3.245 \(\int \frac{x}{(4+x+x^2) \sqrt{5+4 x+4 x^2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{4 x^2+4 x+5}}{\sqrt{11}}\right )}{\sqrt{11}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{11}{15}} (2 x+1)}{\sqrt{4 x^2+4 x+5}}\right )}{\sqrt{165}} \]

[Out]

ArcTan[Sqrt[5 + 4*x + 4*x^2]/Sqrt[11]]/Sqrt[11] - ArcTanh[(Sqrt[11/15]*(1 + 2*x))/Sqrt[5 + 4*x + 4*x^2]]/Sqrt[
165]

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Rubi [A]  time = 0.0541548, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1025, 982, 207, 1024, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{4 x^2+4 x+5}}{\sqrt{11}}\right )}{\sqrt{11}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{11}{15}} (2 x+1)}{\sqrt{4 x^2+4 x+5}}\right )}{\sqrt{165}} \]

Antiderivative was successfully verified.

[In]

Int[x/((4 + x + x^2)*Sqrt[5 + 4*x + 4*x^2]),x]

[Out]

ArcTan[Sqrt[5 + 4*x + 4*x^2]/Sqrt[11]]/Sqrt[11] - ArcTanh[(Sqrt[11/15]*(1 + 2*x))/Sqrt[5 + 4*x + 4*x^2]]/Sqrt[
165]

Rule 1025

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]

Rule 982

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 207

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

Rule 1024

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 204

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

Rubi steps

\begin{align*} \int \frac{x}{\left (4+x+x^2\right ) \sqrt{5+4 x+4 x^2}} \, dx &=\frac{1}{8} \int \frac{4+8 x}{\left (4+x+x^2\right ) \sqrt{5+4 x+4 x^2}} \, dx-\frac{1}{2} \int \frac{1}{\left (4+x+x^2\right ) \sqrt{5+4 x+4 x^2}} \, dx\\ &=4 \operatorname{Subst}\left (\int \frac{1}{-240+11 x^2} \, dx,x,\frac{4+8 x}{\sqrt{5+4 x+4 x^2}}\right )-\operatorname{Subst}\left (\int \frac{1}{-11-x^2} \, dx,x,\sqrt{5+4 x+4 x^2}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{5+4 x+4 x^2}}{\sqrt{11}}\right )}{\sqrt{11}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{11}{15}} (1+2 x)}{\sqrt{5+4 x+4 x^2}}\right )}{\sqrt{165}}\\ \end{align*}

Mathematica [C]  time = 0.10611, size = 114, normalized size = 1.81 \[ \frac{\left (\sqrt{15}-i\right ) \tan ^{-1}\left (\frac{-2 i \sqrt{15} x-i \sqrt{15}+4}{\sqrt{11} \sqrt{4 x^2+4 x+5}}\right )+\left (\sqrt{15}+i\right ) \tan ^{-1}\left (\frac{2 i \sqrt{15} x+i \sqrt{15}+4}{\sqrt{11} \sqrt{4 x^2+4 x+5}}\right )}{2 \sqrt{165}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/((4 + x + x^2)*Sqrt[5 + 4*x + 4*x^2]),x]

[Out]

((-I + Sqrt[15])*ArcTan[(4 - I*Sqrt[15] - (2*I)*Sqrt[15]*x)/(Sqrt[11]*Sqrt[5 + 4*x + 4*x^2])] + (I + Sqrt[15])
*ArcTan[(4 + I*Sqrt[15] + (2*I)*Sqrt[15]*x)/(Sqrt[11]*Sqrt[5 + 4*x + 4*x^2])])/(2*Sqrt[165])

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Maple [A]  time = 0.011, size = 53, normalized size = 0.8 \begin{align*}{\frac{\sqrt{11}}{11}\arctan \left ({\frac{\sqrt{11}}{11}\sqrt{4\,{x}^{2}+4\,x+5}} \right ) }-{\frac{\sqrt{165}}{165}{\it Artanh} \left ({\frac{\sqrt{165} \left ( 8\,x+4 \right ) }{60}{\frac{1}{\sqrt{4\,{x}^{2}+4\,x+5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2+x+4)/(4*x^2+4*x+5)^(1/2),x)

[Out]

1/11*arctan(1/11*(4*x^2+4*x+5)^(1/2)*11^(1/2))*11^(1/2)-1/165*165^(1/2)*arctanh(1/60*165^(1/2)*(8*x+4)/(4*x^2+
4*x+5)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{4 \, x^{2} + 4 \, x + 5}{\left (x^{2} + x + 4\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+x+4)/(4*x^2+4*x+5)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/(sqrt(4*x^2 + 4*x + 5)*(x^2 + x + 4)), x)

