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

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

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Rubi [A]  time = 0.11, antiderivative size = 99, normalized size of antiderivative = 0.40, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {640, 612, 619, 215} \begin {gather*} \frac {1}{3} \left (x^2+\sqrt {2} x+\sqrt {2}+1\right )^{3/2}-\frac {1}{4} \left (\sqrt {2} x+1\right ) \sqrt {x^2+\sqrt {2} x+\sqrt {2}+1}-\frac {1}{8} \left (4+\sqrt {2}\right ) \sinh ^{-1}\left (\sqrt {\frac {1}{7} \left (2 \sqrt {2}-1\right )} \left (\sqrt {2} x+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((1 + Sqrt[2]*x)*Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2]) + (1 + Sqrt[2] + Sqrt[2]*x + x^2)^(3/2)/3 - ((4 + S
qrt[2])*ArcSinh[Sqrt[(-1 + 2*Sqrt[2])/7]*(1 + Sqrt[2]*x)])/8

Rule 215

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 612

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] - Dist[(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]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[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 640

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx &=\frac {1}{3} \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2}-\frac {\int \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2} \, dx}{\sqrt {2}}\\ &=-\frac {1}{4} \left (1+\sqrt {2} x\right ) \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}+\frac {1}{3} \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2}-\frac {1}{8} \left (4+\sqrt {2}\right ) \int \frac {1}{\sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}} \, dx\\ &=-\frac {1}{4} \left (1+\sqrt {2} x\right ) \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}+\frac {1}{3} \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2}-\frac {1}{8} \sqrt {1+2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2 \left (1+2 \sqrt {2}\right )}}} \, dx,x,\sqrt {2}+2 x\right )\\ &=-\frac {1}{4} \left (1+\sqrt {2} x\right ) \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}+\frac {1}{3} \left (1+\sqrt {2}+\sqrt {2} x+x^2\right )^{3/2}-\frac {1}{8} \left (4+\sqrt {2}\right ) \sinh ^{-1}\left (\sqrt {\frac {1}{7} \left (-1+2 \sqrt {2}\right )} \left (1+\sqrt {2} x\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 86, normalized size = 0.35 \begin {gather*} \frac {1}{24} \left (2 \sqrt {x^2+\sqrt {2} x+\sqrt {2}+1} \left (4 x^2+\sqrt {2} x+4 \sqrt {2}+1\right )-3 \left (4+\sqrt {2}\right ) \sinh ^{-1}\left (\sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \left (2 x+\sqrt {2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2]*(1 + 4*Sqrt[2] + Sqrt[2]*x + 4*x^2) - 3*(4 + Sqrt[2])*ArcSinh[Sqrt[(-1
+ 2*Sqrt[2])/14]*(Sqrt[2] + 2*x)])/24

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IntegrateAlgebraic [A]  time = 0.36, size = 91, normalized size = 0.37 \begin {gather*} \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 ) \log \left (\sqrt {2}+2 x-2 \sqrt {1+\sqrt {2}+\sqrt {2} x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2]*(1 + 4*Sqrt[2] + Sqrt[2]*x + 4*x^2))/12 + ((4 + Sqrt[2])*Log[Sqrt[2] + 2*
x - 2*Sqrt[1 + Sqrt[2] + Sqrt[2]*x + x^2]])/8

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fricas [A]  time = 0.58, size = 64, normalized size = 0.26 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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giac [A]  time = 0.16, size = 77, normalized size = 0.31 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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maple [A]  time = 1.05, size = 60, normalized size = 0.24

method	result	size
risch	\(\frac {\left (\sqrt {2}\, x +4 x^{2}+4 \sqrt {2}+1\right ) \sqrt {1+\sqrt {2}+\sqrt {2}\, x +x^{2}}}{12}+\left (-\frac {1}{2}-\frac {\sqrt {2}}{8}\right ) \arcsinh \left (\frac {x +\frac {\sqrt {2}}{2}}{\sqrt {\frac {1}{2}+\sqrt {2}}}\right )\)	\(60\)
default	\(\frac {\left (1+\sqrt {2}+\sqrt {2}\, x +x^{2}\right )^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\frac {\left (\sqrt {2}+2 x \right ) \sqrt {1+\sqrt {2}+\sqrt {2}\, x +x^{2}}}{4}+\frac {\left (2+4 \sqrt {2}\right ) \arcsinh \left (\frac {x +\frac {\sqrt {2}}{2}}{\sqrt {\frac {1}{2}+\sqrt {2}}}\right )}{8}\right )}{2}\)	\(74\)
trager	\(\left (\frac {\sqrt {2}\, x}{12}+\frac {x^{2}}{3}+\frac {\sqrt {2}}{3}+\frac {1}{12}\right ) \sqrt {1+\sqrt {2}+\sqrt {2}\, x +x^{2}}+\frac {\left (3 \sqrt {2}-2\right ) \left (1+\sqrt {2}\right ) \ln \left (-2 x -\sqrt {2}+2 \sqrt {1+\sqrt {2}+\sqrt {2}\, x +x^{2}}\right )}{8}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2^(1/2)*x+4*x^2+4*2^(1/2)+1)*(1+2^(1/2)+2^(1/2)*x+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)))

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maxima [A]  time = 0.52, size = 107, normalized size = 0.43 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + s
qrt(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)

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mupad [B]  time = 1.69, size = 76, normalized size = 0.31 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {x^{2} + \sqrt {2} x + 1 + \sqrt {2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(x**2 + sqrt(2)*x + 1 + sqrt(2)), x)

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