Integrand size = 22, antiderivative size = 240 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {1+2 x} (37+20 x)}{434 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (9227+7920 x)}{94178 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{47089}+\frac {3 \sqrt {\frac {1}{434} \left (-250141922+64681225 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right )}{47089} \] Output:
1/434*(1+2*x)^(1/2)*(37+20*x)/(5*x^2+3*x+2)^2+(1+2*x)^(1/2)*(9227+7920*x)/ (470890*x^2+282534*x+188356)-3/20436626*(108561594148+28071651650*35^(1/2) )^(1/2)*arctan(((20+10*35^(1/2))^(1/2)-10*(1+2*x)^(1/2))/(-20+10*35^(1/2)) ^(1/2))+3/20436626*(108561594148+28071651650*35^(1/2))^(1/2)*arctan(((20+1 0*35^(1/2))^(1/2)+10*(1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2))+3/20436626*(- 108561594148+28071651650*35^(1/2))^(1/2)*arctanh((20+10*35^(1/2))^(1/2)*(1 +2*x)^(1/2)/(5+35^(1/2)+10*x))
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {217 \sqrt {1+2 x} \left (26483+47861 x+69895 x^2+39600 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {217 \left (250141922+52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {217 \left (250141922-52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{10218313} \] Input:
Integrate[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^3),x]
Output:
((217*Sqrt[1 + 2*x]*(26483 + 47861*x + 69895*x^2 + 39600*x^3))/(2*(2 + 3*x + 5*x^2)^2) + 3*Sqrt[217*(250141922 + (52010281*I)*Sqrt[31])]*ArcTan[Sqrt [(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 3*Sqrt[217*(250141922 - (52010281*I )*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/10218313
Time = 0.65 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.46, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1165, 1235, 27, 1197, 27, 1483, 27, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {1}{434} \int \frac {100 x+271}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}dx+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{434} \left (\frac {1}{217} \int \frac {3 (2640 x+8699)}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{434} \left (\frac {3}{217} \int \frac {2640 x+8699}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {1}{434} \left (\frac {6}{217} \int \frac {2 (1320 (2 x+1)+7379)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \int \frac {1320 (2 x+1)+7379}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \left (\frac {\int \frac {5 \left (7379 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (7379-264 \sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (7379-264 \sqrt {35}\right ) \sqrt {2 x+1}+7379 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \left (\frac {5 \int \frac {7379 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (7379-264 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (7379-264 \sqrt {35}\right ) \sqrt {2 x+1}+7379 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \left (\frac {5 \left (\frac {1}{10} \sqrt {2501419220+646812250 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {2501419220+646812250 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \left (\frac {5 \left (\frac {1}{10} \sqrt {2501419220+646812250 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {2501419220+646812250 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \left (\frac {5 \left (\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {2501419220+646812250 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {2501419220+646812250 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \left (\frac {5 \left (\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {2501419220+646812250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {2501419220+646812250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{434} \left (\frac {12}{217} \left (\frac {5 \left (\frac {1}{5} \sqrt {\frac {2501419220+646812250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{5} \sqrt {\frac {2501419220+646812250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{10} \left (7379-264 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (7920 x+9227)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^3),x]
Output:
(Sqrt[1 + 2*x]*(37 + 20*x))/(434*(2 + 3*x + 5*x^2)^2) + ((Sqrt[1 + 2*x]*(9 227 + 7920*x))/(217*(2 + 3*x + 5*x^2)) + (12*((5*((Sqrt[(2501419220 + 6468 12250*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 1 0*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 - ((7379 - 264*Sqrt[35])*Log [Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10))/(2* Sqrt[14*(2 + Sqrt[35])]) + (5*((Sqrt[(2501419220 + 646812250*Sqrt[35])/(10 *(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqr t[10*(-2 + Sqrt[35])]])/5 + ((7379 - 264*Sqrt[35])*Log[Sqrt[35] + Sqrt[10* (2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10))/(2*Sqrt[14*(2 + Sqrt[35 ])])))/217)/434
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Time = 3.80 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.35
| method | result | size |
| pseudoelliptic | \(-\frac {7830000 \left (-\frac {2464 \left (\sqrt {5}\, \sqrt {7}-\frac {39}{4}\right ) \left (x^{3}+\frac {13979}{7920} x^{2}+\frac {4351}{3600} x +\frac {26483}{39600}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {1+2 x}}{83607}+\left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (-\frac {\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (2830555 \sqrt {5}-2042902 \sqrt {7}\right ) \left (\ln \left (\sqrt {35}+5+10 x -\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )-\ln \left (\sqrt {35}+5+10 x +\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{31101804}+\left (\sqrt {5}\, \sqrt {7}-\frac {700}{261}\right ) \left (\arctan \left (\frac {\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )-\arctan \left (\frac {\sqrt {20+10 \sqrt {35}}-10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )\right )\right )\right )}{343 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (4 \sqrt {5}\, \sqrt {7}-39\right ) \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )^{2} \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )^{2}}\) | \(324\) |
