\(\int \frac {x}{(3+x+x^2) \sqrt {5+x+x^2}} \, dx\) [34]

Optimal result

Integrand size = 20, antiderivative size = 56 \[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {\frac {2}{11}} (1+2 x)}{\sqrt {5+x+x^2}}\right )}{\sqrt {22}}-\frac {\text {arctanh}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]

[Out]

-1/2*arctanh(1/2*(x^2+x+5)^(1/2)*2^(1/2))*2^(1/2)-1/22*arctan(1/11*(1+2*x)*22^(1/2)/(x^2+x+5)^(1/2))*22^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1039, 996, 210, 1038, 212} \[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {\frac {2}{11}} (2 x+1)}{\sqrt {x^2+x+5}}\right )}{\sqrt {22}}-\frac {\text {arctanh}\left (\frac {\sqrt {x^2+x+5}}{\sqrt {2}}\right )}{\sqrt {2}} \]

[In]

Int[x/((3 + x + x^2)*Sqrt[5 + x + x^2]),x]

[Out]

-(ArcTan[(Sqrt[2/11]*(1 + 2*x))/Sqrt[5 + x + x^2]]/Sqrt[22]) - ArcTanh[Sqrt[5 + x + x^2]/Sqrt[2]]/Sqrt[2]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 996

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*e, Su
bst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0]

Rule 1038

Int[((g_) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol
] :> Dist[-2*g, Subst[Int[1/(b*d - a*e - b*x^2), x], x, Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, e, f,
g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && EqQ[h*e - 2*g*f, 0]

Rule 1039

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> Dist[-(h*e - 2*g*f)/(2*f), Int[1/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] + Dist[h/(2*f), Int[(
e + 2*f*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2
- 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 0] && NeQ[h*e - 2*g*f, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {1}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx\right )+\frac {1}{2} \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx \\ & = \text {Subst}\left (\int \frac {1}{-11-2 x^2} \, dx,x,\frac {1+2 x}{\sqrt {5+x+x^2}}\right )-\text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {5+x+x^2}\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {2}{11}} (1+2 x)}{\sqrt {5+x+x^2}}\right )}{\sqrt {22}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.64 \[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\text {RootSum}\left [23-2 \text {$\#$1}+3 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-5 \log \left (-x+\sqrt {5+x+x^2}-\text {$\#$1}\right )+\log \left (-x+\sqrt {5+x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-1+3 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[x/((3 + x + x^2)*Sqrt[5 + x + x^2]),x]

[Out]

RootSum[23 - 2*#1 + 3*#1^2 - 2*#1^3 + #1^4 & , (-5*Log[-x + Sqrt[5 + x + x^2] - #1] + Log[-x + Sqrt[5 + x + x^
2] - #1]*#1^2)/(-1 + 3*#1 - 3*#1^2 + 2*#1^3) & ]

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {x^{2}+x +5}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {22}}{11 \sqrt {x^{2}+x +5}}\right ) \sqrt {22}}{22}\)	\(45\)
trager	\(-\operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right ) \ln \left (\frac {133342 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{5} x -34298 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{3} x +2970 \sqrt {x^{2}+x +5}\, \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{2}+29667 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{3}+2100 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right ) x -504 \sqrt {x^{2}+x +5}-4650 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )}{22 x \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{2}-2 x +3}\right )+\frac {22 \ln \left (-\frac {-22990 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{5} x +2079 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{3} x +990 \sqrt {x^{2}+x +5}\, \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{2}+5115 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{3}+106 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right ) x -57 \sqrt {x^{2}+x +5}+186 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )}{22 x \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{2}-3 x -3}\right ) \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{3}}{3}-\frac {5 \ln \left (-\frac {-22990 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{5} x +2079 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{3} x +990 \sqrt {x^{2}+x +5}\, \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{2}+5115 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{3}+106 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right ) x -57 \sqrt {x^{2}+x +5}+186 \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )}{22 x \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )^{2}-3 x -3}\right ) \operatorname {RootOf}\left (484 \textit {\_Z}^{4}-110 \textit {\_Z}^{2}+9\right )}{3}\)	\(492\)

[In]

