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

Optimal. Leaf size=213 \[ -\frac{\sqrt{5 x^2+2 x+3} (272941-813113 x)}{1721104 \left (-7 x^2+4 x+1\right )}+\frac{3 (61 x+3) \sqrt{5 x^2+2 x+3}}{308 \left (-7 x^2+4 x+1\right )^2}-\frac{\sqrt{\frac{6492253020949-11879169071 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{491744}+\frac{\sqrt{\frac{6492253020949+11879169071 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{491744} \]

[Out]

(3*(3 + 61*x)*Sqrt[3 + 2*x + 5*x^2])/(308*(1 + 4*x - 7*x^2)^2) - ((272941 - 813113*x)*Sqrt[3 + 2*x + 5*x^2])/(
1721104*(1 + 4*x - 7*x^2)) - (Sqrt[(6492253020949 - 11879169071*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])])/491744 + (Sqrt[(6492253020949 + 11879169
071*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])])/491744

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Rubi [A]  time = 0.236551, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1054, 1060, 1032, 724, 206} \[ -\frac{\sqrt{5 x^2+2 x+3} (272941-813113 x)}{1721104 \left (-7 x^2+4 x+1\right )}+\frac{3 (61 x+3) \sqrt{5 x^2+2 x+3}}{308 \left (-7 x^2+4 x+1\right )^2}-\frac{\sqrt{\frac{6492253020949-11879169071 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{491744}+\frac{\sqrt{\frac{6492253020949+11879169071 \sqrt{11}}{1397}} \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{491744} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(3 + 61*x)*Sqrt[3 + 2*x + 5*x^2])/(308*(1 + 4*x - 7*x^2)^2) - ((272941 - 813113*x)*Sqrt[3 + 2*x + 5*x^2])/(
1721104*(1 + 4*x - 7*x^2)) - (Sqrt[(6492253020949 - 11879169071*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])])/491744 + (Sqrt[(6492253020949 + 11879169
071*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])])/491744

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 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{\left (2+5 x+x^2\right ) \sqrt{3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac{1}{616} \int \frac{-3012-1564 x-3220 x^2}{\left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}} \, dx\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac{(272941-813113 x) \sqrt{3+2 x+5 x^2}}{1721104 \left (1+4 x-7 x^2\right )}+\frac{\int \frac{47581408+28345408 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{27537664}\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac{(272941-813113 x) \sqrt{3+2 x+5 x^2}}{1721104 \left (1+4 x-7 x^2\right )}+\frac{\left (1391962-1740003 \sqrt{11}\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{1352296}+\frac{\left (1391962+1740003 \sqrt{11}\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{1352296}\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac{(272941-813113 x) \sqrt{3+2 x+5 x^2}}{1721104 \left (1+4 x-7 x^2\right )}+\frac{\left (-1391962+1740003 \sqrt{11}\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 )}{676148}-\frac{\left (1391962+1740003 \sqrt{11}\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 )}{676148}\\ &=\frac{3 (3+61 x) \sqrt{3+2 x+5 x^2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac{(272941-813113 x) \sqrt{3+2 x+5 x^2}}{1721104 \left (1+4 x-7 x^2\right )}-\frac{\sqrt{\frac{6492253020949-11879169071 \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 )}{491744}+\frac{\sqrt{\frac{6492253020949+11879169071 \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 )}{491744}\\ \end{align*}

Mathematica [A]  time = 1.49292, size = 334, normalized size = 1.57 \[ \frac{-\frac{44 \sqrt{5 x^2+2 x+3} \left (813113 x^3-737577 x^2-106279 x+31807\right )}{\left (-7 x^2+4 x+1\right )^2}-\sqrt{\frac{22}{125-17 \sqrt{11}}} \left (126542 \sqrt{11}-1740003\right ) \log \left (49 x^2+14 \left (\sqrt{11}-2\right ) x-4 \sqrt{11}+15\right )+2 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left (1740003+126542 \sqrt{11}\right ) \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )-2 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left (126542 \sqrt{11}-1740003\right ) \tanh ^{-1}\left (\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left (5 \sqrt{11}-17\right ) x+\sqrt{11}-23}\right )-2 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left (1740003+126542 \sqrt{11}\right ) \log \left (-7 x+\sqrt{11}+2\right )+\sqrt{\frac{22}{125-17 \sqrt{11}}} \left (126542 \sqrt{11}-1740003\right ) \log \left (\left (7 x+\sqrt{11}-2\right )^2\right )}{10818368} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-44*Sqrt[3 + 2*x + 5*x^2]*(31807 - 106279*x - 737577*x^2 + 813113*x^3))/(1 + 4*x - 7*x^2)^2 - 2*Sqrt[22/(125
- 17*Sqrt[11])]*(-1740003 + 126542*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-23 + S
qrt[11] + (-17 + 5*Sqrt[11])*x)] - 2*Sqrt[22/(125 + 17*Sqrt[11])]*(1740003 + 126542*Sqrt[11])*Log[2 + Sqrt[11]
- 7*x] + Sqrt[22/(125 - 17*Sqrt[11])]*(-1740003 + 126542*Sqrt[11])*Log[(-2 + Sqrt[11] + 7*x)^2] - Sqrt[22/(12
5 - 17*Sqrt[11])]*(-1740003 + 126542*Sqrt[11])*Log[15 - 4*Sqrt[11] + 14*(-2 + Sqrt[11])*x + 49*x^2] + 2*Sqrt[2
2/(125 + 17*Sqrt[11])]*(1740003 + 126542*Sqrt[11])*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 3
74*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/10818368

________________________________________________________________________________________

Maple [B]  time = 0.127, size = 2342, normalized size = 11. \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)^3,x)

[Out]

-(-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))^(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+1
0/7*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))))-
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))^(3/2)-1/1372*(34/7-10/7*11^(1/2))/(250/4
9-34/49*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))^(3/2)+1/2*(34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(1/7*(2
45*(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*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)))+10/(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))))+5/686/(250/49-34/49*1
1^(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))))-3535/21296*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+3
4/49*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)*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)*(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)))-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))^(3/2)-1/1372*(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))^(3/2)+1/2*(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/(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))
))+5/686/(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))))-(-3535/1936+273/1936*11^(1/2))*(-1/49/(250/49-3
4/49*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/(2
50/49-34/49*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))+250/4
9-34/49*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))))

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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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.25648, size = 1700, normalized size = 7.98 \begin{align*} -\frac{\sqrt{1397}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{11879169071 \, \sqrt{11} + 6492253020949} \log \left (\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{11879169071 \, \sqrt{11} + 6492253020949}{\left (4822219 \, \sqrt{11} - 37335441\right )} + 569071698870455 \, \sqrt{11}{\left (x + 3\right )} + 1707215096611365 \, x - 2845358494352275}{x}\right ) - \sqrt{1397}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{11879169071 \, \sqrt{11} + 6492253020949} \log \left (-\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3} \sqrt{11879169071 \, \sqrt{11} + 6492253020949}{\left (4822219 \, \sqrt{11} - 37335441\right )} - 569071698870455 \, \sqrt{11}{\left (x + 3\right )} - 1707215096611365 \, x + 2845358494352275}{x}\right ) + \sqrt{1397}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{-11879169071 \, \sqrt{11} + 6492253020949} \log \left (-\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (4822219 \, \sqrt{11} + 37335441\right )} \sqrt{-11879169071 \, \sqrt{11} + 6492253020949} + 569071698870455 \, \sqrt{11}{\left (x + 3\right )} - 1707215096611365 \, x + 2845358494352275}{x}\right ) - \sqrt{1397}{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt{-11879169071 \, \sqrt{11} + 6492253020949} \log \left (\frac{\sqrt{1397} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (4822219 \, \sqrt{11} + 37335441\right )} \sqrt{-11879169071 \, \sqrt{11} + 6492253020949} - 569071698870455 \, \sqrt{11}{\left (x + 3\right )} + 1707215096611365 \, x - 2845358494352275}{x}\right ) + 5588 \,{\left (813113 \, x^{3} - 737577 \, x^{2} - 106279 \, x + 31807\right )} \sqrt{5 \, x^{2} + 2 \, x + 3}}{1373932736 \,{\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)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^3,x, algorithm="fricas")

[Out]

-1/1373932736*(sqrt(1397)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(11879169071*sqrt(11) + 6492253020949)*log((
sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*sqrt(11879169071*sqrt(11) + 6492253020949)*(4822219*sqrt(11) - 37335441) + 56
9071698870455*sqrt(11)*(x + 3) + 1707215096611365*x - 2845358494352275)/x) - sqrt(1397)*(49*x^4 - 56*x^3 + 2*x
^2 + 8*x + 1)*sqrt(11879169071*sqrt(11) + 6492253020949)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*sqrt(118791690
71*sqrt(11) + 6492253020949)*(4822219*sqrt(11) - 37335441) - 569071698870455*sqrt(11)*(x + 3) - 17072150966113
65*x + 2845358494352275)/x) + sqrt(1397)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-11879169071*sqrt(11) + 6492
253020949)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(4822219*sqrt(11) + 37335441)*sqrt(-11879169071*sqrt(11) + 6
492253020949) + 569071698870455*sqrt(11)*(x + 3) - 1707215096611365*x + 2845358494352275)/x) - sqrt(1397)*(49*
x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-11879169071*sqrt(11) + 6492253020949)*log((sqrt(1397)*sqrt(5*x^2 + 2*x +
3)*(4822219*sqrt(11) + 37335441)*sqrt(-11879169071*sqrt(11) + 6492253020949) - 569071698870455*sqrt(11)*(x +
3) + 1707215096611365*x - 2845358494352275)/x) + 5588*(813113*x^3 - 737577*x^2 - 106279*x + 31807)*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)*(5*x**2+2*x+3)**(1/2)/(-7*x**2+4*x+1)**3,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)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError