3.247 \(\int \frac{1+2 x}{\sqrt{-1+6 x+x^2} (4+4 x+3 x^2)} \, dx\)

Optimal. Leaf size=70 \[ -\frac{5 \tan ^{-1}\left (\frac{\sqrt{\frac{7}{2}} (2-x)}{2 \sqrt{x^2+6 x-1}}\right )}{6 \sqrt{14}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{7} (x+1)}{\sqrt{x^2+6 x-1}}\right )}{3 \sqrt{7}} \]

[Out]

(-5*ArcTan[(Sqrt[7/2]*(2 - x))/(2*Sqrt[-1 + 6*x + x^2])])/(6*Sqrt[14]) - ArcTanh[(Sqrt[7]*(1 + x))/Sqrt[-1 + 6
*x + x^2]]/(3*Sqrt[7])

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Rubi [A]  time = 0.0638688, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1035, 1029, 207, 203} \[ -\frac{5 \tan ^{-1}\left (\frac{\sqrt{\frac{7}{2}} (2-x)}{2 \sqrt{x^2+6 x-1}}\right )}{6 \sqrt{14}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{7} (x+1)}{\sqrt{x^2+6 x-1}}\right )}{3 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)/(Sqrt[-1 + 6*x + x^2]*(4 + 4*x + 3*x^2)),x]

[Out]

(-5*ArcTan[(Sqrt[7/2]*(2 - x))/(2*Sqrt[-1 + 6*x + x^2])])/(6*Sqrt[14]) - ArcTanh[(Sqrt[7]*(1 + x))/Sqrt[-1 + 6
*x + x^2]]/(3*Sqrt[7])

Rule 1035

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Dist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*
d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - D
ist[1/(2*q), Int[Simp[h*(b*d - a*e) - g*(c*d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, 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] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]

Rule 1029

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[g*b - 2*a*h - (b*h - 2*g*c)*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] && NeQ[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(
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 203

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

Rubi steps

\begin{align*} \int \frac{1+2 x}{\sqrt{-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx &=-\left (\frac{1}{42} \int \frac{-70-70 x}{\sqrt{-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx\right )+\frac{1}{42} \int \frac{-28+14 x}{\sqrt{-1+6 x+x^2} \left (4+4 x+3 x^2\right )} \, dx\\ &=-\left (\frac{896}{3} \operatorname{Subst}\left (\int \frac{1}{-200704+28 x^2} \, dx,x,\frac{-224-224 x}{\sqrt{-1+6 x+x^2}}\right )\right )-\frac{2800}{3} \operatorname{Subst}\left (\int \frac{1}{627200+28 x^2} \, dx,x,\frac{280-140 x}{\sqrt{-1+6 x+x^2}}\right )\\ &=-\frac{5 \tan ^{-1}\left (\frac{\sqrt{\frac{7}{2}} (2-x)}{2 \sqrt{-1+6 x+x^2}}\right )}{6 \sqrt{14}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{7} (1+x)}{\sqrt{-1+6 x+x^2}}\right )}{3 \sqrt{7}}\\ \end{align*}

Mathematica [C]  time = 0.418217, size = 174, normalized size = 2.49 \[ -\frac{\sqrt{7-4 i \sqrt{2}} \left (8 \sqrt{2}+13 i\right ) \tan ^{-1}\left (\frac{\left (-7-2 i \sqrt{2}\right ) x-6 i \sqrt{2}+9}{\sqrt{7 \left (7-4 i \sqrt{2}\right )} \sqrt{x^2+6 x-1}}\right )+\sqrt{7+4 i \sqrt{2}} \left (8 \sqrt{2}-13 i\right ) \tan ^{-1}\left (\frac{\left (-7+2 i \sqrt{2}\right ) x+6 i \sqrt{2}+9}{\sqrt{7 \left (7+4 i \sqrt{2}\right )} \sqrt{x^2+6 x-1}}\right )}{108 \sqrt{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)/(Sqrt[-1 + 6*x + x^2]*(4 + 4*x + 3*x^2)),x]

[Out]

-(Sqrt[7 - (4*I)*Sqrt[2]]*(13*I + 8*Sqrt[2])*ArcTan[(9 - (6*I)*Sqrt[2] + (-7 - (2*I)*Sqrt[2])*x)/(Sqrt[7*(7 -
(4*I)*Sqrt[2])]*Sqrt[-1 + 6*x + x^2])] + Sqrt[7 + (4*I)*Sqrt[2]]*(-13*I + 8*Sqrt[2])*ArcTan[(9 + (6*I)*Sqrt[2]
+ (-7 + (2*I)*Sqrt[2])*x)/(Sqrt[7*(7 + (4*I)*Sqrt[2])]*Sqrt[-1 + 6*x + x^2])])/(108*Sqrt[14])

