Integrand size = 35, antiderivative size = 215 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx=-\frac {76567+22755 x}{19870928 \sqrt {3+2 x+5 x^2}}-\frac {3 (40-371 x)}{5588 \left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}}-\frac {7 \left (541543-5144 \sqrt {11}\right ) \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2838704 \sqrt {22 \left (125-17 \sqrt {11}\right )}}+\frac {7 \left (541543+5144 \sqrt {11}\right ) \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2838704 \sqrt {22 \left (125+17 \sqrt {11}\right )}} \] Output:
-1/19870928*(76567+22755*x)/(5*x^2+2*x+3)^(1/2)-3/5588*(40-371*x)/(-7*x^2+ 4*x+1)/(5*x^2+2*x+3)^(1/2)-7/2838704*(541543-5144*11^(1/2))*arctanh((23-11 ^(1/2)+(17-5*11^(1/2))*x)/(250-34*11^(1/2))^(1/2)/(5*x^2+2*x+3)^(1/2))/(27 50-374*11^(1/2))^(1/2)+7/2838704*(541543+5144*11^(1/2))*arctanh((23+11^(1/ 2)+(17+5*11^(1/2))*x)/(250+34*11^(1/2))^(1/2)/(5*x^2+2*x+3)^(1/2))/(2750+3 74*11^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.95 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.93 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx=\frac {503287-3628805 x-444949 x^2-159285 x^3}{19870928 \sqrt {3+2 x+5 x^2} \left (-1-4 x+7 x^2\right )}+\frac {\text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {116685 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+205710 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+8351 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{258064 \sqrt {5}}-\frac {3 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {746007 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )-1016580 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+42623 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{2838704 \sqrt {5}} \] Input:
Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^2*(3 + 2*x + 5*x^2)^(3/2)),x]
Output:
(503287 - 3628805*x - 444949*x^2 - 159285*x^3)/(19870928*Sqrt[3 + 2*x + 5* x^2]*(-1 - 4*x + 7*x^2)) + RootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5 ]*#1^3 + 7*#1^4 & , (116685*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^ 2] - #1] + 205710*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 8351 *Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1^2)/(-4*Sqrt[5] - 35*#1 + 6*Sqrt[5]*#1^2 + 7*#1^3) & ]/(258064*Sqrt[5]) - (3*RootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (746007*Sqrt[5]*Log[ -(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] - 1016580*Log[-(Sqrt[5]*x) + Sq rt[3 + 2*x + 5*x^2] - #1]*#1 + 42623*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2 *x + 5*x^2] - #1]*#1^2)/(-4*Sqrt[5] - 35*#1 + 6*Sqrt[5]*#1^2 + 7*#1^3) & ] )/(2838704*Sqrt[5])
Time = 0.85 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {2135, 27, 2135, 27, 1365, 27, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2+5 x+2}{\left (-7 x^2+4 x+1\right )^2 \left (5 x^2+2 x+3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle -\frac {\int -\frac {8 \left (11130 x^2+4719 x+6277\right )}{\left (-7 x^2+4 x+1\right ) \left (5 x^2+2 x+3\right )^{3/2}}dx}{44704}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11130 x^2+4719 x+6277}{\left (-7 x^2+4 x+1\right ) \left (5 x^2+2 x+3\right )^{3/2}}dx}{5588}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\frac {\int \frac {112 (36008 x+531255)}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx}{28448}-\frac {22755 x+76567}{3556 \sqrt {5 x^2+2 x+3}}}{5588}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{254} \int \frac {36008 x+531255}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {22755 x+76567}{3556 \sqrt {5 x^2+2 x+3}}}{5588}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {\frac {1}{254} \left (\frac {7}{11} \left (56584-541543 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {7}{11} \left (56584+541543 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {22755 x+76567}{3556 \sqrt {5 x^2+2 x+3}}}{5588}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{254} \left (\frac {7}{22} \left (56584-541543 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {7}{22} \left (56584+541543 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {22755 x+76567}{3556 \sqrt {5 x^2+2 x+3}}}{5588}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {1}{254} \left (-\frac {7}{11} \left (56584-541543 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {7}{11} \left (56584+541543 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )\right )-\frac {22755 x+76567}{3556 \sqrt {5 x^2+2 x+3}}}{5588}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{254} \left (\frac {7 \left (56584-541543 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {7 \left (56584+541543 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}\right )-\frac {22755 x+76567}{3556 \sqrt {5 x^2+2 x+3}}}{5588}-\frac {3 (40-371 x)}{5588 \left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}\) |
Input:
Int[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^2*(3 + 2*x + 5*x^2)^(3/2)),x]
Output:
(-3*(40 - 371*x))/(5588*(1 + 4*x - 7*x^2)*Sqrt[3 + 2*x + 5*x^2]) + (-1/355 6*(76567 + 22755*x)/Sqrt[3 + 2*x + 5*x^2] + ((7*(56584 - 541543*Sqrt[11])* ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])] *Sqrt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 - 17*Sqrt[11])]) + (7*(56584 + 5 41543*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 + 17*Sqrt[11])]) )/254)/5588
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-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]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(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]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 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), x] + Simp[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]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && 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]
Time = 1.06 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(-\frac {159285 x^{3}+444949 x^{2}+3628805 x -503287}{19870928 \left (7 x^{2}-4 x -1\right ) \sqrt {5 x^{2}+2 x +3}}+\frac {7 \left (-541543+5144 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{31225744 \sqrt {250-34 \sqrt {11}}}+\frac {7 \left (541543+5144 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{31225744 \sqrt {250+34 \sqrt {11}}}\) | \(231\) |
| trager | \(\text {Expression too large to display}\) | \(494\) |
| default | \(\text {Expression too large to display}\) | \(1213\) |
Input:
int((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(3/2),x,method=_RETURNVERBO SE)
Output:
-1/19870928*(159285*x^3+444949*x^2+3628805*x-503287)/(7*x^2-4*x-1)/(5*x^2+ 2*x+3)^(1/2)+7/31225744*(-541543+5144*11^(1/2))*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*(34/7-1 0/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^(1/2))+7/31225744*(541 543+5144*11^(1/2))*11^(1/2)/(250+34*11^(1/2))^(1/2)*arctanh(49/2*(500/49+6 8/49*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))
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (162) = 324\).
Time = 0.09 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.71 \[ \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 {7 \, {\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt {\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} \log \left (-\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} {\left (5609479 \, \sqrt {11} + 77949905\right )} + 2050844269271 \, \sqrt {11} {\left (x + 3\right )} - 6152532807813 \, x + 10254221346355}{x}\right ) - 7 \, {\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt {\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} \log \left (\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} {\left (5609479 \, \sqrt {11} + 77949905\right )} - 2050844269271 \, \sqrt {11} {\left (x + 3\right )} + 6152532807813 \, x - 10254221346355}{x}\right ) + 7 \, {\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt {-\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} \log \left (\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5609479 \, \sqrt {11} - 77949905\right )} \sqrt {-\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} + 2050844269271 \, \sqrt {11} {\left (x + 3\right )} + 6152532807813 \, x - 10254221346355}{x}\right ) - 7 \, {\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )} \sqrt {-\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} \log \left (-\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5609479 \, \sqrt {11} - 77949905\right )} \sqrt {-\frac {390372164915}{127} \, \sqrt {11} + \frac {35653135368317}{1397}} - 2050844269271 \, \sqrt {11} {\left (x + 3\right )} - 6152532807813 \, x + 10254221346355}{x}\right ) + 4 \, {\left (159285 \, x^{3} + 444949 \, x^{2} + 3628805 \, x - 503287\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{79483712 \, {\left (35 \, x^{4} - 6 \, x^{3} + 8 \, x^{2} - 14 \, x - 3\right )}} \] Input:
integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(3/2),x, algorithm="f ricas")
Output:
-1/79483712*(7*(35*x^4 - 6*x^3 + 8*x^2 - 14*x - 3)*sqrt(390372164915/127*s qrt(11) + 35653135368317/1397)*log(-(sqrt(5*x^2 + 2*x + 3)*sqrt(3903721649 15/127*sqrt(11) + 35653135368317/1397)*(5609479*sqrt(11) + 77949905) + 205 0844269271*sqrt(11)*(x + 3) - 6152532807813*x + 10254221346355)/x) - 7*(35 *x^4 - 6*x^3 + 8*x^2 - 14*x - 3)*sqrt(390372164915/127*sqrt(11) + 35653135 368317/1397)*log((sqrt(5*x^2 + 2*x + 3)*sqrt(390372164915/127*sqrt(11) + 3 5653135368317/1397)*(5609479*sqrt(11) + 77949905) - 2050844269271*sqrt(11) *(x + 3) + 6152532807813*x - 10254221346355)/x) + 7*(35*x^4 - 6*x^3 + 8*x^ 2 - 14*x - 3)*sqrt(-390372164915/127*sqrt(11) + 35653135368317/1397)*log(( sqrt(5*x^2 + 2*x + 3)*(5609479*sqrt(11) - 77949905)*sqrt(-390372164915/127 *sqrt(11) + 35653135368317/1397) + 2050844269271*sqrt(11)*(x + 3) + 615253 2807813*x - 10254221346355)/x) - 7*(35*x^4 - 6*x^3 + 8*x^2 - 14*x - 3)*sqr t(-390372164915/127*sqrt(11) + 35653135368317/1397)*log(-(sqrt(5*x^2 + 2*x + 3)*(5609479*sqrt(11) - 77949905)*sqrt(-390372164915/127*sqrt(11) + 3565 3135368317/1397) - 2050844269271*sqrt(11)*(x + 3) - 6152532807813*x + 1025 4221346355)/x) + 4*(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)
\[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx=\int \frac {x^{2} + 5 x + 2}{\left (5 x^{2} + 2 x + 3\right )^{\frac {3}{2}} \left (7 x^{2} - 4 x - 1\right )^{2}}\, dx \] Input:
integrate((x**2+5*x+2)/(-7*x**2+4*x+1)**2/(5*x**2+2*x+3)**(3/2),x)
Output:
Integral((x**2 + 5*x + 2)/((5*x**2 + 2*x + 3)**(3/2)*(7*x**2 - 4*x - 1)**2 ), x)
\[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx=\int { \frac {x^{2} + 5 \, x + 2}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{2} {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(3/2),x, algorithm="m axima")
Output:
integrate((x^2 + 5*x + 2)/((7*x^2 - 4*x - 1)^2*(5*x^2 + 2*x + 3)^(3/2)), x )
Time = 0.16 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.37 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx=\frac {25230 \, x + 13397}{903224 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {3 \, {\left (42623 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} + 77302 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} - 275511 \, \sqrt {5} x - 219860 \, \sqrt {5} + 275511 \, \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}}{709676 \, {\left (7 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} - 8 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} - 70 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} + 83\right )}} + 0.0218058276253952 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.0332874364433911 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 0.0218058276253952 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.0332874364433911 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \] Input:
integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(3/2),x, algorithm="g iac")
Output:
1/903224*(25230*x + 13397)/sqrt(5*x^2 + 2*x + 3) + 3/709676*(42623*(sqrt(5 )*x - sqrt(5*x^2 + 2*x + 3))^3 + 77302*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2 *x + 3))^2 - 275511*sqrt(5)*x - 219860*sqrt(5) + 275511*sqrt(5*x^2 + 2*x + 3))/(7*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^4 - 8*sqrt(5)*(sqrt(5)*x - sqr t(5*x^2 + 2*x + 3))^3 - 70*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^2 + 16*sqrt (5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) + 83) + 0.0218058276253952*log(-sq rt(5)*x + sqrt(5*x^2 + 2*x + 3) + 4.41924736459000) - 0.0332874364433911*l og(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.25295163054000) - 0.021805827625 3952*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 1.02258038113000) + 0.033287 4364433911*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 2.09411235400000)
Timed out. \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx=\int \frac {x^2+5\,x+2}{{\left (5\,x^2+2\,x+3\right )}^{3/2}\,{\left (-7\,x^2+4\,x+1\right )}^2} \,d x \] Input:
int((5*x + x^2 + 2)/((2*x + 5*x^2 + 3)^(3/2)*(4*x - 7*x^2 + 1)^2),x)
Output:
int((5*x + x^2 + 2)/((2*x + 5*x^2 + 3)^(3/2)*(4*x - 7*x^2 + 1)^2), x)
Time = 67.23 (sec) , antiderivative size = 2193, normalized size of antiderivative = 10.20 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^2 \left (3+2 x+5 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((x^2+5*x+2)/(-7*x^2+4*x+1)^2/(5*x^2+2*x+3)^(3/2),x)
Output:
( - 2335583145*sqrt(17*sqrt(11) - 125)*sqrt(22)*atan((24*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(22)*x - 19*sqrt(5*x**2 + 2*x + 3)*sqrt( 17*sqrt(11) - 125)*sqrt(22) - 85*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(2)*x - 192*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt( 2))/(8890*x**2 + 3556*x + 5334))*x**4 + 400385682*sqrt(17*sqrt(11) - 125)* sqrt(22)*atan((24*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(22)* x - 19*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(22) - 85*sqrt(5 *x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(2)*x - 192*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(2))/(8890*x**2 + 3556*x + 5334))*x**3 - 533847576*sqrt(17*sqrt(11) - 125)*sqrt(22)*atan((24*sqrt(5*x**2 + 2*x + 3 )*sqrt(17*sqrt(11) - 125)*sqrt(22)*x - 19*sqrt(5*x**2 + 2*x + 3)*sqrt(17*s qrt(11) - 125)*sqrt(22) - 85*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125 )*sqrt(2)*x - 192*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(2))/ (8890*x**2 + 3556*x + 5334))*x**2 + 934233258*sqrt(17*sqrt(11) - 125)*sqrt (22)*atan((24*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(22)*x - 19*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(22) - 85*sqrt(5*x** 2 + 2*x + 3)*sqrt(17*sqrt(11) - 125)*sqrt(2)*x - 192*sqrt(5*x**2 + 2*x + 3 )*sqrt(17*sqrt(11) - 125)*sqrt(2))/(8890*x**2 + 3556*x + 5334))*x + 200192 841*sqrt(17*sqrt(11) - 125)*sqrt(22)*atan((24*sqrt(5*x**2 + 2*x + 3)*sqrt( 17*sqrt(11) - 125)*sqrt(22)*x - 19*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(...