int(x/(x^2+x+3)/(x^2+x+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*arctanh(1/2*(x^2+x+5)^(1/2)*2^(1/2))*2^(1/2)-1/22*arctan(1/11*(1+2*x)*22^(1/2)/(x^2+x+5)^(1/2))*22^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 3.62 \[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\frac {1}{22} \, \sqrt {11} \sqrt {i \, \sqrt {11} + 5} \log \left ({\left (\sqrt {11} - i\right )} \sqrt {i \, \sqrt {11} + 5} - 6 \, x + 3 i \, \sqrt {11} + 6 \, \sqrt {x^{2} + x + 5} - 3\right ) + \frac {1}{22} \, \sqrt {11} \sqrt {i \, \sqrt {11} + 5} \log \left (-{\left (\sqrt {11} - i\right )} \sqrt {i \, \sqrt {11} + 5} - 6 \, x + 3 i \, \sqrt {11} + 6 \, \sqrt {x^{2} + x + 5} - 3\right ) - \frac {1}{22} \, \sqrt {11} \sqrt {-i \, \sqrt {11} + 5} \log \left ({\left (\sqrt {11} + i\right )} \sqrt {-i \, \sqrt {11} + 5} - 6 \, x - 3 i \, \sqrt {11} + 6 \, \sqrt {x^{2} + x + 5} - 3\right ) + \frac {1}{22} \, \sqrt {11} \sqrt {-i \, \sqrt {11} + 5} \log \left (-{\left (\sqrt {11} + i\right )} \sqrt {-i \, \sqrt {11} + 5} - 6 \, x - 3 i \, \sqrt {11} + 6 \, \sqrt {x^{2} + x + 5} - 3\right ) \]

[In]

integrate(x/(x^2+x+3)/(x^2+x+5)^(1/2),x, algorithm="fricas")

[Out]

-1/22*sqrt(11)*sqrt(I*sqrt(11) + 5)*log((sqrt(11) - I)*sqrt(I*sqrt(11) + 5) - 6*x + 3*I*sqrt(11) + 6*sqrt(x^2
+ x + 5) - 3) + 1/22*sqrt(11)*sqrt(I*sqrt(11) + 5)*log(-(sqrt(11) - I)*sqrt(I*sqrt(11) + 5) - 6*x + 3*I*sqrt(1
1) + 6*sqrt(x^2 + x + 5) - 3) - 1/22*sqrt(11)*sqrt(-I*sqrt(11) + 5)*log((sqrt(11) + I)*sqrt(-I*sqrt(11) + 5) -
6*x - 3*I*sqrt(11) + 6*sqrt(x^2 + x + 5) - 3) + 1/22*sqrt(11)*sqrt(-I*sqrt(11) + 5)*log(-(sqrt(11) + I)*sqrt(
-I*sqrt(11) + 5) - 6*x - 3*I*sqrt(11) + 6*sqrt(x^2 + x + 5) - 3)

Sympy [F]

\[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\int \frac {x}{\left (x^{2} + x + 3\right ) \sqrt {x^{2} + x + 5}}\, dx \]

[In]

integrate(x/(x**2+x+3)/(x**2+x+5)**(1/2),x)

[Out]

Integral(x/((x**2 + x + 3)*sqrt(x**2 + x + 5)), x)

Maxima [F]

\[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\int { \frac {x}{\sqrt {x^{2} + x + 5} {\left (x^{2} + x + 3\right )}} \,d x } \]

[In]

integrate(x/(x^2+x+3)/(x^2+x+5)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/(sqrt(x^2 + x + 5)*(x^2 + x + 3)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (44) = 88\).

Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.38 \[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\frac {1}{22} \, \sqrt {11} \sqrt {2} \arctan \left (-\frac {1}{11} \, \sqrt {11} {\left (2 \, x + 2 \, \sqrt {2} - 2 \, \sqrt {x^{2} + x + 5} + 1\right )}\right ) - \frac {1}{22} \, \sqrt {11} \sqrt {2} \arctan \left (-\frac {1}{11} \, \sqrt {11} {\left (2 \, x - 2 \, \sqrt {2} - 2 \, \sqrt {x^{2} + x + 5} + 1\right )}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (324 \, {\left (2 \, x + 2 \, \sqrt {2} - 2 \, \sqrt {x^{2} + x + 5} + 1\right )}^{2} + 3564\right ) - \frac {1}{4} \, \sqrt {2} \log \left (324 \, {\left (2 \, x - 2 \, \sqrt {2} - 2 \, \sqrt {x^{2} + x + 5} + 1\right )}^{2} + 3564\right ) \]

[In]

integrate(x/(x^2+x+3)/(x^2+x+5)^(1/2),x, algorithm="giac")

[Out]

1/22*sqrt(11)*sqrt(2)*arctan(-1/11*sqrt(11)*(2*x + 2*sqrt(2) - 2*sqrt(x^2 + x + 5) + 1)) - 1/22*sqrt(11)*sqrt(
2)*arctan(-1/11*sqrt(11)*(2*x - 2*sqrt(2) - 2*sqrt(x^2 + x + 5) + 1)) + 1/4*sqrt(2)*log(324*(2*x + 2*sqrt(2) -
2*sqrt(x^2 + x + 5) + 1)^2 + 3564) - 1/4*sqrt(2)*log(324*(2*x - 2*sqrt(2) - 2*sqrt(x^2 + x + 5) + 1)^2 + 3564
)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\int \frac {x}{\left (x^2+x+3\right )\,\sqrt {x^2+x+5}} \,d x \]

[In]

int(x/((x + x^2 + 3)*(x + x^2 + 5)^(1/2)),x)

[Out]

int(x/((x + x^2 + 3)*(x + x^2 + 5)^(1/2)), x)