### 3.389 $$\int \frac{2+5 x+x^2}{(1+4 x-7 x^2) \sqrt{3+2 x+5 x^2}} \, dx$$

Optimal. Leaf size=164 $-\frac{3}{14} \sqrt{\frac{4091-1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )+\frac{3}{14} \sqrt{\frac{4091+1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )-\frac{\sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{7 \sqrt{5}}$

[Out]

-ArcSinh[(1 + 5*x)/Sqrt[14]]/(7*Sqrt[5]) - (3*Sqrt[(4091 - 1055*Sqrt[11])/2794]*ArcTanh[(23 - Sqrt[11] + (17 -
5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/14 + (3*Sqrt[(4091 + 1055*Sqrt[11])/2794
]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/14

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Rubi [A]  time = 0.230132, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.171, Rules used = {1076, 619, 215, 1032, 724, 206} $-\frac{3}{14} \sqrt{\frac{4091-1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )+\frac{3}{14} \sqrt{\frac{4091+1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )-\frac{\sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{7 \sqrt{5}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)*Sqrt[3 + 2*x + 5*x^2]),x]

[Out]

-ArcSinh[(1 + 5*x)/Sqrt[14]]/(7*Sqrt[5]) - (3*Sqrt[(4091 - 1055*Sqrt[11])/2794]*ArcTanh[(23 - Sqrt[11] + (17 -
5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/14 + (3*Sqrt[(4091 + 1055*Sqrt[11])/2794
]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/14

Rule 1076

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

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 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 1032

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

Rule 724

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx &=-\left (\frac{1}{7} \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx\right )-\frac{1}{7} \int \frac{-15-39 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{14 \sqrt{70}}+\frac{1}{77} \left (3 \left (143-61 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx+\frac{1}{77} \left (3 \left (143+61 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{\sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{7 \sqrt{5}}-\frac{1}{77} \left (6 \left (143-61 \sqrt{11}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2352+112 \left (4-2 \sqrt{11}\right )+20 \left (4-2 \sqrt{11}\right )^2-x^2} \, dx,x,\frac{-84-2 \left (4-2 \sqrt{11}\right )-\left (28+10 \left (4-2 \sqrt{11}\right )\right ) x}{\sqrt{3+2 x+5 x^2}}\right )-\frac{1}{77} \left (6 \left (143+61 \sqrt{11}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{2352+112 \left (4+2 \sqrt{11}\right )+20 \left (4+2 \sqrt{11}\right )^2-x^2} \, dx,x,\frac{-84-2 \left (4+2 \sqrt{11}\right )-\left (28+10 \left (4+2 \sqrt{11}\right )\right ) x}{\sqrt{3+2 x+5 x^2}}\right )\\ &=-\frac{\sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{7 \sqrt{5}}-\frac{3}{14} \sqrt{\frac{4091-1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{23-\sqrt{11}+\left (17-5 \sqrt{11}\right ) x}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{3+2 x+5 x^2}}\right )+\frac{3}{14} \sqrt{\frac{4091+1055 \sqrt{11}}{2794}} \tanh ^{-1}\left (\frac{23+\sqrt{11}+\left (17+5 \sqrt{11}\right ) x}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{3+2 x+5 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.484516, size = 157, normalized size = 0.96 $-\frac{3 \left (\sqrt{4091-1055 \sqrt{11}} \tanh ^{-1}\left (\frac{-5 \sqrt{11} x+17 x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )-\sqrt{4091+1055 \sqrt{11}} \tanh ^{-1}\left (\frac{5 \sqrt{11} x+17 x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )\right )}{14 \sqrt{2794}}-\frac{\sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{7 \sqrt{5}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)*Sqrt[3 + 2*x + 5*x^2]),x]

[Out]

-ArcSinh[(1 + 5*x)/Sqrt[14]]/(7*Sqrt[5]) - (3*(Sqrt[4091 - 1055*Sqrt[11]]*ArcTanh[(23 - Sqrt[11] + 17*x - 5*Sq
rt[11]*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])] - Sqrt[4091 + 1055*Sqrt[11]]*ArcTanh[(23 + Sqrt[11]
+ 17*x + 5*Sqrt[11]*x)/(Sqrt[250 + 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])]))/(14*Sqrt[2794])

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Maple [A]  time = 0.106, size = 204, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{5}}{35}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{ \left ( 183+39\,\sqrt{11} \right ) \sqrt{11}}{154\,\sqrt{250+34\,\sqrt{11}}}{\it Artanh} \left ({\frac{49}{2\,\sqrt{250+34\,\sqrt{11}}} \left ({\frac{500}{49}}+{\frac{68\,\sqrt{11}}{49}}+ \left ({\frac{34}{7}}+{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-{\frac{2}{7}}-{\frac{\sqrt{11}}{7}} \right ) \right ){\frac{1}{\sqrt{245\, \left ( x-2/7-1/7\,\sqrt{11} \right ) ^{2}+49\, \left ({\frac{34}{7}}+{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-2/7-1/7\,\sqrt{11} \right ) +250+34\,\sqrt{11}}}}} \right ) }+{\frac{ \left ( -183+39\,\sqrt{11} \right ) \sqrt{11}}{154\,\sqrt{250-34\,\sqrt{11}}}{\it Artanh} \left ({\frac{49}{2\,\sqrt{250-34\,\sqrt{11}}} \left ({\frac{500}{49}}-{\frac{68\,\sqrt{11}}{49}}+ \left ({\frac{34}{7}}-{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-{\frac{2}{7}}+{\frac{\sqrt{11}}{7}} \right ) \right ){\frac{1}{\sqrt{245\, \left ( x-2/7+1/7\,\sqrt{11} \right ) ^{2}+49\, \left ({\frac{34}{7}}-{\frac{10\,\sqrt{11}}{7}} \right ) \left ( x-2/7+1/7\,\sqrt{11} \right ) +250-34\,\sqrt{11}}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+5*x+2)/(-7*x^2+4*x+1)/(5*x^2+2*x+3)^(1/2),x)

[Out]

-1/35*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))+3/154*(61+13*11^(1/2))*11^(1/2)/(250+34*11^(1/2))^(1/2)*arctanh(4
9/2*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2)))/(250+34*11^(1/2))^(1/2)/(245*(x-2/7-1/7*
11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250+34*11^(1/2))^(1/2))+3/154*(-61+13*11^(1/2))*11^(1
/2)/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2)+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2)))/(25
0-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^2+49*(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))
^(1/2))

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Maxima [B]  time = 1.65272, size = 628, normalized size = 3.83 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5*x+2)/(-7*x^2+4*x+1)/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")

[Out]

-1/10780*sqrt(11)*(28*sqrt(11)*sqrt(5)*arcsinh(5/14*sqrt(7)*sqrt(2)*x + 1/14*sqrt(7)*sqrt(2)) - 1365*sqrt(11)*
sqrt(2)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sqrt(2)*x/abs(14*x -
2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*
sqrt(11) - 4))/sqrt(17*sqrt(11) + 125) + 390*sqrt(11)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt
(11) - 4) - 17/7*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt
(11) - 4) - 23/7*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4))/sqrt(-34/49*sqrt(11) + 250/49) - 6405*sqrt(2)*arc
sinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11)
- 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) -
4))/sqrt(17*sqrt(11) + 125) - 1830*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) - 17/7*sq
rt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4) - 23/7*sq
rt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4))/sqrt(-34/49*sqrt(11) + 250/49))

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Fricas [B]  time = 1.54786, size = 1126, normalized size = 6.87 \begin{align*} -\frac{3}{78232} \, \sqrt{2794} \sqrt{1055 \, \sqrt{11} + 4091} \log \left (\frac{3 \,{\left (\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{1055 \, \sqrt{11} + 4091}{\left (172 \, \sqrt{11} - 715\right )} + 185801 \, \sqrt{11}{\left (x + 3\right )} + 557403 \, x - 929005\right )}}{x}\right ) + \frac{3}{78232} \, \sqrt{2794} \sqrt{1055 \, \sqrt{11} + 4091} \log \left (-\frac{3 \,{\left (\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{1055 \, \sqrt{11} + 4091}{\left (172 \, \sqrt{11} - 715\right )} - 185801 \, \sqrt{11}{\left (x + 3\right )} - 557403 \, x + 929005\right )}}{x}\right ) - \frac{1}{78232} \, \sqrt{2794} \sqrt{-9495 \, \sqrt{11} + 36819} \log \left (-\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (172 \, \sqrt{11} + 715\right )} \sqrt{-9495 \, \sqrt{11} + 36819} + 557403 \, \sqrt{11}{\left (x + 3\right )} - 1672209 \, x + 2787015}{x}\right ) + \frac{1}{78232} \, \sqrt{2794} \sqrt{-9495 \, \sqrt{11} + 36819} \log \left (\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (172 \, \sqrt{11} + 715\right )} \sqrt{-9495 \, \sqrt{11} + 36819} - 557403 \, \sqrt{11}{\left (x + 3\right )} + 1672209 \, x - 2787015}{x}\right ) + \frac{1}{70} \, \sqrt{5} \log \left (\sqrt{5} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5*x+2)/(-7*x^2+4*x+1)/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")

[Out]

-3/78232*sqrt(2794)*sqrt(1055*sqrt(11) + 4091)*log(3*(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(1055*sqrt(11) + 40
91)*(172*sqrt(11) - 715) + 185801*sqrt(11)*(x + 3) + 557403*x - 929005)/x) + 3/78232*sqrt(2794)*sqrt(1055*sqrt
(11) + 4091)*log(-3*(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(1055*sqrt(11) + 4091)*(172*sqrt(11) - 715) - 185801
*sqrt(11)*(x + 3) - 557403*x + 929005)/x) - 1/78232*sqrt(2794)*sqrt(-9495*sqrt(11) + 36819)*log(-(sqrt(2794)*s
qrt(5*x^2 + 2*x + 3)*(172*sqrt(11) + 715)*sqrt(-9495*sqrt(11) + 36819) + 557403*sqrt(11)*(x + 3) - 1672209*x +
2787015)/x) + 1/78232*sqrt(2794)*sqrt(-9495*sqrt(11) + 36819)*log((sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*(172*sqrt
(11) + 715)*sqrt(-9495*sqrt(11) + 36819) - 557403*sqrt(11)*(x + 3) + 1672209*x - 2787015)/x) + 1/70*sqrt(5)*lo
g(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{5 x}{7 x^{2} \sqrt{5 x^{2} + 2 x + 3} - 4 x \sqrt{5 x^{2} + 2 x + 3} - \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{x^{2}}{7 x^{2} \sqrt{5 x^{2} + 2 x + 3} - 4 x \sqrt{5 x^{2} + 2 x + 3} - \sqrt{5 x^{2} + 2 x + 3}}\, dx - \int \frac{2}{7 x^{2} \sqrt{5 x^{2} + 2 x + 3} - 4 x \sqrt{5 x^{2} + 2 x + 3} - \sqrt{5 x^{2} + 2 x + 3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+5*x+2)/(-7*x**2+4*x+1)/(5*x**2+2*x+3)**(1/2),x)

[Out]

-Integral(5*x/(7*x**2*sqrt(5*x**2 + 2*x + 3) - 4*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x) - Inte
gral(x**2/(7*x**2*sqrt(5*x**2 + 2*x + 3) - 4*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x) - Integral
(2/(7*x**2*sqrt(5*x**2 + 2*x + 3) - 4*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5*x+2)/(-7*x^2+4*x+1)/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError