Integrand size = 35, antiderivative size = 227 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx=-\frac {3 (40-371 x) \sqrt {3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac {7 (409769-1189370 x) \sqrt {3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}-\frac {7 \left (39370231-2538725 \sqrt {11}\right ) \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{124902976 \sqrt {22 \left (125-17 \sqrt {11}\right )}}+\frac {7 \left (39370231+2538725 \sqrt {11}\right ) \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{124902976 \sqrt {22 \left (125+17 \sqrt {11}\right )}} \] Output:
-3/11176*(40-371*x)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^2-7*(409769-1189370 *x)*(5*x^2+2*x+3)^(1/2)/(-437160416*x^2+249805952*x+62451488)-7/124902976* (39370231-2538725*11^(1/2))*arctanh((23-11^(1/2)+(17-5*11^(1/2))*x)/(250-3 4*11^(1/2))^(1/2)/(5*x^2+2*x+3)^(1/2))/(2750-374*11^(1/2))^(1/2)+7/1249029 76*(39370231+2538725*11^(1/2))*arctanh((23+11^(1/2)+(17+5*11^(1/2))*x)/(25 0+34*11^(1/2))^(1/2)/(5*x^2+2*x+3)^(1/2))/(2750+374*11^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.98 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.91 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx=\frac {\frac {235298 \sqrt {3+2 x+5 x^2} \left (-3538943+3071502 x+53381041 x^2-58279130 x^3\right )}{\left (1+4 x-7 x^2\right )^2}-1796775175713 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]+11176 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {10486671792 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+6928653865 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]-3 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {36376673721218 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+26508461599305 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{14694710223424} \] Input:
Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
((235298*Sqrt[3 + 2*x + 5*x^2]*(-3538943 + 3071502*x + 53381041*x^2 - 5827 9130*x^3))/(1 + 4*x - 7*x^2)^2 - 1796775175713*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) & ] + 11176*Ro otSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (10486671 792*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 6928653865 *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) & ] - 3*RootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (36376673721218*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqr t[3 + 2*x + 5*x^2] - #1]*#1 + 26508461599305*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) & ] )/14694710223424
Time = 0.86 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int \frac {x^2+5 x+2}{\left (-7 x^2+4 x+1\right )^3 \sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle -\frac {\int -\frac {8 \left (11130 x^2+10125 x+16253\right )}{\left (-7 x^2+4 x+1\right )^2 \sqrt {5 x^2+2 x+3}}dx}{89408}-\frac {3 \sqrt {5 x^2+2 x+3} (40-371 x)}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11130 x^2+10125 x+16253}{\left (-7 x^2+4 x+1\right )^2 \sqrt {5 x^2+2 x+3}}dx}{11176}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {-\frac {\int -\frac {8 (17771075 x+34292781)}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx}{44704}-\frac {7 \sqrt {5 x^2+2 x+3} (409769-1189370 x)}{5588 \left (-7 x^2+4 x+1\right )}}{11176}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {17771075 x+34292781}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx}{5588}-\frac {7 (409769-1189370 x) \sqrt {5 x^2+2 x+3}}{5588 \left (-7 x^2+4 x+1\right )}}{11176}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {\frac {\frac {7}{11} \left (27925975-39370231 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {7}{11} \left (27925975+39370231 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx}{5588}-\frac {7 (409769-1189370 x) \sqrt {5 x^2+2 x+3}}{5588 \left (-7 x^2+4 x+1\right )}}{11176}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {7}{22} \left (27925975-39370231 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {7}{22} \left (27925975+39370231 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx}{5588}-\frac {7 (409769-1189370 x) \sqrt {5 x^2+2 x+3}}{5588 \left (-7 x^2+4 x+1\right )}}{11176}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {-\frac {7}{11} \left (27925975-39370231 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {7}{11} \left (27925975+39370231 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )}{5588}-\frac {7 (409769-1189370 x) \sqrt {5 x^2+2 x+3}}{5588 \left (-7 x^2+4 x+1\right )}}{11176}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {7 \left (27925975-39370231 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {7 \left (27925975+39370231 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}}{5588}-\frac {7 (409769-1189370 x) \sqrt {5 x^2+2 x+3}}{5588 \left (-7 x^2+4 x+1\right )}}{11176}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}\) |
Input:
Int[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
(-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])/(5588*(1 + 4*x - 7*x^2)) + (( 7*(27925975 - 39370231*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*(27925975 + 39370231*Sqrt[11])*ArcTanh[(23 + Sqrt[1 1] + (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])]))/5588)/11176
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 = 1.10 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.02
| method | result | size |
| risch | \(-\frac {\left (58279130 x^{3}-53381041 x^{2}-3071502 x +3538943\right ) \sqrt {5 x^{2}+2 x +3}}{62451488 \left (7 x^{2}-4 x -1\right )^{2}}+\frac {7 \left (-39370231+2538725 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{1373932736 \sqrt {250-34 \sqrt {11}}}+\frac {7 \left (39370231+2538725 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{1373932736 \sqrt {250+34 \sqrt {11}}}\) | \(231\) |
| trager | \(\text {Expression too large to display}\) | \(482\) |
| default | \(\text {Expression too large to display}\) | \(1194\) |
Input:
int((x^2+5*x+2)/(-7*x^2+4*x+1)^3/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBO SE)
Output:
-1/62451488*(58279130*x^3-53381041*x^2-3071502*x+3538943)/(7*x^2-4*x-1)^2* (5*x^2+2*x+3)^(1/2)+7/1373932736*(-39370231+2538725*11^(1/2))*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))+7 /1373932736*(39370231+2538725*11^(1/2))*11^(1/2)/(250+34*11^(1/2))^(1/2)*a rctanh(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. 370 vs. \(2 (174) = 348\).
Time = 0.10 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.63 \[ \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 {{\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} \log \left (-\frac {2 \, \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} {\left (358684877 \, \sqrt {11} + 2940638404\right )} + 5176915513390201 \, \sqrt {11} {\left (x + 3\right )} - 15530746540170603 \, x + 25884577566951005}{x}\right ) - {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} \log \left (\frac {2 \, \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} {\left (358684877 \, \sqrt {11} + 2940638404\right )} - 5176915513390201 \, \sqrt {11} {\left (x + 3\right )} + 15530746540170603 \, x - 25884577566951005}{x}\right ) + {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} \log \left (\frac {2 \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (358684877 \, \sqrt {11} - 2940638404\right )} \sqrt {-\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} + 5176915513390201 \, \sqrt {11} {\left (x + 3\right )} + 15530746540170603 \, x - 25884577566951005}{x}\right ) - {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} \log \left (-\frac {2 \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (358684877 \, \sqrt {11} - 2940638404\right )} \sqrt {-\frac {116724881545921}{254} \, \sqrt {11} + \frac {82616280769148425}{2794}} - 5176915513390201 \, \sqrt {11} {\left (x + 3\right )} - 15530746540170603 \, x + 25884577566951005}{x}\right ) + 4 \, {\left (58279130 \, x^{3} - 53381041 \, x^{2} - 3071502 \, x + 3538943\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{249805952 \, {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )}} \] Input:
integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^3/(5*x^2+2*x+3)^(1/2),x, algorithm="f ricas")
Output:
-1/249805952*((49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(116724881545921/254 *sqrt(11) + 82616280769148425/2794)*log(-(2*sqrt(5*x^2 + 2*x + 3)*sqrt(116 724881545921/254*sqrt(11) + 82616280769148425/2794)*(358684877*sqrt(11) + 2940638404) + 5176915513390201*sqrt(11)*(x + 3) - 15530746540170603*x + 25 884577566951005)/x) - (49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(11672488154 5921/254*sqrt(11) + 82616280769148425/2794)*log((2*sqrt(5*x^2 + 2*x + 3)*s qrt(116724881545921/254*sqrt(11) + 82616280769148425/2794)*(358684877*sqrt (11) + 2940638404) - 5176915513390201*sqrt(11)*(x + 3) + 15530746540170603 *x - 25884577566951005)/x) + (49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-116 724881545921/254*sqrt(11) + 82616280769148425/2794)*log((2*sqrt(5*x^2 + 2* x + 3)*(358684877*sqrt(11) - 2940638404)*sqrt(-116724881545921/254*sqrt(11 ) + 82616280769148425/2794) + 5176915513390201*sqrt(11)*(x + 3) + 15530746 540170603*x - 25884577566951005)/x) - (49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)* sqrt(-116724881545921/254*sqrt(11) + 82616280769148425/2794)*log(-(2*sqrt( 5*x^2 + 2*x + 3)*(358684877*sqrt(11) - 2940638404)*sqrt(-116724881545921/2 54*sqrt(11) + 82616280769148425/2794) - 5176915513390201*sqrt(11)*(x + 3) - 15530746540170603*x + 25884577566951005)/x) + 4*(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)
\[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx=- \int \frac {5 x}{343 x^{6} \sqrt {5 x^{2} + 2 x + 3} - 588 x^{5} \sqrt {5 x^{2} + 2 x + 3} + 189 x^{4} \sqrt {5 x^{2} + 2 x + 3} + 104 x^{3} \sqrt {5 x^{2} + 2 x + 3} - 27 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 12 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {x^{2}}{343 x^{6} \sqrt {5 x^{2} + 2 x + 3} - 588 x^{5} \sqrt {5 x^{2} + 2 x + 3} + 189 x^{4} \sqrt {5 x^{2} + 2 x + 3} + 104 x^{3} \sqrt {5 x^{2} + 2 x + 3} - 27 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 12 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {2}{343 x^{6} \sqrt {5 x^{2} + 2 x + 3} - 588 x^{5} \sqrt {5 x^{2} + 2 x + 3} + 189 x^{4} \sqrt {5 x^{2} + 2 x + 3} + 104 x^{3} \sqrt {5 x^{2} + 2 x + 3} - 27 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 12 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:
integrate((x**2+5*x+2)/(-7*x**2+4*x+1)**3/(5*x**2+2*x+3)**(1/2),x)
Output:
-Integral(5*x/(343*x**6*sqrt(5*x**2 + 2*x + 3) - 588*x**5*sqrt(5*x**2 + 2* x + 3) + 189*x**4*sqrt(5*x**2 + 2*x + 3) + 104*x**3*sqrt(5*x**2 + 2*x + 3) - 27*x**2*sqrt(5*x**2 + 2*x + 3) - 12*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x **2 + 2*x + 3)), x) - Integral(x**2/(343*x**6*sqrt(5*x**2 + 2*x + 3) - 588 *x**5*sqrt(5*x**2 + 2*x + 3) + 189*x**4*sqrt(5*x**2 + 2*x + 3) + 104*x**3* sqrt(5*x**2 + 2*x + 3) - 27*x**2*sqrt(5*x**2 + 2*x + 3) - 12*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x) - Integral(2/(343*x**6*sqrt(5*x* *2 + 2*x + 3) - 588*x**5*sqrt(5*x**2 + 2*x + 3) + 189*x**4*sqrt(5*x**2 + 2 *x + 3) + 104*x**3*sqrt(5*x**2 + 2*x + 3) - 27*x**2*sqrt(5*x**2 + 2*x + 3) - 12*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x)
\[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx=\int { -\frac {x^{2} + 5 \, x + 2}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{3} \sqrt {5 \, x^{2} + 2 \, x + 3}} \,d x } \] Input:
integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^3/(5*x^2+2*x+3)^(1/2),x, algorithm="m axima")
Output:
-integrate((x^2 + 5*x + 2)/((7*x^2 - 4*x - 1)^3*sqrt(5*x^2 + 2*x + 3)), x)
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (174) = 348\).
Time = 0.17 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.67 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx=\frac {124397525 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{7} + 26796567 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{6} - 3595807617 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{5} - 1719888775 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} + 17096132999 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} + 8328401413 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} - 16383202915 \, \sqrt {5} x - 7800623485 \, \sqrt {5} + 16383202915 \, \sqrt {5 \, x^{2} + 2 \, x + 3}}{31225744 \, {\left (7 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} - 8 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} - 70 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} + 83\right )}^{2}} + 0.0423989586659649 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.0446437606655585 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 0.0423989586659649 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.0446437606655585 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \] Input:
integrate((x^2+5*x+2)/(-7*x^2+4*x+1)^3/(5*x^2+2*x+3)^(1/2),x, algorithm="g iac")
Output:
1/31225744*(124397525*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^7 + 26796567*sqr t(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^6 - 3595807617*(sqrt(5)*x - sqrt( 5*x^2 + 2*x + 3))^5 - 1719888775*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3 ))^4 + 17096132999*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^3 + 8328401413*sqrt (5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^2 - 16383202915*sqrt(5)*x - 780062 3485*sqrt(5) + 16383202915*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*(s qrt(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 + 0.0423989586659649*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 4.41924736459000) - 0.0446437606655585*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.25295163054000) - 0.0423989586659649*log(-sqrt(5)*x + sqrt( 5*x^2 + 2*x + 3) - 1.02258038113000) + 0.0446437606655585*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 2.09411235400000)
Timed out. \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x^2+5\,x+2}{\sqrt {5\,x^2+2\,x+3}\,{\left (-7\,x^2+4\,x+1\right )}^3} \,d x \] 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 = 68.04 (sec) , antiderivative size = 2193, normalized size of antiderivative = 9.66 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx =\text {Too large to display} \] Input:
int((x^2+5*x+2)/(-7*x^2+4*x+1)^3/(5*x^2+2*x+3)^(1/2),x)
Output:
( - 108940163850*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)*sqr t(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)*sqr t(2))/(8890*x**2 + 3556*x + 5334))*x**4 + 124503044400*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*s qrt(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 - 4446537300*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)*sqr t(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)*sqr t(2))/(8890*x**2 + 3556*x + 5334))*x**2 - 17786149200*sqrt(17*sqrt(11) - 1 25)*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*sq rt(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 - 2223268650*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...