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Maple [B]  time = 0.023, size = 158, normalized size = 2.3 \begin{align*} -{\frac{1}{84}\sqrt{-6\,{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+15} \left ( 4\,\sqrt{7}{\it Artanh} \left ( 1/21\,\sqrt{-6\,{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+15}\sqrt{7} \right ) -5\,\sqrt{14}\arctan \left ( 1/4\,{\frac{\sqrt{14} \left ( -2+x \right ) }{-1-x}\sqrt{-6\,{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+15} \left ( 2\,{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}-5 \right ) ^{-1}} \right ) \right ){\frac{1}{\sqrt{-3\,{ \left ( 2\,{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}-5 \right ) \left ( 1+{\frac{-2+x}{-1-x}} \right ) ^{-2}}}}} \left ( 1+{\frac{-2+x}{-1-x}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)/(3*x^2+4*x+4)/(x^2+6*x-1)^(1/2),x)

[Out]

-1/84*(-6*(-2+x)^2/(-1-x)^2+15)^(1/2)*(4*7^(1/2)*arctanh(1/21*(-6*(-2+x)^2/(-1-x)^2+15)^(1/2)*7^(1/2))-5*14^(1
/2)*arctan(1/4*14^(1/2)*(-6*(-2+x)^2/(-1-x)^2+15)^(1/2)/(2*(-2+x)^2/(-1-x)^2-5)*(-2+x)/(-1-x)))/(-3*(2*(-2+x)^
2/(-1-x)^2-5)/(1+(-2+x)/(-1-x))^2)^(1/2)/(1+(-2+x)/(-1-x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x + 1}{{\left (3 \, x^{2} + 4 \, x + 4\right )} \sqrt{x^{2} + 6 \, x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)/(3*x^2+4*x+4)/(x^2+6*x-1)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x + 1)/((3*x^2 + 4*x + 4)*sqrt(x^2 + 6*x - 1)), x)

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Fricas [B]  time = 2.84163, size = 1110, normalized size = 15.86 \begin{align*} \frac{1}{84} \, \sqrt{14} \sqrt{2} \log \left (13068 \, \sqrt{14} \sqrt{2}{\left (x - 2\right )} + 78408 \, x^{2} - 13068 \, \sqrt{x^{2} + 6 \, x - 1}{\left (\sqrt{14} \sqrt{2} + 6 \, x + 4\right )} + 287496 \, x + 287496\right ) - \frac{1}{84} \, \sqrt{14} \sqrt{2} \log \left (-13068 \, \sqrt{14} \sqrt{2}{\left (x - 2\right )} + 78408 \, x^{2} + 13068 \, \sqrt{x^{2} + 6 \, x - 1}{\left (\sqrt{14} \sqrt{2} - 6 \, x - 4\right )} + 287496 \, x + 287496\right ) - \frac{5}{42} \, \sqrt{14} \arctan \left (\frac{1}{24} \, \sqrt{3} \sqrt{\sqrt{14} \sqrt{2}{\left (x - 2\right )} + 6 \, x^{2} - \sqrt{x^{2} + 6 \, x - 1}{\left (\sqrt{14} \sqrt{2} + 6 \, x + 4\right )} + 22 \, x + 22}{\left (\sqrt{14} + \sqrt{2}\right )} + \frac{1}{8} \, \sqrt{2}{\left (x + 3\right )} + \frac{1}{8} \, \sqrt{14}{\left (x + 1\right )} - \frac{1}{8} \, \sqrt{x^{2} + 6 \, x - 1}{\left (\sqrt{14} + \sqrt{2}\right )}\right ) - \frac{5}{42} \, \sqrt{14} \arctan \left (-\frac{1}{8} \, \sqrt{2}{\left (x + 3\right )} + \frac{1}{8} \, \sqrt{14}{\left (x + 1\right )} + \frac{1}{1584} \, \sqrt{-13068 \, \sqrt{14} \sqrt{2}{\left (x - 2\right )} + 78408 \, x^{2} + 13068 \, \sqrt{x^{2} + 6 \, x - 1}{\left (\sqrt{14} \sqrt{2} - 6 \, x - 4\right )} + 287496 \, x + 287496}{\left (\sqrt{14} - \sqrt{2}\right )} - \frac{1}{8} \, \sqrt{x^{2} + 6 \, x - 1}{\left (\sqrt{14} - \sqrt{2}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)/(3*x^2+4*x+4)/(x^2+6*x-1)^(1/2),x, algorithm="fricas")

[Out]

1/84*sqrt(14)*sqrt(2)*log(13068*sqrt(14)*sqrt(2)*(x - 2) + 78408*x^2 - 13068*sqrt(x^2 + 6*x - 1)*(sqrt(14)*sqr
t(2) + 6*x + 4) + 287496*x + 287496) - 1/84*sqrt(14)*sqrt(2)*log(-13068*sqrt(14)*sqrt(2)*(x - 2) + 78408*x^2 +
13068*sqrt(x^2 + 6*x - 1)*(sqrt(14)*sqrt(2) - 6*x - 4) + 287496*x + 287496) - 5/42*sqrt(14)*arctan(1/24*sqrt(
3)*sqrt(sqrt(14)*sqrt(2)*(x - 2) + 6*x^2 - sqrt(x^2 + 6*x - 1)*(sqrt(14)*sqrt(2) + 6*x + 4) + 22*x + 22)*(sqrt
(14) + sqrt(2)) + 1/8*sqrt(2)*(x + 3) + 1/8*sqrt(14)*(x + 1) - 1/8*sqrt(x^2 + 6*x - 1)*(sqrt(14) + sqrt(2))) -
5/42*sqrt(14)*arctan(-1/8*sqrt(2)*(x + 3) + 1/8*sqrt(14)*(x + 1) + 1/1584*sqrt(-13068*sqrt(14)*sqrt(2)*(x - 2
) + 78408*x^2 + 13068*sqrt(x^2 + 6*x - 1)*(sqrt(14)*sqrt(2) - 6*x - 4) + 287496*x + 287496)*(sqrt(14) - sqrt(2
)) - 1/8*sqrt(x^2 + 6*x - 1)*(sqrt(14) - sqrt(2)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 x + 1}{\sqrt{x^{2} + 6 x - 1} \left (3 x^{2} + 4 x + 4\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)/(3*x**2+4*x+4)/(x**2+6*x-1)**(1/2),x)

[Out]

Integral((2*x + 1)/(sqrt(x**2 + 6*x - 1)*(3*x**2 + 4*x + 4)), x)

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Giac [C]  time = 1.12576, size = 261, normalized size = 3.73 \begin{align*} \frac{1}{168} \, \sqrt{7}{\left (-5 i \, \sqrt{2} - 4\right )} \log \left (-\left (8 i - 4\right ) \, \sqrt{7} \sqrt{2} - \left (6 i + 12\right ) \, x + \left (2 i + 4\right ) \, \sqrt{7} - \left (8 i - 4\right ) \, \sqrt{2} + \left (6 i + 12\right ) \, \sqrt{x^{2} + 6 \, x - 1} - 4 i - 8\right ) - \frac{1}{168} \, \sqrt{7}{\left (5 i \, \sqrt{2} - 4\right )} \log \left (-\left (8 i - 4\right ) \, \sqrt{7} \sqrt{2} - \left (6 i + 12\right ) \, x - \left (2 i + 4\right ) \, \sqrt{7} + \left (8 i - 4\right ) \, \sqrt{2} + \left (6 i + 12\right ) \, \sqrt{x^{2} + 6 \, x - 1} - 4 i - 8\right ) + \frac{1}{168} \, \sqrt{7}{\left (5 i \, \sqrt{2} - 4\right )} \log \left (\left (4 i - 8\right ) \, \sqrt{7} \sqrt{2} - \left (12 i + 6\right ) \, x + \left (4 i + 2\right ) \, \sqrt{7} + \left (4 i - 8\right ) \, \sqrt{2} + \left (12 i + 6\right ) \, \sqrt{x^{2} + 6 \, x - 1} - 8 i - 4\right ) - \frac{1}{168} \, \sqrt{7}{\left (-5 i \, \sqrt{2} - 4\right )} \log \left (\left (4 i - 8\right ) \, \sqrt{7} \sqrt{2} - \left (12 i + 6\right ) \, x - \left (4 i + 2\right ) \, \sqrt{7} - \left (4 i - 8\right ) \, \sqrt{2} + \left (12 i + 6\right ) \, \sqrt{x^{2} + 6 \, x - 1} - 8 i - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)/(3*x^2+4*x+4)/(x^2+6*x-1)^(1/2),x, algorithm="giac")

[Out]

1/168*sqrt(7)*(-5*I*sqrt(2) - 4)*log(-(8*I - 4)*sqrt(7)*sqrt(2) - (6*I + 12)*x + (2*I + 4)*sqrt(7) - (8*I - 4)
*sqrt(2) + (6*I + 12)*sqrt(x^2 + 6*x - 1) - 4*I - 8) - 1/168*sqrt(7)*(5*I*sqrt(2) - 4)*log(-(8*I - 4)*sqrt(7)*
sqrt(2) - (6*I + 12)*x - (2*I + 4)*sqrt(7) + (8*I - 4)*sqrt(2) + (6*I + 12)*sqrt(x^2 + 6*x - 1) - 4*I - 8) + 1
/168*sqrt(7)*(5*I*sqrt(2) - 4)*log((4*I - 8)*sqrt(7)*sqrt(2) - (12*I + 6)*x + (4*I + 2)*sqrt(7) + (4*I - 8)*sq
rt(2) + (12*I + 6)*sqrt(x^2 + 6*x - 1) - 8*I - 4) - 1/168*sqrt(7)*(-5*I*sqrt(2) - 4)*log((4*I - 8)*sqrt(7)*sqr
t(2) - (12*I + 6)*x - (4*I + 2)*sqrt(7) - (4*I - 8)*sqrt(2) + (12*I + 6)*sqrt(x^2 + 6*x - 1) - 8*I - 4)