, x + \sqrt {2}}{\sqrt {4 \, \sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1} x + \frac {1}{3} \, {\left (x^{2} + \sqrt {2} x + \sqrt {2} + 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {2} \operatorname {arsinh}\left (\frac {2 \, x + \sqrt {2}}{\sqrt {4 \, \sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1} \]
-1/4*sqrt(2)*(sqrt(2) + 1)*arcsinh((2*x + sqrt(2))/sqrt(4*sqrt(2) + 2)) - 1/4*sqrt(2)*sqrt(x^2 + sqrt(2)*x + sqrt(2) + 1)*x + 1/3*(x^2 + sqrt(2)*x + sqrt(2) + 1)^(3/2) + 1/8*sqrt(2)*arcsinh((2*x + sqrt(2))/sqrt(4*sqrt(2) + 2)) - 1/4*sqrt(x^2 + sqrt(2)*x + sqrt(2) + 1)
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.31 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {1}{8} \, \sqrt {2} {\left (2 \, \sqrt {2} + 1\right )} \log \left (-\sqrt {2} {\left (x - \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1}\right )} - 1\right ) + \frac {1}{24} \, {\left (2 \, {\left (4 \, x + \sqrt {2}\right )} x + \sqrt {2} {\left (\sqrt {2} + 8\right )}\right )} \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1} \]
1/8*sqrt(2)*(2*sqrt(2) + 1)*log(-sqrt(2)*(x - sqrt(x^2 + sqrt(2)*x + sqrt( 2) + 1)) - 1) + 1/24*(2*(4*x + sqrt(2))*x + sqrt(2)*(sqrt(2) + 8))*sqrt(x^ 2 + sqrt(2)*x + sqrt(2) + 1)
Time = 6.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.31 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {\left (8\,x^2+2\,\sqrt {2}\,x+8\,\sqrt {2}+2\right )\,\sqrt {x^2+\sqrt {2}\,x+\sqrt {2}+1}}{24}+\ln \left (x+\sqrt {x^2+\sqrt {2}\,x+\sqrt {2}+1}+\frac {\sqrt {2}}{2}\right )\,\left (\frac {\sqrt {2}}{8}-\frac {\sqrt {2}\,\left (\sqrt {2}+1\right )}{4}\right ) \]