3.28.16 \(\int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx\) [2716]

3.28.16.1 Optimal result

Integrand size = 23, antiderivative size = 249 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {234639860780381988229271893709089+599643393101900122105957184226864 x+1129650644023239063399012556878492 x^2+1387443210853276135686753647625176 x^3+1378491402159939739304774333547654 x^4+1027228435068267876872457367335328 x^5+628530823497779895944692654279120 x^6+287852425478772976908575001303368 x^7+105092569428910116088968496479929 x^8+24945598143999361560228474400848 x^9+4150400378441530192773231110852 x^{10}+\sqrt {2} \left (165915436694475540805635728313976+424011909556064190889798701120701 x+798783630760582994655573305295524 x^2+981070502905588448116241178725596 x^3+974740618274645593869948214329596 x^4+726360192264417334196307996315058 x^5+444438407480045173855448430747052 x^6+203542402037035701060244675374692 x^7+74311668495500397843180232944316 x^8+17639201608376503319286232221121 x^9+2934776252235219007085702870688 x^{10}\right )}{12 \sqrt {2} \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2} \left (4276282378397179351700892708442+5009976441455984103472318426114 x+5009976441455984103472318426114 x^2+2075200189220765096386615555426 x^3+733694063058804751771425717672 x^4\right )+12 \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2} \left (6047576536066366651665626203827+7085176630676749199858933981540 x+7085176630676749199858933981540 x^2+2934776252235219007085702870688 x^3+1037600094610382548193307777713 x^4\right )}+\left (\frac {1}{2}+\frac {1}{4 \sqrt {2}}\right ) \log \left (\sqrt {2}+2 x-2 \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}\right ) \]

output

(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[phone] 
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[phone] 
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) 
)
 

3.28.16.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.38 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {1}{12} \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \left (1+4 \sqrt {2}+\sqrt {2} x+4 x^2\right )+\frac {1}{8} \left (-4-\sqrt {2}\right ) \text {arctanh}\left (\frac {\sqrt {2}+2 x}{2 \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}}\right ) \]

input

Integrate[x*Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2],x]
 

output

(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
 

3.28.16.3 Rubi [A] (verified)

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

\(\displaystyle \frac {1}{3} \left (x^2+\sqrt {2} x+\sqrt {2}+1\right )^{3/2}-\frac {\frac {1}{4} \left (1+2 \sqrt {2}\right ) \text {arcsinh}\left (\sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (2 x+\sqrt {2}\right )\right )+\frac {\sqrt {x^2+\sqrt {2} x+\sqrt {2}+1} \left (\sqrt {2} x+1\right )}{2 \sqrt {2}}}{\sqrt {2}}\)

input

Int[x*Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2],x]
 

output

(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

rule 222

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]
 

rule 1087

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])
 

rule 1090

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]
 

rule 1160

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]
 

3.28.16.4 Maple [A] (verified)

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	\(\left (\frac {x \sqrt {2}}{12}+\frac {x^{2}}{3}+\frac {\sqrt {2}}{3}+\frac {1}{12}\right ) \sqrt {1+\sqrt {2}+x \sqrt {2}+x^{2}}+\frac {\left (3 \sqrt {2}-2\right ) \left (1+\sqrt {2}\right ) \ln \left (-2 x +2 \sqrt {1+\sqrt {2}+x \sqrt {2}+x^{2}}-\sqrt {2}\right )}{8}\)	\(77\)

input

int(x*(1+2^(1/2)+x*2^(1/2)+x^2)^(1/2),x,method=_RETURNVERBOSE)
 

output

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)))
 

3.28.16.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {1}{8} \, {\left (\sqrt {2} + 4\right )} \log \left (-2 \, x - \sqrt {2} + 2 \, \sqrt {x^{2} + \sqrt {2} {\left (x + 1\right )} + 1}\right ) + \frac {1}{12} \, {\left (4 \, x^{2} + \sqrt {2} {\left (x + 4\right )} + 1\right )} \sqrt {x^{2} + \sqrt {2} {\left (x + 1\right )} + 1} \]

input

integrate(x*(1+2^(1/2)+x*2^(1/2)+x^2)^(1/2),x, algorithm="fricas")
 

output

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)
 

3.28.16.6 Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.39 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\left (\frac {x^{2}}{3} + \frac {\sqrt {2} x}{12} + \frac {1}{12} + \frac {\sqrt {2}}{3}\right ) \sqrt {x^{2} + \sqrt {2} x + 1 + \sqrt {2}} + \left (- \frac {\sqrt {2} \cdot \left (\frac {1}{12} + \frac {\sqrt {2}}{3}\right )}{2} - \frac {\sqrt {2} \cdot \left (1 + \sqrt {2}\right )}{12}\right ) \operatorname {asinh}{\left (\frac {x + \frac {\sqrt {2}}{2}}{\sqrt {\frac {1}{2} + \sqrt {2}}} \right )} \]

input

integrate(x*(1+2**(1/2)+x*2**(1/2)+x**2)**(1/2),x)
 

output

(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)))
 

3.28.16.7 Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.43 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=-\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, x + \sqrt {2}}{\sqrt {4 \, \sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1} x + \frac {1}{3} \, {\left (x^{2} + \sqrt {2} x + \sqrt {2} + 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {2} \operatorname {arsinh}\left (\frac {2 \, x + \sqrt {2}}{\sqrt {4 \, \sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1} \]

input

integrate(x*(1+2^(1/2)+x*2^(1/2)+x^2)^(1/2),x, algorithm="maxima")
 

output

-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)
 

3.28.16.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.31 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {1}{8} \, \sqrt {2} {\left (2 \, \sqrt {2} + 1\right )} \log \left (-\sqrt {2} {\left (x - \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1}\right )} - 1\right ) + \frac {1}{24} \, {\left (2 \, {\left (4 \, x + \sqrt {2}\right )} x + \sqrt {2} {\left (\sqrt {2} + 8\right )}\right )} \sqrt {x^{2} + \sqrt {2} x + \sqrt {2} + 1} \]

input

integrate(x*(1+2^(1/2)+x*2^(1/2)+x^2)^(1/2),x, algorithm="giac")
 

output

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)
 

3.28.16.9 Mupad [B] (verification not implemented)

Time = 6.46 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.31 \[ \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx=\frac {\left (8\,x^2+2\,\sqrt {2}\,x+8\,\sqrt {2}+2\right )\,\sqrt {x^2+\sqrt {2}\,x+\sqrt {2}+1}}{24}+\ln \left (x+\sqrt {x^2+\sqrt {2}\,x+\sqrt {2}+1}+\frac {\sqrt {2}}{2}\right )\,\left (\frac {\sqrt {2}}{8}-\frac {\sqrt {2}\,\left (\sqrt {2}+1\right )}{4}\right ) \]

input

int(x*(2^(1/2)*x + 2^(1/2) + x^2 + 1)^(1/2),x)
 

output

((2*2^(1/2)*x + 8*2^(1/2) + 8*x^2 + 2)*(2^(1/2)*x + 2^(1/2) + x^2 + 1)^(1/ 
2))/24 + log(x + (2^(1/2)*x + 2^(1/2) + x^2 + 1)^(1/2) + 2^(1/2)/2)*(2^(1/ 
2)/8 - (2^(1/2)*(2^(1/2) + 1))/4)