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

Optimal. Leaf size=215 \[ -\frac{3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt{5 x^2+2 x+3}}-\frac{22755 x+76567}{19870928 \sqrt{5 x^2+2 x+3}}-\frac{7 \left (541543-5144 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2838704 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (541543+5144 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2838704 \sqrt{22 \left (125+17 \sqrt{11}\right )}} \]

[Out]

-(76567 + 22755*x)/(19870928*Sqrt[3 + 2*x + 5*x^2]) - (3*(40 - 371*x))/(5588*(1 + 4*x - 7*x^2)*Sqrt[3 + 2*x +
5*x^2]) - (7*(541543 - 5144*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11]
)]*Sqrt[3 + 2*x + 5*x^2])])/(2838704*Sqrt[22*(125 - 17*Sqrt[11])]) + (7*(541543 + 5144*Sqrt[11])*ArcTanh[(23 +
Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(2838704*Sqrt[22*(125 +
17*Sqrt[11])])

________________________________________________________________________________________

Rubi [A]  time = 0.316306, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {1060, 1032, 724, 206} \[ -\frac{3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt{5 x^2+2 x+3}}-\frac{22755 x+76567}{19870928 \sqrt{5 x^2+2 x+3}}-\frac{7 \left (541543-5144 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2838704 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (541543+5144 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{2838704 \sqrt{22 \left (125+17 \sqrt{11}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(76567 + 22755*x)/(19870928*Sqrt[3 + 2*x + 5*x^2]) - (3*(40 - 371*x))/(5588*(1 + 4*x - 7*x^2)*Sqrt[3 + 2*x +
5*x^2]) - (7*(541543 - 5144*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11]
)]*Sqrt[3 + 2*x + 5*x^2])])/(2838704*Sqrt[22*(125 - 17*Sqrt[11])]) + (7*(541543 + 5144*Sqrt[11])*ArcTanh[(23 +
Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(2838704*Sqrt[22*(125 +
17*Sqrt[11])])

Rule 1060

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*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

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 )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx &=-\frac{3 (40-371 x)}{5588 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}-\frac{\int \frac{-50216-37752 x-89040 x^2}{\left (1+4 x-7 x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}} \, dx}{44704}\\ &=-\frac{76567+22755 x}{19870928 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{5588 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}-\frac{\int \frac{-476004480-32263168 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{1271739392}\\ &=-\frac{76567+22755 x}{19870928 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{5588 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}+\frac{\left (7 \left (56584-541543 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{15612872}+\frac{\left (7 \left (56584+541543 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{15612872}\\ &=-\frac{76567+22755 x}{19870928 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{5588 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}-\frac{\left (7 \left (56584-541543 \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 )}{7806436}-\frac{\left (7 \left (56584+541543 \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 )}{7806436}\\ &=-\frac{76567+22755 x}{19870928 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{5588 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}-\frac{7 \left (541543-5144 \sqrt{11}\right ) \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 )}{2838704 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (541543+5144 \sqrt{11}\right ) \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 )}{2838704 \sqrt{22 \left (125+17 \sqrt{11}\right )}}\\ \end{align*}

Mathematica [A]  time = 1.19835, size = 351, normalized size = 1.63 \[ \frac{\frac{5084772 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{24422640 x}{7 \sqrt{5 x^2+2 x+3}}+\frac{12968296}{7 \sqrt{5 x^2+2 x+3}}+\frac{1672044 \sqrt{5 x^2+2 x+3}}{7 x^2-4 x-1}+7581602 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+792176 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+14 \sqrt{\frac{2}{125-17 \sqrt{11}}} \left (541543 \sqrt{11}-56584\right ) \tanh ^{-1}\left (\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left (5 \sqrt{11}-17\right ) x+\sqrt{11}-23}\right )-14 \sqrt{\frac{2}{125+17 \sqrt{11}}} \left (56584+541543 \sqrt{11}\right ) \log \left (-7 x+\sqrt{11}+2\right )}{124902976} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12968296/(7*Sqrt[3 + 2*x + 5*x^2]) + (24422640*x)/(7*Sqrt[3 + 2*x + 5*x^2]) + (5084772*x*Sqrt[3 + 2*x + 5*x^2
])/(1 + 4*x - 7*x^2) + (1672044*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) + 14*Sqrt[2/(125 - 17*Sqrt[11])]*(-5
6584 + 541543*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqr
t[11])*x)] - 14*Sqrt[2/(125 + 17*Sqrt[11])]*(56584 + 541543*Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + 792176*Sqrt[2/
(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]] + 7581602*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqr
t[11]]*Sqrt[3 + 2*x + 5*x^2]])/124902976

________________________________________________________________________________________

Maple [B]  time = 0.111, size = 1214, normalized size = 5.7 \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)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(3/2),x)

[Out]

-161/484*11^(1/2)*(1/7/(250/49+34/49*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))^(1/2)-1/7*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))*(10*x+2)/(5000/49+680/49*11
^(1/2)-(34/7+10/7*11^(1/2))^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+34/4
9*11^(1/2))^(1/2)-1/(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))^(1/2
)-3/98*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))*(1/(250/49+34/49*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))^(1/2)-(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))
*(10*x+2)/(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^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)-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)))-20/49/(250/49+34/49*11^(1/2))*(10
*x+2)/(5000/49+680/49*11^(1/2)-(34/7+10/7*11^(1/2))^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))+161/484*11^(1/2)*(1/7/(250/49-34/49*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))^(1/2)-1/7*(34/7-10/7*11^(1/2))/(250/49-3
4/49*11^(1/2))*(10*x+2)/(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^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/(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))^(1/2)-3/98*(34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(1/(250/49-34/49
*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))^(1/2)-(3
4/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(10*x+2)/(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^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)-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)))/(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)))-20/49/(250/49-34/49*11^(1/2))*(10*x+2)/(5000/49-680/49*11^(1/2)-(34/7-10/7*11^(1/2))^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))

________________________________________________________________________________________

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

[Out]

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

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Fricas [B]  time = 1.48231, size = 1759, normalized size = 8.18 \begin{align*} -\frac{7 \, \sqrt{1397}{\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt{4294093814065 \, \sqrt{11} + 35653135368317} \log \left (-\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{4294093814065 \, \sqrt{11} + 35653135368317}{\left (5609479 \, \sqrt{11} + 77949905\right )} + 2865029444171587 \, \sqrt{11}{\left (x + 3\right )} - 8595088332514761 \, x + 14325147220857935}{x}\right ) - 7 \, \sqrt{1397}{\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt{4294093814065 \, \sqrt{11} + 35653135368317} \log \left (\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{4294093814065 \, \sqrt{11} + 35653135368317}{\left (5609479 \, \sqrt{11} + 77949905\right )} - 2865029444171587 \, \sqrt{11}{\left (x + 3\right )} + 8595088332514761 \, x - 14325147220857935}{x}\right ) + 7 \, \sqrt{1397}{\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt{-4294093814065 \, \sqrt{11} + 35653135368317} \log \left (\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5609479 \, \sqrt{11} - 77949905\right )} \sqrt{-4294093814065 \, \sqrt{11} + 35653135368317} + 2865029444171587 \, \sqrt{11}{\left (x + 3\right )} + 8595088332514761 \, x - 14325147220857935}{x}\right ) - 7 \, \sqrt{1397}{\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt{-4294093814065 \, \sqrt{11} + 35653135368317} \log \left (-\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5609479 \, \sqrt{11} - 77949905\right )} \sqrt{-4294093814065 \, \sqrt{11} + 35653135368317} - 2865029444171587 \, \sqrt{11}{\left (x + 3\right )} - 8595088332514761 \, x + 14325147220857935}{x}\right ) + 5588 \,{\left (159285 \, x^{3} + 444949 \, x^{2} + 3628805 \, x - 503287\right )} \sqrt{5 \, x^{2} + 2 \, x + 3}}{111038745664 \,{\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\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)^2/(5*x^2+2*x+3)^(3/2),x, algorithm="fricas")

[Out]

-1/111038745664*(7*sqrt(1397)*(35*x^4 - 6*x^3 + 8*x^2 - 14*x - 3)*sqrt(4294093814065*sqrt(11) + 35653135368317
)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*sqrt(4294093814065*sqrt(11) + 35653135368317)*(5609479*sqrt(11) + 779
49905) + 2865029444171587*sqrt(11)*(x + 3) - 8595088332514761*x + 14325147220857935)/x) - 7*sqrt(1397)*(35*x^4
- 6*x^3 + 8*x^2 - 14*x - 3)*sqrt(4294093814065*sqrt(11) + 35653135368317)*log((sqrt(1397)*sqrt(5*x^2 + 2*x +
3)*sqrt(4294093814065*sqrt(11) + 35653135368317)*(5609479*sqrt(11) + 77949905) - 2865029444171587*sqrt(11)*(x
+ 3) + 8595088332514761*x - 14325147220857935)/x) + 7*sqrt(1397)*(35*x^4 - 6*x^3 + 8*x^2 - 14*x - 3)*sqrt(-429
4093814065*sqrt(11) + 35653135368317)*log((sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(5609479*sqrt(11) - 77949905)*sqrt
(-4294093814065*sqrt(11) + 35653135368317) + 2865029444171587*sqrt(11)*(x + 3) + 8595088332514761*x - 14325147
220857935)/x) - 7*sqrt(1397)*(35*x^4 - 6*x^3 + 8*x^2 - 14*x - 3)*sqrt(-4294093814065*sqrt(11) + 35653135368317
)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(5609479*sqrt(11) - 77949905)*sqrt(-4294093814065*sqrt(11) + 35653135
368317) - 2865029444171587*sqrt(11)*(x + 3) - 8595088332514761*x + 14325147220857935)/x) + 5588*(159285*x^3 +
444949*x^2 + 3628805*x - 503287)*sqrt(5*x^2 + 2*x + 3))/(35*x^4 - 6*x^3 + 8*x^2 - 14*x - 3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**2/(5*x**2+2*x+3)**(3/2),x)

[Out]

Timed out

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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)^2/(5*x^2+2*x+3)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError