Integrand size = 35, antiderivative size = 213 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac {(272941-813113 x) \sqrt {3+2 x+5 x^2}}{1721104 \left (1+4 x-7 x^2\right )}-\frac {\sqrt {\frac {6492253020949-11879169071 \sqrt {11}}{1397}} \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{491744}+\frac {\sqrt {\frac {6492253020949+11879169071 \sqrt {11}}{1397}} \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{491744} \] Output:
3/308*(3+61*x)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^2-(272941-813113*x)*(5*x ^2+2*x+3)^(1/2)/(-12047728*x^2+6884416*x+1721104)-1/686966368*(90696774702 65753-16595199192187*11^(1/2))^(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))+1/686966368*(9069677470265 753+16595199192187*11^(1/2))^(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))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.12 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.83 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\frac {\frac {5764801 \sqrt {3+2 x+5 x^2} \left (-31807+106279 x+737577 x^2-813113 x^3\right )}{\left (1+4 x-7 x^2\right )^2}-60545521580434 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]+20661853520 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {-465 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+7 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]+22 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {3751778663030 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2597308755559 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]-6 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {-11648778057271 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+13372446682211 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+9645047011740 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{1417403151472} \] Input:
Integrate[((2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^3,x]
Output:
((5764801*Sqrt[3 + 2*x + 5*x^2]*(-31807 + 106279*x + 737577*x^2 - 813113*x ^3))/(1 + 4*x - 7*x^2)^2 - 60545521580434*RootSum[83 - 16*Sqrt[5]*#1 - 70* #1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2 ] - #1]/(-4*Sqrt[5] - 35*#1 + 6*Sqrt[5]*#1^2 + 7*#1^3) & ] + 20661853520*R ootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (-465*Lo g[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] + 7*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1)/(-4*Sqrt[5] - 35*#1 + 6*Sqrt[5]*#1^2 + 7 *#1^3) & ] + 22*RootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7* #1^4 & , (3751778663030*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 2597308755559*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) & ] - 6*RootSum[83 - 16* Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (-11648778057271*Log[-( Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] + 13372446682211*Sqrt[5]*Log[-(Sq rt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 9645047011740*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) & ])/1417403151472
Time = 0.86 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {2132, 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.
| \(\displaystyle \int \frac {\left (x^2+5 x+2\right ) \sqrt {5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2132 |
| \(\displaystyle \frac {3 (61 x+3) \sqrt {5 x^2+2 x+3}}{308 \left (-7 x^2+4 x+1\right )^2}-\frac {1}{616} \int -\frac {4 \left (805 x^2+391 x+753\right )}{\left (-7 x^2+4 x+1\right )^2 \sqrt {5 x^2+2 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{154} \int \frac {805 x^2+391 x+753}{\left (-7 x^2+4 x+1\right )^2 \sqrt {5 x^2+2 x+3}}dx+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{154} \left (-\frac {\int -\frac {56 (126542 x+212417)}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx}{44704}-\frac {\sqrt {5 x^2+2 x+3} (272941-813113 x)}{11176 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{154} \left (\frac {7 \int \frac {126542 x+212417}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx}{5588}-\frac {(272941-813113 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{154} \left (\frac {7 \left (\frac {1}{11} \left (1391962-1740003 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{11} \left (1391962+1740003 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )}{5588}-\frac {(272941-813113 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{154} \left (\frac {7 \left (\frac {1}{22} \left (1391962-1740003 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{22} \left (1391962+1740003 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )}{5588}-\frac {(272941-813113 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{154} \left (\frac {7 \left (-\frac {1}{11} \left (1391962-1740003 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {1}{11} \left (1391962+1740003 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )\right )}{5588}-\frac {(272941-813113 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{154} \left (\frac {7 \left (\frac {\left (1391962-1740003 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {\left (1391962+1740003 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}\right )}{5588}-\frac {(272941-813113 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{308 \left (-7 x^2+4 x+1\right )^2}\) |
Input:
Int[((2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^3,x]
Output:
(3*(3 + 61*x)*Sqrt[3 + 2*x + 5*x^2])/(308*(1 + 4*x - 7*x^2)^2) + (-1/11176 *((272941 - 813113*x)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (7*(((139 1962 - 1740003*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sq rt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 - 17*Sq rt[11])]) + ((1391962 + 1740003*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*Sq rt[2*(125 + 17*Sqrt[11])])))/5588)/154
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*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] - Simp[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}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
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 = 1.10 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(-\frac {\left (813113 x^{3}-737577 x^{2}-106279 x +31807\right ) \sqrt {5 x^{2}+2 x +3}}{245872 \left (7 x^{2}-4 x -1\right )^{2}}+\frac {\left (-1740003+126542 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{2704592 \sqrt {250-34 \sqrt {11}}}+\frac {\left (1740003+126542 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{2704592 \sqrt {250+34 \sqrt {11}}}\) | \(231\) |
| trager | \(\text {Expression too large to display}\) | \(483\) |
| default | \(\text {Expression too large to display}\) | \(2341\) |
Input:
int((x^2+5*x+2)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^3,x,method=_RETURNVERBO SE)
Output:
-1/245872*(813113*x^3-737577*x^2-106279*x+31807)/(7*x^2-4*x-1)^2*(5*x^2+2* x+3)^(1/2)+1/2704592*(-1740003+126542*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))+1/2704592*(1740 003+126542*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+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. 366 vs. \(2 (160) = 320\).
Time = 0.11 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.72 \[ \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 {{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} \log \left (\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} {\left (4822219 \, \sqrt {11} - 37335441\right )} + 407352683515 \, \sqrt {11} {\left (x + 3\right )} + 1222058050545 \, x - 2036763417575}{x}\right ) - {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} \log \left (-\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} {\left (4822219 \, \sqrt {11} - 37335441\right )} - 407352683515 \, \sqrt {11} {\left (x + 3\right )} - 1222058050545 \, x + 2036763417575}{x}\right ) + {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} \log \left (-\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (4822219 \, \sqrt {11} + 37335441\right )} \sqrt {-\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} + 407352683515 \, \sqrt {11} {\left (x + 3\right )} - 1222058050545 \, x + 2036763417575}{x}\right ) - {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} \log \left (\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (4822219 \, \sqrt {11} + 37335441\right )} \sqrt {-\frac {1079924461}{127} \, \sqrt {11} + \frac {6492253020949}{1397}} - 407352683515 \, \sqrt {11} {\left (x + 3\right )} + 1222058050545 \, x - 2036763417575}{x}\right ) + 4 \, {\left (813113 \, x^{3} - 737577 \, x^{2} - 106279 \, x + 31807\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{983488 \, {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )}} \] Input:
integrate((x^2+5*x+2)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^3,x, algorithm="f ricas")
Output:
-1/983488*((49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(1079924461/127*sqrt(11 ) + 6492253020949/1397)*log((sqrt(5*x^2 + 2*x + 3)*sqrt(1079924461/127*sqr t(11) + 6492253020949/1397)*(4822219*sqrt(11) - 37335441) + 407352683515*s qrt(11)*(x + 3) + 1222058050545*x - 2036763417575)/x) - (49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(1079924461/127*sqrt(11) + 6492253020949/1397)*log(- (sqrt(5*x^2 + 2*x + 3)*sqrt(1079924461/127*sqrt(11) + 6492253020949/1397)* (4822219*sqrt(11) - 37335441) - 407352683515*sqrt(11)*(x + 3) - 1222058050 545*x + 2036763417575)/x) + (49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-1079 924461/127*sqrt(11) + 6492253020949/1397)*log(-(sqrt(5*x^2 + 2*x + 3)*(482 2219*sqrt(11) + 37335441)*sqrt(-1079924461/127*sqrt(11) + 6492253020949/13 97) + 407352683515*sqrt(11)*(x + 3) - 1222058050545*x + 2036763417575)/x) - (49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-1079924461/127*sqrt(11) + 6492 253020949/1397)*log((sqrt(5*x^2 + 2*x + 3)*(4822219*sqrt(11) + 37335441)*s qrt(-1079924461/127*sqrt(11) + 6492253020949/1397) - 407352683515*sqrt(11) *(x + 3) + 1222058050545*x - 2036763417575)/x) + 4*(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)
\[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx=- \int \frac {2 \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {5 x \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {x^{2} \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx \] Input:
integrate((x**2+5*x+2)*(5*x**2+2*x+3)**(1/2)/(-7*x**2+4*x+1)**3,x)
Output:
-Integral(2*sqrt(5*x**2 + 2*x + 3)/(343*x**6 - 588*x**5 + 189*x**4 + 104*x **3 - 27*x**2 - 12*x - 1), x) - Integral(5*x*sqrt(5*x**2 + 2*x + 3)/(343*x **6 - 588*x**5 + 189*x**4 + 104*x**3 - 27*x**2 - 12*x - 1), x) - Integral( x**2*sqrt(5*x**2 + 2*x + 3)/(343*x**6 - 588*x**5 + 189*x**4 + 104*x**3 - 2 7*x**2 - 12*x - 1), x)
\[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\int { -\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (x^{2} + 5 \, x + 2\right )}}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{3}} \,d x } \] Input:
integrate((x^2+5*x+2)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^3,x, algorithm="m axima")
Output:
-integrate(sqrt(5*x^2 + 2*x + 3)*(x^2 + 5*x + 2)/(7*x^2 - 4*x - 1)^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (160) = 320\).
Time = 0.17 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.77 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\frac {6200558 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{7} - 835775 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{6} - 190947036 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{5} - 92732607 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} + 816321374 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} + 419437335 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} - 765111048 \, \sqrt {5} x - 376983161 \, \sqrt {5} + 765111048 \, \sqrt {5 \, x^{2} + 2 \, x + 3}}{430276 \, {\left (7 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} - 8 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} - 70 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} + 83\right )}^{2}} + 0.139051039089282 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.138209741946053 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 0.139051039089282 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.138209741946053 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \] Input:
integrate((x^2+5*x+2)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^3,x, algorithm="g iac")
Output:
1/430276*(6200558*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^7 - 835775*sqrt(5)*( sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^6 - 190947036*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^5 - 92732607*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^4 + 81 6321374*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^3 + 419437335*sqrt(5)*(sqrt(5) *x - sqrt(5*x^2 + 2*x + 3))^2 - 765111048*sqrt(5)*x - 376983161*sqrt(5) + 765111048*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 - sqrt(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)^ 2 + 0.139051039089282*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 4.419247364 59000) - 0.138209741946053*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.2529 5163054000) - 0.139051039089282*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 1 .02258038113000) + 0.138209741946053*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3 ) - 2.09411235400000)
Timed out. \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\int \frac {\left (x^2+5\,x+2\right )\,\sqrt {5\,x^2+2\,x+3}}{{\left (-7\,x^2+4\,x+1\right )}^3} \,d x \] Input:
int(((5*x + x^2 + 2)*(2*x + 5*x^2 + 3)^(1/2))/(4*x - 7*x^2 + 1)^3,x)
Output:
int(((5*x + x^2 + 2)*(2*x + 5*x^2 + 3)^(1/2))/(4*x - 7*x^2 + 1)^3, x)
Time = 69.45 (sec) , antiderivative size = 2193, normalized size of antiderivative = 10.30 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((x^2+5*x+2)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^3,x)
Output:
( - 1356859147*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 + 1550696168*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 - 55382006*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 - 221528024*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 - 276910 03*sqrt(17*sqrt(11) - 125)*sqrt(22)*atan((24*sqrt(5*x**2 + 2*x + 3)*sqrt(1 7*sqrt(11) - 125)*sqrt(22)*x - 19*sqrt(5*x**2 + 2*x + 3)*sqrt(17*sqrt(1...