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

Optimal. Leaf size=187 \[ -\frac{1}{490} \sqrt{5 x^2+2 x+3} (35 x+397)-\frac{3}{343} \sqrt{\frac{1}{11} \left (497041-146555 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )+\frac{3}{343} \sqrt{\frac{1}{11} \left (497041+146555 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )-\frac{8233 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1715 \sqrt{5}} \]

[Out]

-((397 + 35*x)*Sqrt[3 + 2*x + 5*x^2])/490 - (8233*ArcSinh[(1 + 5*x)/Sqrt[14]])/(1715*Sqrt[5]) - (3*Sqrt[(49704
1 - 146555*Sqrt[11])/11]*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2
*x + 5*x^2])])/343 + (3*Sqrt[(497041 + 146555*Sqrt[11])/11]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqr
t[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/343

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Rubi [A]  time = 0.35472, antiderivative size = 187, 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 = {1066, 1076, 619, 215, 1032, 724, 206} \[ -\frac{1}{490} \sqrt{5 x^2+2 x+3} (35 x+397)-\frac{3}{343} \sqrt{\frac{1}{11} \left (497041-146555 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )+\frac{3}{343} \sqrt{\frac{1}{11} \left (497041+146555 \sqrt{11}\right )} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )-\frac{8233 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{1715 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((397 + 35*x)*Sqrt[3 + 2*x + 5*x^2])/490 - (8233*ArcSinh[(1 + 5*x)/Sqrt[14]])/(1715*Sqrt[5]) - (3*Sqrt[(49704
1 - 146555*Sqrt[11])/11]*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2
*x + 5*x^2])])/343 + (3*Sqrt[(497041 + 146555*Sqrt[11])/11]*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqr
t[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/343

Rule 1066

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*
(a + b*x + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), x] - Dist[1/(2*c*f^2*(p
+ q + 1)*(2*p + 2*q + 3)), Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*(b*d - a*e)*(C*(c*e - b*f)
*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*
(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (
p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q +
3))))*x + (p*(c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 -
4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; F
reeQ[{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] && GtQ[p, 0] && NeQ[p +
q + 1, 0] && NeQ[2*p + 2*q + 3, 0] &&  !IGtQ[p, 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}}{1+4 x-7 x^2} \, dx &=-\frac{1}{490} (397+35 x) \sqrt{3+2 x+5 x^2}-\frac{1}{490} \int \frac{-3442-13408 x-16466 x^2}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx\\ &=-\frac{1}{490} (397+35 x) \sqrt{3+2 x+5 x^2}+\frac{\int \frac{40560+159720 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{3430}-\frac{8233 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{1715}\\ &=-\frac{1}{490} (397+35 x) \sqrt{3+2 x+5 x^2}-\frac{8233 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{3430 \sqrt{70}}+\frac{\left (12 \left (14641-5028 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{3773}+\frac{\left (12 \left (14641+5028 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{3773}\\ &=-\frac{1}{490} (397+35 x) \sqrt{3+2 x+5 x^2}-\frac{8233 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{1715 \sqrt{5}}-\frac{\left (24 \left (14641-5028 \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 )}{3773}-\frac{\left (24 \left (14641+5028 \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 )}{3773}\\ &=-\frac{1}{490} (397+35 x) \sqrt{3+2 x+5 x^2}-\frac{8233 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{1715 \sqrt{5}}-\frac{3 \sqrt{5467451-1612105 \sqrt{11}} \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 )}{3773}+\frac{3 \sqrt{5467451+1612105 \sqrt{11}} \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 )}{3773}\\ \end{align*}

Mathematica [A]  time = 0.91179, size = 189, normalized size = 1.01 \[ \frac{-385 \sqrt{5 x^2+2 x+3} (35 x+397)-75 \sqrt{250-34 \sqrt{11}} \left (61 \sqrt{11}-143\right ) \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )+75 \sqrt{250+34 \sqrt{11}} \left (143+61 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{250+34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}\right )-181126 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{188650} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-385*(397 + 35*x)*Sqrt[3 + 2*x + 5*x^2] - 181126*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]] - 75*Sqrt[250 - 34*Sqrt[
11]]*(-143 + 61*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x
+ 5*x^2])] + 75*Sqrt[250 + 34*Sqrt[11]]*(143 + 61*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqr
t[250 + 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])])/188650

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Maple [B]  time = 0.142, size = 403, normalized size = 2.2 \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),x)

[Out]

-1/140*(10*x+2)*(5*x^2+2*x+3)^(1/2)-1/25*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))-3/154*(61+13*11^(1/2))*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)))-3/154*(-61+13*11^(1/2))*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)))

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Maxima [B]  time = 1.77033, size = 675, normalized size = 3.61 \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)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1),x, algorithm="maxima")

[Out]

1/188650*sqrt(11)*(975*sqrt(11)*sqrt(2)*sqrt(17*sqrt(11) + 125)*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)) - 1225*sqrt(11)*sqrt(5*x^2 + 2*x + 3)*x
- 16466*sqrt(11)*sqrt(5)*arcsinh(5/14*sqrt(7)*sqrt(2)*x + 1/14*sqrt(7)*sqrt(2)) - 6825*sqrt(11)*sqrt(-34/49*s
qrt(11) + 250/49)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) - 17/7*sqrt(7)*sqrt(2)*x/a
bs(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)) + 4575*sqrt(2)*sqrt(17*sqrt(11) + 125)*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)) + 32025*sqrt(-34/49*sqrt(11) + 250/49)*
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(1
1) - 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)) - 13895*sqrt(11)*sqrt(5*x^2 + 2*x + 3))

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Fricas [B]  time = 1.23336, size = 1148, normalized size = 6.14 \begin{align*} \frac{3}{7546} \, \sqrt{11} \sqrt{146555 \, \sqrt{11} + 497041} \log \left (\frac{6 \,{\left (\sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{146555 \, \sqrt{11} + 497041}{\left (87 \, \sqrt{11} - 265\right )} + 6517 \, \sqrt{11}{\left (x + 3\right )} + 19551 \, x - 32585\right )}}{x}\right ) - \frac{3}{7546} \, \sqrt{11} \sqrt{146555 \, \sqrt{11} + 497041} \log \left (-\frac{6 \,{\left (\sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{146555 \, \sqrt{11} + 497041}{\left (87 \, \sqrt{11} - 265\right )} - 6517 \, \sqrt{11}{\left (x + 3\right )} - 19551 \, x + 32585\right )}}{x}\right ) - \frac{1}{15092} \, \sqrt{11} \sqrt{-5275980 \, \sqrt{11} + 17893476} \log \left (-\frac{\sqrt{5 \, x^{2} + 2 \, x + 3}{\left (87 \, \sqrt{11} + 265\right )} \sqrt{-5275980 \, \sqrt{11} + 17893476} + 39102 \, \sqrt{11}{\left (x + 3\right )} - 117306 \, x + 195510}{x}\right ) + \frac{1}{15092} \, \sqrt{11} \sqrt{-5275980 \, \sqrt{11} + 17893476} \log \left (\frac{\sqrt{5 \, x^{2} + 2 \, x + 3}{\left (87 \, \sqrt{11} + 265\right )} \sqrt{-5275980 \, \sqrt{11} + 17893476} - 39102 \, \sqrt{11}{\left (x + 3\right )} + 117306 \, x - 195510}{x}\right ) - \frac{1}{490} \, \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (35 \, x + 397\right )} + \frac{8233}{17150} \, \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)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1),x, algorithm="fricas")

[Out]

3/7546*sqrt(11)*sqrt(146555*sqrt(11) + 497041)*log(6*(sqrt(5*x^2 + 2*x + 3)*sqrt(146555*sqrt(11) + 497041)*(87
*sqrt(11) - 265) + 6517*sqrt(11)*(x + 3) + 19551*x - 32585)/x) - 3/7546*sqrt(11)*sqrt(146555*sqrt(11) + 497041
)*log(-6*(sqrt(5*x^2 + 2*x + 3)*sqrt(146555*sqrt(11) + 497041)*(87*sqrt(11) - 265) - 6517*sqrt(11)*(x + 3) - 1
9551*x + 32585)/x) - 1/15092*sqrt(11)*sqrt(-5275980*sqrt(11) + 17893476)*log(-(sqrt(5*x^2 + 2*x + 3)*(87*sqrt(
11) + 265)*sqrt(-5275980*sqrt(11) + 17893476) + 39102*sqrt(11)*(x + 3) - 117306*x + 195510)/x) + 1/15092*sqrt(
11)*sqrt(-5275980*sqrt(11) + 17893476)*log((sqrt(5*x^2 + 2*x + 3)*(87*sqrt(11) + 265)*sqrt(-5275980*sqrt(11) +
17893476) - 39102*sqrt(11)*(x + 3) + 117306*x - 195510)/x) - 1/490*sqrt(5*x^2 + 2*x + 3)*(35*x + 397) + 8233/
17150*sqrt(5)*log(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{2 \sqrt{5 x^{2} + 2 x + 3}}{7 x^{2} - 4 x - 1}\, dx - \int \frac{5 x \sqrt{5 x^{2} + 2 x + 3}}{7 x^{2} - 4 x - 1}\, dx - \int \frac{x^{2} \sqrt{5 x^{2} + 2 x + 3}}{7 x^{2} - 4 x - 1}\, 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),x)

[Out]

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

[Out]

Exception raised: TypeError