3.121 \(\int \frac{1}{\sqrt{-7+2 x+5 x^2} (8+12 x+5 x^2)} \, dx\)

Optimal. Leaf size=51 \[ \frac{1}{10} \tan ^{-1}\left (\frac{5 (x+2)}{2 \sqrt{5 x^2+2 x-7}}\right )+\frac{1}{5} \tanh ^{-1}\left (\frac{5 (x+1)}{\sqrt{5 x^2+2 x-7}}\right ) \]

[Out]

ArcTan[(5*(2 + x))/(2*Sqrt[-7 + 2*x + 5*x^2])]/10 + ArcTanh[(5*(1 + x))/Sqrt[-7 + 2*x + 5*x^2]]/5

________________________________________________________________________________________

Rubi [A]  time = 0.0656829, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {986, 1029, 203, 207} \[ \frac{1}{10} \tan ^{-1}\left (\frac{5 (x+2)}{2 \sqrt{5 x^2+2 x-7}}\right )+\frac{1}{5} \tanh ^{-1}\left (\frac{5 (x+1)}{\sqrt{5 x^2+2 x-7}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-7 + 2*x + 5*x^2]*(8 + 12*x + 5*x^2)),x]

[Out]

ArcTan[(5*(2 + x))/(2*Sqrt[-7 + 2*x + 5*x^2])]/10 + ArcTanh[(5*(1 + x))/Sqrt[-7 + 2*x + 5*x^2]]/5

Rule 986

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

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])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-7+2 x+5 x^2} \left (8+12 x+5 x^2\right )} \, dx &=-\left (\frac{1}{50} \int \frac{-100-50 x}{\sqrt{-7+2 x+5 x^2} \left (8+12 x+5 x^2\right )} \, dx\right )+\frac{1}{50} \int \frac{-50-50 x}{\sqrt{-7+2 x+5 x^2} \left (8+12 x+5 x^2\right )} \, dx\\ &=400 \operatorname{Subst}\left (\int \frac{1}{160000+100 x^2} \, dx,x,\frac{200+100 x}{\sqrt{-7+2 x+5 x^2}}\right )+1600 \operatorname{Subst}\left (\int \frac{1}{-640000+100 x^2} \, dx,x,\frac{-400-400 x}{\sqrt{-7+2 x+5 x^2}}\right )\\ &=\frac{1}{10} \tan ^{-1}\left (\frac{5 (2+x)}{2 \sqrt{-7+2 x+5 x^2}}\right )+\frac{1}{5} \tanh ^{-1}\left (\frac{5 (1+x)}{\sqrt{-7+2 x+5 x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0402311, size = 81, normalized size = 1.59 \[ \left (\frac{1}{10}-\frac{i}{20}\right ) \tanh ^{-1}\left (\frac{\left (\frac{1}{100}+\frac{i}{50}\right ) ((100-40 i) x+(164-8 i))}{\sqrt{5 x^2+2 x-7}}\right )-\left (\frac{1}{20}-\frac{i}{10}\right ) \tan ^{-1}\left (\frac{\left (\frac{1}{50}+\frac{i}{100}\right ) ((-100-40 i) x-(164+8 i))}{\sqrt{5 x^2+2 x-7}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-7 + 2*x + 5*x^2]*(8 + 12*x + 5*x^2)),x]

[Out]

(-1/20 + I/10)*ArcTan[((1/50 + I/100)*((-164 - 8*I) - (100 + 40*I)*x))/Sqrt[-7 + 2*x + 5*x^2]] + (1/10 - I/20)
*ArcTanh[((1/100 + I/50)*((164 - 8*I) + (100 - 40*I)*x))/Sqrt[-7 + 2*x + 5*x^2]]

________________________________________________________________________________________

Maple [B]  time = 0.108, size = 144, normalized size = 2.8 \begin{align*} -{\frac{1}{10}\sqrt{-4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+9} \left ( 2\,{\it Artanh} \left ( 1/5\,\sqrt{-4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+9} \right ) +\arctan \left ({\frac{10+5\,x}{-2-2\,x}\sqrt{-4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}+9} \left ( 4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}-9 \right ) ^{-1}} \right ) \right ){\frac{1}{\sqrt{-{ \left ( 4\,{\frac{ \left ( 2+x \right ) ^{2}}{ \left ( -1-x \right ) ^{2}}}-9 \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/(5*x^2+12*x+8)/(5*x^2+2*x-7)^(1/2),x)

[Out]

-1/10*(-4*(2+x)^2/(-1-x)^2+9)^(1/2)*(2*arctanh(1/5*(-4*(2+x)^2/(-1-x)^2+9)^(1/2))+arctan(5/2*(-4*(2+x)^2/(-1-x
)^2+9)^(1/2)/(4*(2+x)^2/(-1-x)^2-9)*(2+x)/(-1-x)))/(-(4*(2+x)^2/(-1-x)^2-9)/(1+(2+x)/(-1-x))^2)^(1/2)/(1+(2+x)
/(-1-x))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x^{2} + 12 \, x + 8\right )} \sqrt{5 \, x^{2} + 2 \, x - 7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+12*x+8)/(5*x^2+2*x-7)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((5*x^2 + 12*x + 8)*sqrt(5*x^2 + 2*x - 7)), x)

________________________________________________________________________________________

Fricas [B]  time = 1.91039, size = 424, normalized size = 8.31 \begin{align*} \frac{1}{20} \, \arctan \left (\frac{27 \, x^{2} + 20 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 2\right )} + 36 \, x}{31 \, x^{2} + 16 \, x - 56}\right ) + \frac{1}{20} \, \arctan \left (-\frac{27 \, x^{2} - 20 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 2\right )} + 36 \, x}{31 \, x^{2} + 16 \, x - 56}\right ) + \frac{1}{20} \, \log \left (\frac{15 \, x^{2} + 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 1\right )} + 26 \, x + 9}{x^{2}}\right ) - \frac{1}{20} \, \log \left (\frac{15 \, x^{2} - 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7}{\left (x + 1\right )} + 26 \, x + 9}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+12*x+8)/(5*x^2+2*x-7)^(1/2),x, algorithm="fricas")

[Out]

1/20*arctan((27*x^2 + 20*sqrt(5*x^2 + 2*x - 7)*(x + 2) + 36*x)/(31*x^2 + 16*x - 56)) + 1/20*arctan(-(27*x^2 -
20*sqrt(5*x^2 + 2*x - 7)*(x + 2) + 36*x)/(31*x^2 + 16*x - 56)) + 1/20*log((15*x^2 + 5*sqrt(5*x^2 + 2*x - 7)*(x
+ 1) + 26*x + 9)/x^2) - 1/20*log((15*x^2 - 5*sqrt(5*x^2 + 2*x - 7)*(x + 1) + 26*x + 9)/x^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (x - 1\right ) \left (5 x + 7\right )} \left (5 x^{2} + 12 x + 8\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x**2+12*x+8)/(5*x**2+2*x-7)**(1/2),x)

[Out]

Integral(1/(sqrt((x - 1)*(5*x + 7))*(5*x**2 + 12*x + 8)), x)

________________________________________________________________________________________

Giac [B]  time = 1.49367, size = 277, normalized size = 5.43 \begin{align*} -\frac{1}{10} \, \arctan \left (-\frac{5 \, \sqrt{5} x + 6 \, \sqrt{5} - 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7} + 5}{2 \,{\left (\sqrt{5} + 5\right )}}\right ) - \frac{1}{10} \, \arctan \left (\frac{5 \, \sqrt{5} x + 6 \, \sqrt{5} - 5 \, \sqrt{5 \, x^{2} + 2 \, x - 7} - 5}{2 \,{\left (\sqrt{5} - 5\right )}}\right ) + \frac{1}{10} \, \log \left (5 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}^{2} + 2 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}{\left (6 \, \sqrt{5} + 5\right )} + 20 \, \sqrt{5} + 65\right ) - \frac{1}{10} \, \log \left (5 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}^{2} + 2 \,{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x - 7}\right )}{\left (6 \, \sqrt{5} - 5\right )} - 20 \, \sqrt{5} + 65\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+12*x+8)/(5*x^2+2*x-7)^(1/2),x, algorithm="giac")

[Out]

-1/10*arctan(-1/2*(5*sqrt(5)*x + 6*sqrt(5) - 5*sqrt(5*x^2 + 2*x - 7) + 5)/(sqrt(5) + 5)) - 1/10*arctan(1/2*(5*
sqrt(5)*x + 6*sqrt(5) - 5*sqrt(5*x^2 + 2*x - 7) - 5)/(sqrt(5) - 5)) + 1/10*log(5*(sqrt(5)*x - sqrt(5*x^2 + 2*x
- 7))^2 + 2*(sqrt(5)*x - sqrt(5*x^2 + 2*x - 7))*(6*sqrt(5) + 5) + 20*sqrt(5) + 65) - 1/10*log(5*(sqrt(5)*x -
sqrt(5*x^2 + 2*x - 7))^2 + 2*(sqrt(5)*x - sqrt(5*x^2 + 2*x - 7))*(6*sqrt(5) - 5) - 20*sqrt(5) + 65)