Integrand size = 35, antiderivative size = 234 \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^3} \, dx=-\frac {(9495-37088 x) \sqrt {3+2 x+5 x^2}}{23716 \left (1+4 x-7 x^2\right )}+\frac {3 (3+61 x) \left (3+2 x+5 x^2\right )^{3/2}}{308 \left (1+4 x-7 x^2\right )^2}-\frac {5}{343} \sqrt {5} \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )-\frac {\sqrt {\frac {62294197250171-2085440742055 \sqrt {11}}{2794}} \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{332024}+\frac {\sqrt {\frac {62294197250171+2085440742055 \sqrt {11}}{2794}} \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{332024} \]
3/308*(3+61*x)*(5*x^2+2*x+3)^(3/2)/(-7*x^2+4*x+1)^2-5/343*arcsinh(1/14*(1+ 5*x)*14^(1/2))*5^(1/2)-1/23716*(9495-37088*x)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+ 4*x+1)-1/927675056*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))*(174049987116977774-5826721433301670*11^(1/2 ))^(1/2)+1/927675056*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))*(174049987116977774+5826721433301670*11^(1 /2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.81 (sec) , antiderivative size = 636, normalized size of antiderivative = 2.72 \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (-7416+42767 x+246464 x^2-189161 x^3\right )}{23716 \left (-1-4 x+7 x^2\right )^2}+\frac {5}{343} \sqrt {5} \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )-\frac {\text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {4506829 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )-1320270 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+64435 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{33614 \sqrt {5}}+\frac {\text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {-16323208013227 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+151120773150070 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+21832390993791 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{71748713246 \sqrt {5}}-\frac {3 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {-4192656948824863 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+24518831643829090 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+3523608887504055 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{34726377211064 \sqrt {5}} \]
(Sqrt[3 + 2*x + 5*x^2]*(-7416 + 42767*x + 246464*x^2 - 189161*x^3))/(23716 *(-1 - 4*x + 7*x^2)^2) + (5*Sqrt[5]*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/343 - RootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*# 1^4 & , (4506829*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] - 1320270*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 64435*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) & ]/(33614*Sqrt[5]) + RootSum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (-16323208013227*Sqrt[5]*Log[-(Sq rt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] + 151120773150070*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 21832390993791*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) & ]/(71748713246*Sqrt[5]) - (3*RootSum[83 - 16*Sqrt[5]*#1 - 70*# 1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (-4192656948824863*Sqrt[5]*Log[-(Sqrt[5] *x) + Sqrt[3 + 2*x + 5*x^2] - #1] + 24518831643829090*Log[-(Sqrt[5]*x) + S qrt[3 + 2*x + 5*x^2] - #1]*#1 + 3523608887504055*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) & ])/(34726377211064*Sqrt[5])
Time = 0.65 (sec) , antiderivative size = 266, 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, 2132, 27, 2143, 25, 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.
| \(\displaystyle \int \frac {\left (x^2+5 x+2\right ) \left (5 x^2+2 x+3\right )^{3/2}}{\left (-7 x^2+4 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2132 |
| \(\displaystyle \frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}-\frac {1}{616} \int -\frac {4 \left (-110 x^2+163 x+744\right ) \sqrt {5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{154} \int \frac {\left (-110 x^2+163 x+744\right ) \sqrt {5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^2}dx+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 2132 |
| \(\displaystyle \frac {1}{154} \left (-\frac {1}{308} \int -\frac {2 \left (12100 x^2+89403 x+128019\right )}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {\sqrt {5 x^2+2 x+3} (9495-37088 x)}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \int \frac {12100 x^2+89403 x+128019}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (-\frac {12100}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx-\frac {1}{7} \int -\frac {674221 x+908233}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (\frac {1}{7} \int \frac {674221 x+908233}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {12100}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (\frac {1}{7} \int \frac {674221 x+908233}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {605}{7} \sqrt {\frac {10}{7}} \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (\frac {1}{7} \int \frac {674221 x+908233}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {2420}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (\frac {1}{7} \left (\frac {1}{11} \left (7416431-7706073 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{11} \left (7416431+7706073 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {2420}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (\frac {1}{7} \left (\frac {1}{22} \left (7416431-7706073 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{22} \left (7416431+7706073 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {2420}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (\frac {1}{7} \left (-\frac {1}{11} \left (7416431-7706073 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {1}{11} \left (7416431+7706073 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )\right )-\frac {2420}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{154} \left (\frac {1}{154} \left (\frac {1}{7} \left (\frac {\left (7416431-7706073 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {\left (7416431+7706073 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}\right )-\frac {2420}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )-\frac {(9495-37088 x) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}\right )+\frac {3 (61 x+3) \left (5 x^2+2 x+3\right )^{3/2}}{308 \left (-7 x^2+4 x+1\right )^2}\) |
(3*(3 + 61*x)*(3 + 2*x + 5*x^2)^(3/2))/(308*(1 + 4*x - 7*x^2)^2) + (-1/154 *((9495 - 37088*x)*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + ((-2420*Sqrt [5]*ArcSinh[(2 + 10*x)/(2*Sqrt[14])])/7 + (((7416431 - 7706073*Sqrt[11])*A rcTanh[(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])]) + ((7416431 + 77 06073*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)/154)/154
3.4.85.3.1 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)*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 = 0.53 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(-\frac {\left (189161 x^{3}-246464 x^{2}-42767 x +7416\right ) \sqrt {5 x^{2}+2 x +3}}{23716 \left (7 x^{2}-4 x -1\right )^{2}}-\frac {5 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{343}+\frac {\left (7706073+674221 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{3652264 \sqrt {250+34 \sqrt {11}}}+\frac {\left (-7706073+674221 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{3652264 \sqrt {250-34 \sqrt {11}}}\) | \(245\) |
| trager | \(\text {Expression too large to display}\) | \(524\) |
| default | \(\text {Expression too large to display}\) | \(3828\) |
-1/23716*(189161*x^3-246464*x^2-42767*x+7416)/(7*x^2-4*x-1)^2*(5*x^2+2*x+3 )^(1/2)-5/343*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))+1/3652264*(7706073+67 4221*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))+1/3652264*(-7706073+674221*11^(1/2))*11^(1/2)/(2 50-34*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. 447 vs. \(2 (176) = 352\).
Time = 0.29 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.91 \[ \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 )^3} \, dx=-\frac {\sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} {\left (11840590 \, \sqrt {11} - 83479737\right )} + 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} + 16280013607680207 \, x - 27133356012800345}{x}\right ) - \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {2085440742055 \, \sqrt {11} + 62294197250171} {\left (11840590 \, \sqrt {11} - 83479737\right )} - 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} - 16280013607680207 \, x + 27133356012800345}{x}\right ) + \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (11840590 \, \sqrt {11} + 83479737\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} + 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} - 16280013607680207 \, x + 27133356012800345}{x}\right ) - \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (11840590 \, \sqrt {11} + 83479737\right )} \sqrt {-2085440742055 \, \sqrt {11} + 62294197250171} - 5426671202560069 \, \sqrt {11} {\left (x + 3\right )} + 16280013607680207 \, x - 27133356012800345}{x}\right ) - 13522960 \, \sqrt {5} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, 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 ) + 78232 \, {\left (189161 \, x^{3} - 246464 \, x^{2} - 42767 \, x + 7416\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{1855350112 \, {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )}} \]
-1/1855350112*(sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(2085440 742055*sqrt(11) + 62294197250171)*log((sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*sq rt(2085440742055*sqrt(11) + 62294197250171)*(11840590*sqrt(11) - 83479737) + 5426671202560069*sqrt(11)*(x + 3) + 16280013607680207*x - 2713335601280 0345)/x) - sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(20854407420 55*sqrt(11) + 62294197250171)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt( 2085440742055*sqrt(11) + 62294197250171)*(11840590*sqrt(11) - 83479737) - 5426671202560069*sqrt(11)*(x + 3) - 16280013607680207*x + 2713335601280034 5)/x) + sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-2085440742055 *sqrt(11) + 62294197250171)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*(118405 90*sqrt(11) + 83479737)*sqrt(-2085440742055*sqrt(11) + 62294197250171) + 5 426671202560069*sqrt(11)*(x + 3) - 16280013607680207*x + 27133356012800345 )/x) - sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-2085440742055* sqrt(11) + 62294197250171)*log((sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*(11840590 *sqrt(11) + 83479737)*sqrt(-2085440742055*sqrt(11) + 62294197250171) - 542 6671202560069*sqrt(11)*(x + 3) + 16280013607680207*x - 27133356012800345)/ x) - 13522960*sqrt(5)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*log(sqrt(5)*sqrt (5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) + 78232*(189161*x^3 - 246 464*x^2 - 42767*x + 7416)*sqrt(5*x^2 + 2*x + 3))/(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)
\[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^3} \, dx=- \int \frac {6 \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {19 x \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {23 x^{2} \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {27 x^{3} \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx - \int \frac {5 x^{4} \sqrt {5 x^{2} + 2 x + 3}}{343 x^{6} - 588 x^{5} + 189 x^{4} + 104 x^{3} - 27 x^{2} - 12 x - 1}\, dx \]
-Integral(6*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(19*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 (23*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), x) - Integral(27*x**3*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**4*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)
\[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\int { -\frac {{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} {\left (x^{2} + 5 \, x + 2\right )}}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (176) = 352\).
Time = 0.33 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.76 \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\frac {5}{343} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) + \frac {264327 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{7} - 3224225 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{6} - 87069759 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{5} - 36535763 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} + 416818149 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} + 204858869 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} - 411908789 \, \sqrt {5} x - 187277977 \, \sqrt {5} + 411908789 \, \sqrt {5 \, x^{2} + 2 \, x + 3}}{83006 \, {\left (7 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} - 8 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} - 70 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} + 83\right )}^{2}} + 0.474028359166807 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.424017987131739 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 0.474028359166807 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.424017987131739 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \]
5/343*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1) + 1/83 006*(264327*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^7 - 3224225*sqrt(5)*(sqrt( 5)*x - sqrt(5*x^2 + 2*x + 3))^6 - 87069759*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^5 - 36535763*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^4 + 41681814 9*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^3 + 204858869*sqrt(5)*(sqrt(5)*x - s qrt(5*x^2 + 2*x + 3))^2 - 411908789*sqrt(5)*x - 187277977*sqrt(5) + 411908 789*sqrt(5*x^2 + 2*x + 3))/(7*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^4 - 8*sq rt(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. 474028359166807*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 4.41924736459000) - 0.424017987131739*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.2529516305 4000) - 0.474028359166807*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 1.02258 038113000) + 0.424017987131739*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 2. 09411235400000)
Timed out. \[ \int \frac {\left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2}}{\left (1+4 x-7 x^2\right )^3} \, dx=\int \frac {\left (x^2+5\,x+2\right )\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{{\left (-7\,x^2+4\,x+1\right )}^3} \,d x \]