| trager | \(\frac {\left (39600 x^{3}+69895 x^{2}+47861 x +26483\right ) \sqrt {1+2 x}}{94178 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {6 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right ) \ln \left (-\frac {996641321984 x \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{5}-18716074679124032 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{3} x -119436855328569856 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{3}+7008069554162945520 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2} \sqrt {1+2 x}-458990644030336842880 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right ) x -1676273005765163471872 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )+5368987869213016478877125 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2} x +406172765 x +208041124}\right )}{47089}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2}+27140398537\right ) \ln \left (\frac {71188665856 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2}+27140398537\right ) \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{4} x +21852150152006112 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2}+27140398537\right ) x +8531203952040704 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2}+27140398537\right )+217250156179051311120 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2} \sqrt {1+2 x}+1637878602708070755480 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2}+27140398537\right ) x +1109535657155862471360 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2}+27140398537\right )-135134838449389184174582135 \sqrt {1+2 x}}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+248140786624 \textit {\_Z}^{2}+20918304337503125\right )^{2} x +94111079 x -208041124}\right )}{10218313}\) | \(458\) |
| risch | \(\frac {\left (39600 x^{3}+69895 x^{2}+47861 x +26483\right ) \sqrt {1+2 x}}{94178 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {23721 \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{5839036}+\frac {35997 \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626}-\frac {118605 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {35997 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {44274 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {23721 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{5839036}-\frac {35997 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626}-\frac {118605 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {35997 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {44274 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(643\) |
| derivativedivides | \(\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-91048526818200 \sqrt {5}+65791327714000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{27651265893650 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}+320925328420550\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-123371070933600 \sqrt {7}+152992435939000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{55302531787300 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x -\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}\right )^{2}}+\frac {\frac {3 \left (14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x -\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}+\frac {15 \left (-17842422 \sqrt {35}+64049720+\frac {\left (14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}-\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-91048526818200 \sqrt {5}+65791327714000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{20436626 \left (-2638398750+270605000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}+320925328420550\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-123371070933600 \sqrt {7}+152992435939000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626 \left (-5276797500+541210000 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x +\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}\right )^{2}}+\frac {\frac {3 \left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x +\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}+\frac {15 \left (-17842422 \sqrt {35}+64049720-\frac {\left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\) | \(691\) |
| default | \(\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-91048526818200 \sqrt {5}+65791327714000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{27651265893650 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}+320925328420550\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-123371070933600 \sqrt {7}+152992435939000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{55302531787300 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x -\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}\right )^{2}}+\frac {-\frac {3 \left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x -\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}-\frac {15 \left (17842422 \sqrt {35}-64049720+\frac {\left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}-\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {3 \left (-6045943503600+620096769600 \sqrt {5}\, \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{2765126589365 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-91048526818200 \sqrt {5}+65791327714000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{20436626 \left (-2638398750+270605000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-59423591568600 \sqrt {5}\, \sqrt {7}+320925328420550\right ) \sqrt {1+2 x}}{13825632946825 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-123371070933600 \sqrt {7}+152992435939000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626 \left (-5276797500+541210000 \sqrt {5}\, \sqrt {7}\right )}}{\left (\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x +\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}\right )^{2}}+\frac {\frac {3 \left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x +\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{40873252}+\frac {15 \left (-17842422 \sqrt {35}+64049720-\frac {\left (-14152775 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+10214510 \sqrt {7}\, \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{10218313 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\) | \(691\) |
Input:
int(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
-7830000/343*(-2464/83607*(5^(1/2)*7^(1/2)-39/4)*(x^3+13979/7920*x^2+4351/ 3600*x+26483/39600)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(1+2*x)^(1/2)+(x^2+3/5*x +2/5)^2*(-1/31101804*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2830555*5^(1/2)-204290 2*7^(1/2))*(ln(35^(1/2)+5+10*x-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))-ln(35 ^(1/2)+5+10*x+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2)))*(2*5^(1/2)*7^(1/2)+4) ^(1/2)+(5^(1/2)*7^(1/2)-700/261)*(arctan(((20+10*35^(1/2))^(1/2)+10*(1+2*x )^(1/2))/(-20+10*35^(1/2))^(1/2))-arctan(((20+10*35^(1/2))^(1/2)-10*(1+2*x )^(1/2))/(-20+10*35^(1/2))^(1/2)))))/(10*5^(1/2)*7^(1/2)-20)^(1/2)/(4*5^(1 /2)*7^(1/2)-39)/(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2 )*7^(1/2)+5+10*x)^2/(5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*( 1+2*x)^(1/2)+5+10*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (177) = 354\).
Time = 0.09 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {6 \, \sqrt {\frac {1}{434}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {452768575 \, \sqrt {\frac {5}{7}} + 250141922} \arctan \left (\frac {14}{52010281} \, \sqrt {\frac {1}{434}} {\left (\sqrt {\frac {1}{434}} \sqrt {452768575 \, \sqrt {\frac {5}{7}} - 250141922} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} + \sqrt {2 \, x + 1} {\left (7379 \, \sqrt {\frac {5}{7}} - 1320\right )}\right )} \sqrt {452768575 \, \sqrt {\frac {5}{7}} + 250141922}\right ) - 6 \, \sqrt {\frac {1}{434}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {452768575 \, \sqrt {\frac {5}{7}} + 250141922} \arctan \left (\frac {14}{52010281} \, \sqrt {\frac {1}{434}} {\left (\sqrt {\frac {1}{434}} \sqrt {452768575 \, \sqrt {\frac {5}{7}} - 250141922} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} - \sqrt {2 \, x + 1} {\left (7379 \, \sqrt {\frac {5}{7}} - 1320\right )}\right )} \sqrt {452768575 \, \sqrt {\frac {5}{7}} + 250141922}\right ) + 3 \, \sqrt {\frac {1}{434}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {452768575 \, \sqrt {\frac {5}{7}} - 250141922} \log \left (42 \, \sqrt {\frac {1}{434}} \sqrt {2 \, x + 1} \sqrt {452768575 \, \sqrt {\frac {5}{7}} - 250141922} {\left (23998 \, \sqrt {\frac {5}{7}} + 39535\right )} + 1560308430 \, x + 1092215901 \, \sqrt {\frac {5}{7}} + 780154215\right ) - 3 \, \sqrt {\frac {1}{434}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {452768575 \, \sqrt {\frac {5}{7}} - 250141922} \log \left (-42 \, \sqrt {\frac {1}{434}} \sqrt {2 \, x + 1} \sqrt {452768575 \, \sqrt {\frac {5}{7}} - 250141922} {\left (23998 \, \sqrt {\frac {5}{7}} + 39535\right )} + 1560308430 \, x + 1092215901 \, \sqrt {\frac {5}{7}} + 780154215\right ) + {\left (39600 \, x^{3} + 69895 \, x^{2} + 47861 \, x + 26483\right )} \sqrt {2 \, x + 1}}{94178 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/94178*(6*sqrt(1/434)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(45276857 5*sqrt(5/7) + 250141922)*arctan(14/52010281*sqrt(1/434)*(sqrt(1/434)*sqrt( 452768575*sqrt(5/7) - 250141922)*(7*sqrt(5/7) + 2) + sqrt(2*x + 1)*(7379*s qrt(5/7) - 1320))*sqrt(452768575*sqrt(5/7) + 250141922)) - 6*sqrt(1/434)*( 25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(452768575*sqrt(5/7) + 250141922) *arctan(14/52010281*sqrt(1/434)*(sqrt(1/434)*sqrt(452768575*sqrt(5/7) - 25 0141922)*(7*sqrt(5/7) + 2) - sqrt(2*x + 1)*(7379*sqrt(5/7) - 1320))*sqrt(4 52768575*sqrt(5/7) + 250141922)) + 3*sqrt(1/434)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(452768575*sqrt(5/7) - 250141922)*log(42*sqrt(1/434)*sqrt (2*x + 1)*sqrt(452768575*sqrt(5/7) - 250141922)*(23998*sqrt(5/7) + 39535) + 1560308430*x + 1092215901*sqrt(5/7) + 780154215) - 3*sqrt(1/434)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(452768575*sqrt(5/7) - 250141922)*log(- 42*sqrt(1/434)*sqrt(2*x + 1)*sqrt(452768575*sqrt(5/7) - 250141922)*(23998* sqrt(5/7) + 39535) + 1560308430*x + 1092215901*sqrt(5/7) + 780154215) + (3 9600*x^3 + 69895*x^2 + 47861*x + 26483)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
\[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\sqrt {2 x + 1} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \] Input:
integrate(1/(1+2*x)**(1/2)/(5*x**2+3*x+2)**3,x)
Output:
Integral(1/(sqrt(2*x + 1)*(5*x**2 + 3*x + 2)**3), x)
\[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} \sqrt {2 \, x + 1}} \,d x } \] Input:
integrate(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x + 1)), x)
Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (177) = 354\).
Time = 0.59 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.68 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
Output:
3/175244067950*sqrt(31)*(13860*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt (-140*sqrt(35) + 2450) - 66*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3 /2) + 132*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 27720*(7/5)^(3/4)*sqrt (140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1807855*sqrt(31)*(7/5)^(1/4)*sqr t(-140*sqrt(35) + 2450) + 3615710*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*a rctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 3/175244067950*sqrt(31)*(13860*sqrt(31)* (7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 66*sqrt(31)*(7/ 5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 132*(7/5)^(3/4)*(140*sqrt(35) + 24 50)^(3/2) + 27720*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 3615710*(7/5)^ (1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt (1/35*sqrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 3/350 488135900*sqrt(31)*(66*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 13860*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2 7720*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 132*(7/5)^ (3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1807855*sqrt(31)*(7/5)^(1/4)*sqrt(140 *sqrt(35) + 2450) - 3615710*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(2* (7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 3/350488135900*sqrt(31)*(66*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 245...
Time = 5.62 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {\frac {3446\,\sqrt {2\,x+1}}{33635}+\frac {30664\,{\left (2\,x+1\right )}^{3/2}}{1177225}+\frac {4198\,{\left (2\,x+1\right )}^{5/2}}{235445}+\frac {1584\,{\left (2\,x+1\right )}^{7/2}}{47089}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}+\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{665489348040125\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}-\frac {46760544\,\sqrt {31}\,\sqrt {217}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{20630169789243875\,\left (-\frac {561079767456}{95069906862875}+\frac {\sqrt {31}\,172523027088{}\mathrm {i}}{95069906862875}\right )}\right )\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{10218313} \] Input:
int(1/((2*x + 1)^(1/2)*(3*x + 5*x^2 + 2)^3),x)
Output:
(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(665489348040125*((31^(1/2)*172523027088i)/9506990686 2875 + 561079767456/95069906862875)) + (46760544*31^(1/2)*217^(1/2)*(- 31^ (1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2))/(20630169789243875*((3 1^(1/2)*172523027088i)/95069906862875 + 561079767456/95069906862875)))*(- 31^(1/2)*52010281i - 250141922)^(1/2)*3i)/10218313 - ((3446*(2*x + 1)^(1/2 ))/33635 + (30664*(2*x + 1)^(3/2))/1177225 + (4198*(2*x + 1)^(5/2))/235445 + (1584*(2*x + 1)^(7/2))/47089)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2 *x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (217^(1/2)*atan((217^(1/2)*(31^(1/2)* 52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(665489348040125*( (31^(1/2)*172523027088i)/95069906862875 - 561079767456/95069906862875)) - (46760544*31^(1/2)*217^(1/2)*(31^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2))/(20630169789243875*((31^(1/2)*172523027088i)/95069906862875 - 5 61079767456/95069906862875)))*(31^(1/2)*52010281i - 250141922)^(1/2)*3i)/1 0218313
Time = 0.20 (sec) , antiderivative size = 1653, normalized size of antiderivative = 6.89 \[ \int \frac {1}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x)
Output:
( - 3599700*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**4 - 4319640*sqr t(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1 )*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**3 - 4175652*sqrt(sqrt(35) - 2) *sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqr t(sqrt(35) - 2)*sqrt(2)))*x**2 - 1727856*sqrt(sqrt(35) - 2)*sqrt(14)*atan( (sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2) *sqrt(2)))*x - 575952*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2) *sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2))) - 830235 0*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2* x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**4 - 9962820*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5)) /(sqrt(sqrt(35) - 2)*sqrt(2)))*x**3 - 9630726*sqrt(sqrt(35) - 2)*sqrt(10)* atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**2 - 3985128*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt (35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2))) *x - 1328376*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2))) + 3599700*sqrt(sq rt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1)*sq rt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**4 + 4319640*sqrt(sqrt(35) - 2)*...