3.391 \(\int \frac{2+5 x+x^2}{(1+4 x-7 x^2)^3 \sqrt{3+2 x+5 x^2}} \, dx\)
Optimal. Leaf size=227 \[ -\frac{7 \sqrt{5 x^2+2 x+3} (409769-1189370 x)}{62451488 \left (-7 x^2+4 x+1\right )}-\frac{3 (40-371 x) \sqrt{5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}-\frac{7 \left (39370231-2538725 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{124902976 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (39370231+2538725 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{124902976 \sqrt{22 \left (125+17 \sqrt{11}\right )}} \]
[Out]
(-3*(40 - 371*x)*Sqrt[3 + 2*x + 5*x^2])/(11176*(1 + 4*x - 7*x^2)^2) - (7*(409769 - 1189370*x)*Sqrt[3 + 2*x + 5
*x^2])/(62451488*(1 + 4*x - 7*x^2)) - (7*(39370231 - 2538725*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[1
1])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(124902976*Sqrt[22*(125 - 17*Sqrt[11])]) + (7*(39
370231 + 2538725*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 +
2*x + 5*x^2])])/(124902976*Sqrt[22*(125 + 17*Sqrt[11])])
________________________________________________________________________________________
Rubi [A] time = 0.271138, antiderivative size = 227, 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} \[ -\frac{7 \sqrt{5 x^2+2 x+3} (409769-1189370 x)}{62451488 \left (-7 x^2+4 x+1\right )}-\frac{3 (40-371 x) \sqrt{5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}-\frac{7 \left (39370231-2538725 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{124902976 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (39370231+2538725 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{124902976 \sqrt{22 \left (125+17 \sqrt{11}\right )}} \]
Antiderivative was successfully verified.
[In]
Int[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*Sqrt[3 + 2*x + 5*x^2]),x]
[Out]
(-3*(40 - 371*x)*Sqrt[3 + 2*x + 5*x^2])/(11176*(1 + 4*x - 7*x^2)^2) - (7*(409769 - 1189370*x)*Sqrt[3 + 2*x + 5
*x^2])/(62451488*(1 + 4*x - 7*x^2)) - (7*(39370231 - 2538725*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[1
1])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(124902976*Sqrt[22*(125 - 17*Sqrt[11])]) + (7*(39
370231 + 2538725*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 +
2*x + 5*x^2])])/(124902976*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 )^3 \sqrt{3+2 x+5 x^2}} \, dx &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac{\int \frac{-130024-81000 x-89040 x^2}{\left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}} \, dx}{89408}\\ &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac{7 (409769-1189370 x) \sqrt{3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}+\frac{\int \frac{2194737984+1137348800 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{3996895232}\\ &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac{7 (409769-1189370 x) \sqrt{3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}+\frac{\left (7 \left (27925975-39370231 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{686966368}+\frac{\left (7 \left (27925975+39370231 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{686966368}\\ &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac{7 (409769-1189370 x) \sqrt{3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}-\frac{\left (7 \left (27925975-39370231 \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 )}{343483184}-\frac{\left (7 \left (27925975+39370231 \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 )}{343483184}\\ &=-\frac{3 (40-371 x) \sqrt{3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac{7 (409769-1189370 x) \sqrt{3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}-\frac{7 \left (39370231-2538725 \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 )}{124902976 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (39370231+2538725 \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 )}{124902976 \sqrt{22 \left (125+17 \sqrt{11}\right )}}\\ \end{align*}
Mathematica [A] time = 1.28187, size = 371, normalized size = 1.63 \[ \frac{\frac{732651920 \sqrt{5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac{547311072 \sqrt{5 x^2+2 x+3} x}{\left (-7 x^2+4 x+1\right )^2}-\frac{59009280 \sqrt{5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^2}+\frac{252417704 \sqrt{5 x^2+2 x+3}}{7 x^2-4 x-1}+551183234 \sqrt{\frac{22}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+390963650 \sqrt{\frac{2}{125+17 \sqrt{11}}} \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )+14 \sqrt{\frac{2}{125-17 \sqrt{11}}} \left (39370231 \sqrt{11}-27925975\right ) \tanh ^{-1}\left (\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left (5 \sqrt{11}-17\right ) x+\sqrt{11}-23}\right )-14 \sqrt{\frac{2}{125+17 \sqrt{11}}} \left (27925975+39370231 \sqrt{11}\right ) \log \left (-7 x+\sqrt{11}+2\right )}{5495730944} \]
Antiderivative was successfully verified.
[In]
Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*Sqrt[3 + 2*x + 5*x^2]),x]
[Out]
((-59009280*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2 + (547311072*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)
^2 + (732651920*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (252417704*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x
^2) + 14*Sqrt[2/(125 - 17*Sqrt[11])]*(-27925975 + 39370231*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 +
2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 14*Sqrt[2/(125 + 17*Sqrt[11])]*(27925975 + 39370231*
Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + 390963650*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]] + 551183234*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 2
3*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/5495730944
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Maple [B] time = 0.131, size = 1194, normalized size = 5.3 \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)^3/(5*x^2+2*x+3)^(1/2),x)
[Out]
3535/21296*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))-21/968*(-61+13*11^(1/2))*11^(1/2)*(-1/686/(250/49-34/49*11^(1/2))/(x-2/7+1/7*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)-3/1372*(34
/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(-1/(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)+7/2*(34/7-10/7*11^(1/2))/(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*1
1^(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)))+5/98/(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)))-21/968*(61+13*11^(1/2))*11^(1/2)*(-1/686/(250/49+34/49
*11^(1/2))/(x-2/7-1/7*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+3
4/49*11^(1/2))^(1/2)-3/1372*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))*(-1/(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)+7
/2*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))/(250+34*11^(1/2))^(1/2)*arctanh(49/2*(500/49+68/49*11^(1/2)+(3
4/7+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)))+5/98/(250/49+34/49*11^(1/2))/(250+34*11^(1/2))^(1/2)*arct
anh(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)))-3535/21296*11^(1/2)/(25
0-34*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))
-(-3535/1936-273/1936*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)+1/14*(34/7+10/7*11^(1/2))/(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)))-(-3535/1936+273/1936*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)+1/14*(34/7-10/7*11^(1/2)
)/(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 [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 )}^{3} \sqrt{5 \, x^{2} + 2 \, x + 3}}\,{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)^3/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
[Out]
-integrate((x^2 + 5*x + 2)/((7*x^2 - 4*x - 1)^3*sqrt(5*x^2 + 2*x + 3)), x)
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Fricas [B] time = 1.56052, size = 1897, normalized size = 8.36 \begin{align*} -\frac{\sqrt{2794}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{1283973697005131 \, \sqrt{11} + 82616280769148425} \log \left (-\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{1283973697005131 \, \sqrt{11} + 82616280769148425}{\left (358684877 \, \sqrt{11} + 2940638404\right )} + 7232150972206110797 \, \sqrt{11}{\left (x + 3\right )} - 21696452916618332391 \, x + 36160754861030553985}{x}\right ) - \sqrt{2794}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{1283973697005131 \, \sqrt{11} + 82616280769148425} \log \left (\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{1283973697005131 \, \sqrt{11} + 82616280769148425}{\left (358684877 \, \sqrt{11} + 2940638404\right )} - 7232150972206110797 \, \sqrt{11}{\left (x + 3\right )} + 21696452916618332391 \, x - 36160754861030553985}{x}\right ) + \sqrt{2794}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{-1283973697005131 \, \sqrt{11} + 82616280769148425} \log \left (\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (358684877 \, \sqrt{11} - 2940638404\right )} \sqrt{-1283973697005131 \, \sqrt{11} + 82616280769148425} + 7232150972206110797 \, \sqrt{11}{\left (x + 3\right )} + 21696452916618332391 \, x - 36160754861030553985}{x}\right ) - \sqrt{2794}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{-1283973697005131 \, \sqrt{11} + 82616280769148425} \log \left (-\frac{\sqrt{2794} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (358684877 \, \sqrt{11} - 2940638404\right )} \sqrt{-1283973697005131 \, \sqrt{11} + 82616280769148425} - 7232150972206110797 \, \sqrt{11}{\left (x + 3\right )} - 21696452916618332391 \, x + 36160754861030553985}{x}\right ) + 11176 \,{\left (58279130 \, x^{3} - 53381041 \, x^{2} - 3071502 \, x + 3538943\right )} \sqrt{5 \, x^{2} + 2 \, x + 3}}{697957829888 \,{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\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)^3/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
[Out]
-1/697957829888*(sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(1283973697005131*sqrt(11) + 8261628076914
8425)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(1283973697005131*sqrt(11) + 82616280769148425)*(358684877*sq
rt(11) + 2940638404) + 7232150972206110797*sqrt(11)*(x + 3) - 21696452916618332391*x + 36160754861030553985)/x
) - sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(1283973697005131*sqrt(11) + 82616280769148425)*log((sq
rt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(1283973697005131*sqrt(11) + 82616280769148425)*(358684877*sqrt(11) + 29406
38404) - 7232150972206110797*sqrt(11)*(x + 3) + 21696452916618332391*x - 36160754861030553985)/x) + sqrt(2794)
*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425)*log((sqrt(2794)*sqrt
(5*x^2 + 2*x + 3)*(358684877*sqrt(11) - 2940638404)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425) + 723
2150972206110797*sqrt(11)*(x + 3) + 21696452916618332391*x - 36160754861030553985)/x) - sqrt(2794)*(49*x^4 - 5
6*x^3 + 2*x^2 + 8*x + 1)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*
x + 3)*(358684877*sqrt(11) - 2940638404)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425) - 72321509722061
10797*sqrt(11)*(x + 3) - 21696452916618332391*x + 36160754861030553985)/x) + 11176*(58279130*x^3 - 53381041*x^
2 - 3071502*x + 3538943)*sqrt(5*x^2 + 2*x + 3))/(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)
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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)**3/(5*x**2+2*x+3)**(1/2),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)^3/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError