### 3.397 $$\int \frac{2+5 x+x^2}{(1+4 x-7 x^2)^3 (3+2 x+5 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=250 $-\frac{2701733-9148874 x}{62451488 \left (-7 x^2+4 x+1\right ) \sqrt{5 x^2+2 x+3}}-\frac{5 (1118731375 x+461370781)}{222077491328 \sqrt{5 x^2+2 x+3}}-\frac{3 (40-371 x)}{11176 \left (-7 x^2+4 x+1\right )^2 \sqrt{5 x^2+2 x+3}}-\frac{7 \left (2792860024-84865895 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{31725355904 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (2792860024+84865895 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{31725355904 \sqrt{22 \left (125+17 \sqrt{11}\right )}}$

[Out]

(-5*(461370781 + 1118731375*x))/(222077491328*Sqrt[3 + 2*x + 5*x^2]) - (3*(40 - 371*x))/(11176*(1 + 4*x - 7*x^
2)^2*Sqrt[3 + 2*x + 5*x^2]) - (2701733 - 9148874*x)/(62451488*(1 + 4*x - 7*x^2)*Sqrt[3 + 2*x + 5*x^2]) - (7*(2
792860024 - 84865895*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt
[3 + 2*x + 5*x^2])])/(31725355904*Sqrt[22*(125 - 17*Sqrt[11])]) + (7*(2792860024 + 84865895*Sqrt[11])*ArcTanh[
(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(31725355904*Sqrt[
22*(125 + 17*Sqrt[11])])

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Rubi [A]  time = 0.323972, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.114, Rules used = {1060, 1032, 724, 206} $-\frac{2701733-9148874 x}{62451488 \left (-7 x^2+4 x+1\right ) \sqrt{5 x^2+2 x+3}}-\frac{5 (1118731375 x+461370781)}{222077491328 \sqrt{5 x^2+2 x+3}}-\frac{3 (40-371 x)}{11176 \left (-7 x^2+4 x+1\right )^2 \sqrt{5 x^2+2 x+3}}-\frac{7 \left (2792860024-84865895 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17-5 \sqrt{11}\right ) x-\sqrt{11}+23}{\sqrt{2 \left (125-17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{31725355904 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (2792860024+84865895 \sqrt{11}\right ) \tanh ^{-1}\left (\frac{\left (17+5 \sqrt{11}\right ) x+\sqrt{11}+23}{\sqrt{2 \left (125+17 \sqrt{11}\right )} \sqrt{5 x^2+2 x+3}}\right )}{31725355904 \sqrt{22 \left (125+17 \sqrt{11}\right )}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*(461370781 + 1118731375*x))/(222077491328*Sqrt[3 + 2*x + 5*x^2]) - (3*(40 - 371*x))/(11176*(1 + 4*x - 7*x^
2)^2*Sqrt[3 + 2*x + 5*x^2]) - (2701733 - 9148874*x)/(62451488*(1 + 4*x - 7*x^2)*Sqrt[3 + 2*x + 5*x^2]) - (7*(2
792860024 - 84865895*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt
[3 + 2*x + 5*x^2])])/(31725355904*Sqrt[22*(125 - 17*Sqrt[11])]) + (7*(2792860024 + 84865895*Sqrt[11])*ArcTanh[
(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(31725355904*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 \left (3+2 x+5 x^2\right )^{3/2}} \, dx &=-\frac{3 (40-371 x)}{11176 \left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}}-\frac{\int \frac{-128104-89208 x-178080 x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx}{89408}\\ &=-\frac{3 (40-371 x)}{11176 \left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}}-\frac{2701733-9148874 x}{62451488 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}+\frac{\int \frac{1722335552+835857088 x+5855279360 x^2}{\left (1+4 x-7 x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}} \, dx}{3996895232}\\ &=-\frac{5 (461370781+1118731375 x)}{222077491328 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{11176 \left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}}-\frac{2701733-9148874 x}{62451488 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}+\frac{\int \frac{18802583181312+4258231147520 x}{\left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}} \, dx}{113703675559936}\\ &=-\frac{5 (461370781+1118731375 x)}{222077491328 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{11176 \left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}}-\frac{2701733-9148874 x}{62451488 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}+\frac{\left (7 \left (933524845-2792860024 \sqrt{11}\right )\right ) \int \frac{1}{\left (4-2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{174489457472}+\frac{\left (7 \left (933524845+2792860024 \sqrt{11}\right )\right ) \int \frac{1}{\left (4+2 \sqrt{11}-14 x\right ) \sqrt{3+2 x+5 x^2}} \, dx}{174489457472}\\ &=-\frac{5 (461370781+1118731375 x)}{222077491328 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{11176 \left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}}-\frac{2701733-9148874 x}{62451488 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}-\frac{\left (7 \left (933524845-2792860024 \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 )}{87244728736}-\frac{\left (7 \left (933524845+2792860024 \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 )}{87244728736}\\ &=-\frac{5 (461370781+1118731375 x)}{222077491328 \sqrt{3+2 x+5 x^2}}-\frac{3 (40-371 x)}{11176 \left (1+4 x-7 x^2\right )^2 \sqrt{3+2 x+5 x^2}}-\frac{2701733-9148874 x}{62451488 \left (1+4 x-7 x^2\right ) \sqrt{3+2 x+5 x^2}}-\frac{7 \left (2792860024-84865895 \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 )}{31725355904 \sqrt{22 \left (125-17 \sqrt{11}\right )}}+\frac{7 \left (2792860024+84865895 \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 )}{31725355904 \sqrt{22 \left (125+17 \sqrt{11}\right )}}\\ \end{align*}

Mathematica [A]  time = 1.57375, size = 381, normalized size = 1.52 $\frac{\frac{44 \sqrt{5 x^2+2 x+3} (507770113-1167248019 x)}{7 x^2-4 x-1}+\frac{737616 (38521 x-12667) \sqrt{5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^2}+\frac{21296 (501205 x+1702037)}{7 \sqrt{5 x^2+2 x+3}}-7 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left (84865895 \sqrt{11}-2792860024\right ) \log \left (49 x^2+14 \left (\sqrt{11}-2\right ) x-4 \sqrt{11}+15\right )+14 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left (2792860024+84865895 \sqrt{11}\right ) \log \left (\sqrt{2750+374 \sqrt{11}} \sqrt{5 x^2+2 x+3}+\left (55+17 \sqrt{11}\right ) x+23 \sqrt{11}+11\right )-14 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left (84865895 \sqrt{11}-2792860024\right ) \tanh ^{-1}\left (\frac{\sqrt{250-34 \sqrt{11}} \sqrt{5 x^2+2 x+3}}{\left (5 \sqrt{11}-17\right ) x+\sqrt{11}-23}\right )-14 \sqrt{\frac{22}{125+17 \sqrt{11}}} \left (2792860024+84865895 \sqrt{11}\right ) \log \left (-7 x+\sqrt{11}+2\right )+7 \sqrt{\frac{22}{125-17 \sqrt{11}}} \left (84865895 \sqrt{11}-2792860024\right ) \log \left (\left (7 x+\sqrt{11}-2\right )^2\right )}{1395915659776}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((21296*(1702037 + 501205*x))/(7*Sqrt[3 + 2*x + 5*x^2]) + (737616*(-12667 + 38521*x)*Sqrt[3 + 2*x + 5*x^2])/(1
+ 4*x - 7*x^2)^2 + (44*(507770113 - 1167248019*x)*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) - 14*Sqrt[22/(125
- 17*Sqrt[11])]*(-2792860024 + 84865895*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2])/(-2
3 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 14*Sqrt[22/(125 + 17*Sqrt[11])]*(2792860024 + 84865895*Sqrt[11])*Log[2
+ Sqrt[11] - 7*x] + 7*Sqrt[22/(125 - 17*Sqrt[11])]*(-2792860024 + 84865895*Sqrt[11])*Log[(-2 + Sqrt[11] + 7*x
)^2] - 7*Sqrt[22/(125 - 17*Sqrt[11])]*(-2792860024 + 84865895*Sqrt[11])*Log[15 - 4*Sqrt[11] + 14*(-2 + Sqrt[11
])*x + 49*x^2] + 14*Sqrt[22/(125 + 17*Sqrt[11])]*(2792860024 + 84865895*Sqrt[11])*Log[11 + 23*Sqrt[11] + (55 +
17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/1395915659776