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Fricas [B]  time = 2.23113, size = 1057, normalized size = 16.78 \begin{align*} \frac{2}{165} \, \sqrt{165} \sqrt{15} \arctan \left (\frac{1}{60} \, \sqrt{2} \sqrt{4 \, x^{2} - \sqrt{4 \, x^{2} + 4 \, x + 5}{\left (2 \, x + 1\right )} + 4 \, x - \sqrt{165} + 16}{\left (\sqrt{165} \sqrt{15} + 15 \, \sqrt{15}\right )} + \frac{1}{60} \, \sqrt{165} \sqrt{15}{\left (2 \, x + 1\right )} - \frac{1}{60} \, \sqrt{4 \, x^{2} + 4 \, x + 5}{\left (\sqrt{165} \sqrt{15} + 15 \, \sqrt{15}\right )} + \frac{1}{4} \, \sqrt{15}{\left (2 \, x + 1\right )}\right ) + \frac{2}{165} \, \sqrt{165} \sqrt{15} \arctan \left (\frac{1}{60} \, \sqrt{2} \sqrt{4 \, x^{2} - \sqrt{4 \, x^{2} + 4 \, x + 5}{\left (2 \, x + 1\right )} + 4 \, x + \sqrt{165} + 16}{\left (\sqrt{165} \sqrt{15} - 15 \, \sqrt{15}\right )} + \frac{1}{60} \, \sqrt{165} \sqrt{15}{\left (2 \, x + 1\right )} - \frac{1}{60} \, \sqrt{4 \, x^{2} + 4 \, x + 5}{\left (\sqrt{165} \sqrt{15} - 15 \, \sqrt{15}\right )} - \frac{1}{4} \, \sqrt{15}{\left (2 \, x + 1\right )}\right ) - \frac{1}{330} \, \sqrt{165} \log \left (460800 \, x^{2} - 115200 \, \sqrt{4 \, x^{2} + 4 \, x + 5}{\left (2 \, x + 1\right )} + 460800 \, x + 115200 \, \sqrt{165} + 1843200\right ) + \frac{1}{330} \, \sqrt{165} \log \left (460800 \, x^{2} - 115200 \, \sqrt{4 \, x^{2} + 4 \, x + 5}{\left (2 \, x + 1\right )} + 460800 \, x - 115200 \, \sqrt{165} + 1843200\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+x+4)/(4*x^2+4*x+5)^(1/2),x, algorithm="fricas")

[Out]

2/165*sqrt(165)*sqrt(15)*arctan(1/60*sqrt(2)*sqrt(4*x^2 - sqrt(4*x^2 + 4*x + 5)*(2*x + 1) + 4*x - sqrt(165) +
16)*(sqrt(165)*sqrt(15) + 15*sqrt(15)) + 1/60*sqrt(165)*sqrt(15)*(2*x + 1) - 1/60*sqrt(4*x^2 + 4*x + 5)*(sqrt(
165)*sqrt(15) + 15*sqrt(15)) + 1/4*sqrt(15)*(2*x + 1)) + 2/165*sqrt(165)*sqrt(15)*arctan(1/60*sqrt(2)*sqrt(4*x
^2 - sqrt(4*x^2 + 4*x + 5)*(2*x + 1) + 4*x + sqrt(165) + 16)*(sqrt(165)*sqrt(15) - 15*sqrt(15)) + 1/60*sqrt(16
5)*sqrt(15)*(2*x + 1) - 1/60*sqrt(4*x^2 + 4*x + 5)*(sqrt(165)*sqrt(15) - 15*sqrt(15)) - 1/4*sqrt(15)*(2*x + 1)
) - 1/330*sqrt(165)*log(460800*x^2 - 115200*sqrt(4*x^2 + 4*x + 5)*(2*x + 1) + 460800*x + 115200*sqrt(165) + 18
43200) + 1/330*sqrt(165)*log(460800*x^2 - 115200*sqrt(4*x^2 + 4*x + 5)*(2*x + 1) + 460800*x - 115200*sqrt(165)
+ 1843200)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (x^{2} + x + 4\right ) \sqrt{4 x^{2} + 4 x + 5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2+x+4)/(4*x**2+4*x+5)**(1/2),x)

[Out]

Integral(x/((x**2 + x + 4)*sqrt(4*x**2 + 4*x + 5)), x)

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Giac [C]  time = 1.1623, size = 228, normalized size = 3.62 \begin{align*} -\frac{1}{330} \, \sqrt{165}{\left (-i \, \sqrt{15} + 1\right )} \log \left (-600 \, x + 300 i \, \sqrt{15} + 300 i \, \sqrt{11} + 300 \, \sqrt{4 \, x^{2} + 4 \, x + 5} - 300\right ) + \frac{1}{330} \, \sqrt{165}{\left (-i \, \sqrt{15} + 1\right )} \log \left (-600 \, x + 300 i \, \sqrt{15} - 300 i \, \sqrt{11} + 300 \, \sqrt{4 \, x^{2} + 4 \, x + 5} - 300\right ) + \frac{1}{330} \, \sqrt{165}{\left (i \, \sqrt{15} + 1\right )} \log \left (-600 \, x - 300 i \, \sqrt{15} + 300 i \, \sqrt{11} + 300 \, \sqrt{4 \, x^{2} + 4 \, x + 5} - 300\right ) - \frac{1}{330} \, \sqrt{165}{\left (i \, \sqrt{15} + 1\right )} \log \left (-600 \, x - 300 i \, \sqrt{15} - 300 i \, \sqrt{11} + 300 \, \sqrt{4 \, x^{2} + 4 \, x + 5} - 300\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^2+x+4)/(4*x^2+4*x+5)^(1/2),x, algorithm="giac")

[Out]

-1/330*sqrt(165)*(-I*sqrt(15) + 1)*log(-600*x + 300*I*sqrt(15) + 300*I*sqrt(11) + 300*sqrt(4*x^2 + 4*x + 5) -
300) + 1/330*sqrt(165)*(-I*sqrt(15) + 1)*log(-600*x + 300*I*sqrt(15) - 300*I*sqrt(11) + 300*sqrt(4*x^2 + 4*x +
5) - 300) + 1/330*sqrt(165)*(I*sqrt(15) + 1)*log(-600*x - 300*I*sqrt(15) + 300*I*sqrt(11) + 300*sqrt(4*x^2 +
4*x + 5) - 300) - 1/330*sqrt(165)*(I*sqrt(15) + 1)*log(-600*x - 300*I*sqrt(15) - 300*I*sqrt(11) + 300*sqrt(4*x
^2 + 4*x + 5) - 300)