Integrand size = 35, antiderivative size = 222 \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\frac {(5826+3395 x) \sqrt {3+2 x+5 x^2}}{3773}+\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{154 \left (1+4 x-7 x^2\right )}+\frac {16691 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{2401 \sqrt {5}}-\frac {\sqrt {\frac {1}{22} \left (52175400311-13155376531 \sqrt {11}\right )} \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{26411}-\frac {\sqrt {\frac {1}{22} \left (52175400311+13155376531 \sqrt {11}\right )} \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{26411} \]
3/154*(3+61*x)*(5*x^2+2*x+3)^(3/2)/(-7*x^2+4*x+1)+16691/12005*arcsinh(1/14 *(1+5*x)*14^(1/2))*5^(1/2)+1/3773*(5826+3395*x)*(5*x^2+2*x+3)^(1/2)-1/5810 42*arctanh((23+x*(17-5*11^(1/2))-11^(1/2))/(5*x^2+2*x+3)^(1/2)/(250-34*11^ (1/2))^(1/2))*(1147858806842-289418283682*11^(1/2))^(1/2)-1/581042*arctanh ((23+11^(1/2)+x*(17+5*11^(1/2)))/(5*x^2+2*x+3)^(1/2)/(250+34*11^(1/2))^(1/ 2))*(1147858806842+289418283682*11^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.70 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.01 \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\frac {\frac {1715 \sqrt {3+2 x+5 x^2} \left (-12975-81181 x+34265 x^2+2695 x^3\right )}{-1-4 x+7 x^2}-17992898 \sqrt {5} \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )+44 \sqrt {5} \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {25954129 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )-19416530 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2717099 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]-6 \sqrt {5} \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {225782939 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )-137400830 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+7775369 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{12941390} \]
((1715*Sqrt[3 + 2*x + 5*x^2]*(-12975 - 81181*x + 34265*x^2 + 2695*x^3))/(- 1 - 4*x + 7*x^2) - 17992898*Sqrt[5]*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]] + 44*Sqrt[5]*RootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (25954129*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] - 19416530*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 271709 9*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) & ] - 6*Sqrt[5]*RootSum[83 - 16*Sqrt[5 ]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (225782939*Sqrt[5]*Log[-(Sqrt [5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] - 137400830*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 7775369*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) & ])/ 12941390
Time = 0.64 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {2132, 27, 2138, 27, 2143, 27, 1090, 222, 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.
| \(\displaystyle \int \frac {\left (x^2+5 x+2\right ) \left (5 x^2+2 x+3\right )^{3/2}}{\left (-7 x^2+4 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2132 |
| \(\displaystyle \frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}-\frac {1}{308} \int -\frac {4 \left (-970 x^2-181 x+228\right ) \sqrt {5 x^2+2 x+3}}{-7 x^2+4 x+1}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \int \frac {\left (-970 x^2-181 x+228\right ) \sqrt {5 x^2+2 x+3}}{-7 x^2+4 x+1}dx+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} (3395 x+5826) \sqrt {5 x^2+2 x+3}-\frac {1}{490} \int -\frac {10 \left (-183601 x^2-107622 x+17505\right )}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \int \frac {-183601 x^2-107622 x+17505}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx-\frac {1}{7} \int \frac {2 (743879 x+30533)}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx-\frac {2}{7} \int \frac {743879 x+30533}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601 \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)}{14 \sqrt {70}}-\frac {2}{7} \int \frac {743879 x+30533}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}-\frac {2}{7} \int \frac {743879 x+30533}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}-\frac {2}{7} \left (\frac {1}{11} \left (8182669-1701489 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{11} \left (8182669+1701489 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}-\frac {2}{7} \left (\frac {1}{22} \left (8182669-1701489 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{22} \left (8182669+1701489 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}-\frac {2}{7} \left (-\frac {1}{11} \left (8182669-1701489 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {1}{11} \left (8182669+1701489 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )\right )\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{49} \left (\frac {183601 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}-\frac {2}{7} \left (\frac {\left (8182669-1701489 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {\left (8182669+1701489 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}\right )\right )+\frac {1}{49} \sqrt {5 x^2+2 x+3} (3395 x+5826)\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{154 \left (-7 x^2+4 x+1\right )}\) |
(3*(3 + 61*x)*(3 + 2*x + 5*x^2)^(3/2))/(154*(1 + 4*x - 7*x^2)) + (((5826 + 3395*x)*Sqrt[3 + 2*x + 5*x^2])/49 + ((183601*ArcSinh[(2 + 10*x)/(2*Sqrt[1 4])])/(7*Sqrt[5]) - (2*(((8182669 - 1701489*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])]) + ((8182669 + 1701489*Sqrt[11])*Arc Tanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sq rt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 + 17*Sqrt[11])])))/7)/49)/77
3.4.84.3.1 Defintions of rubi rules used
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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 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[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Si mp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2* A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*( c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C* d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2* p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ .)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 1/c Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x ^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Time = 0.51 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.10
| method | result | size |
| risch | \(\frac {\left (2695 x^{3}+34265 x^{2}-81181 x -12975\right ) \sqrt {5 x^{2}+2 x +3}}{52822 x^{2}-30184 x -7546}+\frac {16691 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{12005}-\frac {\left (1701489+743879 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{290521 \sqrt {250+34 \sqrt {11}}}-\frac {\left (-1701489+743879 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{290521 \sqrt {250-34 \sqrt {11}}}\) | \(245\) |
| trager | \(\text {Expression too large to display}\) | \(522\) |
| default | \(\text {Expression too large to display}\) | \(1828\) |
1/7546*(2695*x^3+34265*x^2-81181*x-12975)/(7*x^2-4*x-1)*(5*x^2+2*x+3)^(1/2 )+16691/12005*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))-1/290521*(1701489+743 879*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))-1/290521*(-1701489+743879*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. 378 vs. \(2 (164) = 328\).
Time = 0.28 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.70 \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\frac {5 \, \sqrt {11} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {26310753062 \, \sqrt {11} + 104350800622} \log \left (\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {26310753062 \, \sqrt {11} + 104350800622} {\left (16206 \, \sqrt {11} - 68441\right )} + 1795191685 \, \sqrt {11} {\left (x + 3\right )} + 5385575055 \, x - 8975958425}{x}\right ) - 5 \, \sqrt {11} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {26310753062 \, \sqrt {11} + 104350800622} \log \left (-\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {26310753062 \, \sqrt {11} + 104350800622} {\left (16206 \, \sqrt {11} - 68441\right )} - 1795191685 \, \sqrt {11} {\left (x + 3\right )} - 5385575055 \, x + 8975958425}{x}\right ) - 5 \, \sqrt {11} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-26310753062 \, \sqrt {11} + 104350800622} \log \left (-\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (16206 \, \sqrt {11} + 68441\right )} \sqrt {-26310753062 \, \sqrt {11} + 104350800622} + 1795191685 \, \sqrt {11} {\left (x + 3\right )} - 5385575055 \, x + 8975958425}{x}\right ) + 5 \, \sqrt {11} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-26310753062 \, \sqrt {11} + 104350800622} \log \left (\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (16206 \, \sqrt {11} + 68441\right )} \sqrt {-26310753062 \, \sqrt {11} + 104350800622} - 1795191685 \, \sqrt {11} {\left (x + 3\right )} + 5385575055 \, x - 8975958425}{x}\right ) + 4039222 \, \sqrt {5} {\left (7 \, x^{2} - 4 \, x - 1\right )} \log \left (-\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) + 770 \, {\left (2695 \, x^{3} + 34265 \, x^{2} - 81181 \, x - 12975\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{5810420 \, {\left (7 \, x^{2} - 4 \, x - 1\right )}} \]
1/5810420*(5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(26310753062*sqrt(11) + 104350 800622)*log((sqrt(5*x^2 + 2*x + 3)*sqrt(26310753062*sqrt(11) + 10435080062 2)*(16206*sqrt(11) - 68441) + 1795191685*sqrt(11)*(x + 3) + 5385575055*x - 8975958425)/x) - 5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(26310753062*sqrt(11) + 104350800622)*log(-(sqrt(5*x^2 + 2*x + 3)*sqrt(26310753062*sqrt(11) + 104 350800622)*(16206*sqrt(11) - 68441) - 1795191685*sqrt(11)*(x + 3) - 538557 5055*x + 8975958425)/x) - 5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(-26310753062*s qrt(11) + 104350800622)*log(-(sqrt(5*x^2 + 2*x + 3)*(16206*sqrt(11) + 6844 1)*sqrt(-26310753062*sqrt(11) + 104350800622) + 1795191685*sqrt(11)*(x + 3 ) - 5385575055*x + 8975958425)/x) + 5*sqrt(11)*(7*x^2 - 4*x - 1)*sqrt(-263 10753062*sqrt(11) + 104350800622)*log((sqrt(5*x^2 + 2*x + 3)*(16206*sqrt(1 1) + 68441)*sqrt(-26310753062*sqrt(11) + 104350800622) - 1795191685*sqrt(1 1)*(x + 3) + 5385575055*x - 8975958425)/x) + 4039222*sqrt(5)*(7*x^2 - 4*x - 1)*log(-sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) + 7 70*(2695*x^3 + 34265*x^2 - 81181*x - 12975)*sqrt(5*x^2 + 2*x + 3))/(7*x^2 - 4*x - 1)
\[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\int \frac {\left (x^{2} + 5 x + 2\right ) \left (5 x^{2} + 2 x + 3\right )^{\frac {3}{2}}}{\left (7 x^{2} - 4 x - 1\right )^{2}}\, dx \]
\[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\int { \frac {{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} {\left (x^{2} + 5 \, x + 2\right )}}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{63274455776,[8]%%%}+%%%{%%{[144627327488,0]:[1,0,-5]%%},[7 ]%%%}+%%%
Timed out. \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\int \frac {\left (x^2+5\,x+2\right )\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{{\left (-7\,x^2+4\,x+1\right )}^2} \,d x \]