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Maple [B]  time = 0.12, size = 2600, normalized size = 10.4 \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)^(3/2),x)

[Out]

-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)-5/1372*(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))^(1/2)-3/2*(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/(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))-15/686/(250/49+34/4
9*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))))-3535/21296*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+34/49*11^(1/2))^(1/2)-1/(250/49+34/49*11^(1/2))/(250+34*11^(1/2))^(1/2)*ar
ctanh(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)-5/1372*(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)-3/2*(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/
(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+(3
4/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250/49-34/49*11^(1/2))^(1/2))-15/686/(250/49-34/49*11^(1/2))*(1/(250/4
9-34/49*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))))+3535/21296*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))+250/49-34/49*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)))-(-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))+2
50/49+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))-(-3535/1936+273/1936*11^(1/2))*(-1/49/(25
0/49-34/49*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+(3
4/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250/49-34/49*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}{\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)^3/(5*x^2+2*x+3)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.54705, size = 2311, normalized size = 9.24 \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)/(-7*x^2+4*x+1)^3/(5*x^2+2*x+3)^(3/2),x, algorithm="fricas")

[Out]

-1/1240969021540864*(7*sqrt(1397)*(245*x^6 - 182*x^5 + 45*x^4 - 124*x^3 + 27*x^2 + 26*x + 3)*sqrt(746933147106
39641467*sqrt(11) + 896266498377233657855)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*sqrt(74693314710639641467*sq
rt(11) + 896266498377233657855)*(37271563201*sqrt(11) + 407780707037) + 75502120686844055144479*sqrt(11)*(x +
3) - 226506362060532165433437*x + 377510603434220275722395)/x) - 7*sqrt(1397)*(245*x^6 - 182*x^5 + 45*x^4 - 12
4*x^3 + 27*x^2 + 26*x + 3)*sqrt(74693314710639641467*sqrt(11) + 896266498377233657855)*log((sqrt(1397)*sqrt(5*
x^2 + 2*x + 3)*sqrt(74693314710639641467*sqrt(11) + 896266498377233657855)*(37271563201*sqrt(11) + 40778070703
7) - 75502120686844055144479*sqrt(11)*(x + 3) + 226506362060532165433437*x - 377510603434220275722395)/x) + 7*
sqrt(1397)*(245*x^6 - 182*x^5 + 45*x^4 - 124*x^3 + 27*x^2 + 26*x + 3)*sqrt(-74693314710639641467*sqrt(11) + 89
6266498377233657855)*log((sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(37271563201*sqrt(11) - 407780707037)*sqrt(-7469331
4710639641467*sqrt(11) + 896266498377233657855) + 75502120686844055144479*sqrt(11)*(x + 3) + 22650636206053216
5433437*x - 377510603434220275722395)/x) - 7*sqrt(1397)*(245*x^6 - 182*x^5 + 45*x^4 - 124*x^3 + 27*x^2 + 26*x
+ 3)*sqrt(-74693314710639641467*sqrt(11) + 896266498377233657855)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(3727
1563201*sqrt(11) - 407780707037)*sqrt(-74693314710639641467*sqrt(11) + 896266498377233657855) - 75502120686844
055144479*sqrt(11)*(x + 3) - 226506362060532165433437*x + 377510603434220275722395)/x) + 5588*(274089186875*x^
5 - 200208943655*x^4 + 109737266678*x^3 - 148022158802*x^2 + 7828199499*x + 14298727813)*sqrt(5*x^2 + 2*x + 3)
)/(245*x^6 - 182*x^5 + 45*x^4 - 124*x^3 + 27*x^2 + 26*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)**3/(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)^3/(5*x^2+2*x+3)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError