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

Optimal. Leaf size=199 \[ \frac{3 \sqrt{5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}-\frac{\sqrt{\frac{325022311+39132731 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2156}+\frac{\sqrt{\frac{325022311-39132731 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2156}+\frac{1}{49} \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right ) \]

[Out]

(3*(3 + 61*x)*Sqrt[3 + 2*x + 5*x^2])/(154*(1 + 4*x - 7*x^2)) + (Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/49 - (Sqr
t[(325022311 + 39132731*Sqrt[11])/1397]*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[1
1])]*Sqrt[3 + 2*x + 5*x^2])])/2156 + (Sqrt[(325022311 - 39132731*Sqrt[11])/1397]*ArcTanh[(23 + Sqrt[11] + (17
+ 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/2156

________________________________________________________________________________________

Rubi [A]  time = 0.25007, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1054, 1076, 619, 215, 1032, 724, 206} \[ \frac{3 \sqrt{5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}-\frac{\sqrt{\frac{325022311+39132731 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2156}+\frac{\sqrt{\frac{325022311-39132731 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2156}+\frac{1}{49} \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(3 + 61*x)*Sqrt[3 + 2*x + 5*x^2])/(154*(1 + 4*x - 7*x^2)) + (Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/49 - (Sqr
t[(325022311 + 39132731*Sqrt[11])/1397]*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[1
1])]*Sqrt[3 + 2*x + 5*x^2])])/2156 + (Sqrt[(325022311 - 39132731*Sqrt[11])/1397]*ArcTanh[(23 + Sqrt[11] + (17
+ 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/2156

Rule 1054

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((A*b*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x
^2)^(p + 1)*(d + e*x + f*x^2)^q)/(c*(b^2 - 4*a*c)*(p + 1)), x] - Dist[1/(c*(b^2 - 4*a*c)*(p + 1)), Int[(a + b*
x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[e*q*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3)
+ C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(
q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*
x^2, x], 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] && LtQ[
p, -1] && GtQ[q, 0] &&  !IGtQ[q, 0]

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{\left (2+5 x+x^2\right ) \sqrt{3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}-\frac{1}{308} \int \frac{-948-188 x+220 x^2}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac{\int \frac{6416+436 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{2156}+\frac{5}{49} \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac{1}{98} \sqrt{\frac{5}{14}} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )+\frac{\left (1199-11446 \sqrt{11}\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{5929}+\frac{\left (1199+11446 \sqrt{11}\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{5929}\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac{1}{49} \sqrt{5} \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )-\frac{\left (2 \left (1199-11446 \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 )}{5929}-\frac{\left (2 \left (1199+11446 \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 )}{5929}\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac{1}{49} \sqrt{5} \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )-\frac{\sqrt{\frac{325022311+39132731 \sqrt{11}}{1397}} \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 )}{2156}+\frac{\sqrt{\frac{325022311-39132731 \sqrt{11}}{1397}} \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 )}{2156}\\ \end{align*}

Mathematica [A]  time = 1.29952, size = 354, normalized size = 1.78 \[ \frac{\frac{56364 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{2772 \sqrt{5 x^2+2 x+3}}{-7 x^2+4 x+1}+22892 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+2398 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+2 \sqrt{\frac{2}{125-17 \sqrt{11}}} \left (11446 \sqrt{11}-1199\right ) \tanh ^{-1}\left (\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{5 \sqrt{11} x-17 x+\sqrt{11}-23}\right )-22892 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left (-7 x+\sqrt{11}+2\right )-2398 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left (-7 x+\sqrt{11}+2\right )+968 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{47432} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2772*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (56364*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + 968*Sqrt
[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] + 2*Sqrt[2/(125 - 17*Sqrt[11])]*(-1199 + 11446*Sqrt[11])*ArcTanh[(Sqrt[250 - 3
4*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] - 17*x + 5*Sqrt[11]*x)] - 2398*Sqrt[2/(125 + 17*Sqrt[11])]*
Log[2 + Sqrt[11] - 7*x] - 22892*Sqrt[22/(125 + 17*Sqrt[11])]*Log[2 + Sqrt[11] - 7*x] + 2398*Sqrt[2/(125 + 17*S
qrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]] + 228
92*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3
+ 2*x + 5*x^2]])/47432

________________________________________________________________________________________

Maple [B]  time = 0.124, size = 1084, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-161/484*11^(1/2)*(1/49*(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)+1/70*(34/7+10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34/7+10/7*11^(1/2))^2
)^(1/2)*(x+1/5))-(250/49+34/49*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)))/(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)))+(183/44+39/44*11^(1/2))*(-1/49/(250/49+34/49*11^(1/2))/(x-2/7-1/
7*11^(1/2))*(5*(x-2/7-1/7*11^(1/2))^2+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250/49+34/49*11^(1/2))^(3/2)+1
/98*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))*(1/7*(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)+1/10*(34/7+10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/
2)-1/20*(34/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(250/49+34/49*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)))/(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)))+10/49/(250/49+34/49*11^(1/2))*(1
/20*(10*x+2)*(5*(x-2/7-1/7*11^(1/2))^2+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250/49+34/49*11^(1/2))^(1/2)+
1/200*(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^(1/2)/(250/49+34/49*11^(1/2)-1/20*(34
/7+10/7*11^(1/2))^2)^(1/2)*(x+1/5))))+161/484*11^(1/2)*(1/49*(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)+1/70*(34/7-10/7*11^(1/2))*5^(1/2)*arcsinh(5^(1/2)/(250/49-34/49
*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5))-(250/49-34/49*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)))/(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)))+(183/44-39/44*11^(1/2))*(-1
/49/(250/49-34/49*11^(1/2))/(x-2/7+1/7*11^(1/2))*(5*(x-2/7+1/7*11^(1/2))^2+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^
(1/2))+250/49-34/49*11^(1/2))^(3/2)+1/98*(34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(1/7*(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)+1/10*(34/7-10/7*11^(1/2))*5^(1/2)
*arcsinh(5^(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5))-7*(250/49-34/49*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)))/(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)))+10/49/(250/49-34/49*11^(1/2))*(1/20*(10*x+2)*(5*(x-2/7+1/7*11^(1/2))^2+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11
^(1/2))+250/49-34/49*11^(1/2))^(1/2)+1/200*(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^2)*5^(1/2)*arcsinh(5^
(1/2)/(250/49-34/49*11^(1/2)-1/20*(34/7-10/7*11^(1/2))^2)^(1/2)*(x+1/5))))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{5 \, x^{2} + 2 \, x + 3}{\left (x^{2} + 5 \, x + 2\right )}}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.17428, size = 1499, normalized size = 7.53 \begin{align*} -\frac{\sqrt{1397}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{39132731 \, \sqrt{11} + 325022311} \log \left (-\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{39132731 \, \sqrt{11} + 325022311}{\left (16943 \, \sqrt{11} + 235367\right )} + 26119953475 \, \sqrt{11}{\left (x + 3\right )} - 78359860425 \, x + 130599767375}{x}\right ) - \sqrt{1397}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{39132731 \, \sqrt{11} + 325022311} \log \left (\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{39132731 \, \sqrt{11} + 325022311}{\left (16943 \, \sqrt{11} + 235367\right )} - 26119953475 \, \sqrt{11}{\left (x + 3\right )} + 78359860425 \, x - 130599767375}{x}\right ) + \sqrt{1397}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{-39132731 \, \sqrt{11} + 325022311} \log \left (\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (16943 \, \sqrt{11} - 235367\right )} \sqrt{-39132731 \, \sqrt{11} + 325022311} + 26119953475 \, \sqrt{11}{\left (x + 3\right )} + 78359860425 \, x - 130599767375}{x}\right ) - \sqrt{1397}{\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt{-39132731 \, \sqrt{11} + 325022311} \log \left (-\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (16943 \, \sqrt{11} - 235367\right )} \sqrt{-39132731 \, \sqrt{11} + 325022311} - 26119953475 \, \sqrt{11}{\left (x + 3\right )} - 78359860425 \, x + 130599767375}{x}\right ) - 61468 \, \sqrt{5}{\left (7 \, x^{2} - 4 \, x - 1\right )} \log \left (-\sqrt{5} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) + 117348 \, \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (61 \, x + 3\right )}}{6023864 \,{\left (7 \, x^{2} - 4 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6023864*(sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(39132731*sqrt(11) + 325022311)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x
+ 3)*sqrt(39132731*sqrt(11) + 325022311)*(16943*sqrt(11) + 235367) + 26119953475*sqrt(11)*(x + 3) - 783598604
25*x + 130599767375)/x) - sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(39132731*sqrt(11) + 325022311)*log((sqrt(1397)*sqr
t(5*x^2 + 2*x + 3)*sqrt(39132731*sqrt(11) + 325022311)*(16943*sqrt(11) + 235367) - 26119953475*sqrt(11)*(x + 3
) + 78359860425*x - 130599767375)/x) + sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(-39132731*sqrt(11) + 325022311)*log((
sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(16943*sqrt(11) - 235367)*sqrt(-39132731*sqrt(11) + 325022311) + 26119953475*
sqrt(11)*(x + 3) + 78359860425*x - 130599767375)/x) - sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(-39132731*sqrt(11) + 3
25022311)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(16943*sqrt(11) - 235367)*sqrt(-39132731*sqrt(11) + 325022311
) - 26119953475*sqrt(11)*(x + 3) - 78359860425*x + 130599767375)/x) - 61468*sqrt(5)*(7*x^2 - 4*x - 1)*log(-sqr
t(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) + 117348*sqrt(5*x^2 + 2*x + 3)*(61*x + 3))/(7*x^2 -
4*x - 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x^{2} + 5 x + 2\right ) \sqrt{5 x^{2} + 2 x + 3}}{\left (7 x^{2} - 4 x - 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError