Integrand size = 22, antiderivative size = 240 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {1+2 x} (3+10 x)}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (599+1790 x)}{13454 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{6727}+\frac {\sqrt {\frac {1}{434} \left (9651062+1806875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{6727}-\frac {\sqrt {\frac {1}{434} \left (-9651062+1806875 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right )}{6727} \] Output:
1/62*(1+2*x)^(1/2)*(3+10*x)/(5*x^2+3*x+2)^2+(1+2*x)^(1/2)*(599+1790*x)/(67 270*x^2+40362*x+26908)-1/2919518*(4188560908+784183750*35^(1/2))^(1/2)*arc tan(((20+10*35^(1/2))^(1/2)-10*(1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2))+1/2 919518*(4188560908+784183750*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))-1/2919518*(-4188560908+784183 750*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 = 0.79 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {217 \sqrt {1+2 x} \left (1849+7547 x+8365 x^2+8950 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+\sqrt {217 \left (9651062-825499 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {217 \left (9651062+825499 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{1459759} \] Input:
Integrate[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^3,x]
Output:
((217*Sqrt[1 + 2*x]*(1849 + 7547*x + 8365*x^2 + 8950*x^3))/(2*(2 + 3*x + 5 *x^2)^2) + Sqrt[217*(9651062 - (825499*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*S qrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[217*(9651062 + (825499*I)*Sqrt[31])]*Arc Tan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/1459759
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 = {1163, 25, 1235, 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.
| \(\displaystyle \int \frac {\sqrt {2 x+1}}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1163 |
| \(\displaystyle \frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}-\frac {1}{62} \int -\frac {50 x+27}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{62} \int \frac {50 x+27}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}dx+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {1}{62} \left (\frac {1}{217} \int \frac {1790 x+1439}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {1}{62} \left (\frac {2}{217} \int \frac {2 (895 (2 x+1)+544)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \int \frac {895 (2 x+1)+544}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \left (\frac {\int \frac {5 \left (544 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (544-179 \sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (544-179 \sqrt {35}\right ) \sqrt {2 x+1}+544 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \left (\frac {5 \int \frac {544 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (544-179 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (544-179 \sqrt {35}\right ) \sqrt {2 x+1}+544 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \left (\frac {5 \left (\frac {1}{10} \sqrt {96510620+18068750 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {96510620+18068750 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \left (\frac {5 \left (\frac {1}{10} \sqrt {96510620+18068750 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {96510620+18068750 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \left (\frac {5 \left (\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {96510620+18068750 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {96510620+18068750 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \left (\frac {5 \left (\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {96510620+18068750 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (544-179 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {96510620+18068750 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{62} \left (\frac {4}{217} \left (\frac {5 \left (\frac {1}{5} \sqrt {\frac {96510620+18068750 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (544-179 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{5} \sqrt {\frac {96510620+18068750 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{10} \left (544-179 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (1790 x+599)}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^3,x]
Output:
(Sqrt[1 + 2*x]*(3 + 10*x))/(62*(2 + 3*x + 5*x^2)^2) + ((Sqrt[1 + 2*x]*(599 + 1790*x))/(217*(2 + 3*x + 5*x^2)) + (4*((5*((Sqrt[(96510620 + 18068750*S qrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[ 1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 - ((544 - 179*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[(96510620 + 18068750*Sqrt[35])/(10*(-2 + Sqrt [35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 + ((544 - 179*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)/ 62
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*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* (b^2 - 4*a*c))), x] - Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1 )*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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 = 3.60 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.35
| method | result | size |
| pseudoelliptic | \(-\frac {40000 \left (-\frac {1253 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (\sqrt {5}\, \sqrt {7}-\frac {39}{4}\right ) \left (x^{3}+\frac {1673}{1790} x^{2}+\frac {7547}{8950} x +\frac {1849}{8950}\right ) \sqrt {1+2 x}}{961}+\left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \left (-\frac {\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (452130 \sqrt {5}-413047 \sqrt {7}\right ) \left (\ln \left (\sqrt {35}+5+10 x -\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )-\ln \left (\sqrt {35}+5+10 x +\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{476656}+\left (\sqrt {5}\, \sqrt {7}-\frac {175}{4}\right ) \left (\arctan \left (\frac {\sqrt {20+10 \sqrt {35}}-10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )-\arctan \left (\frac {\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )\right )\right )\right )}{49 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (4 \sqrt {5}\, \sqrt {7}-39\right ) \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )^{2} \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )^{2}}\) | \(324\) |
| trager | \(\frac {\left (8950 x^{3}+8365 x^{2}+7547 x +1849\right ) \sqrt {1+2 x}}{13454 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {2 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right ) \ln \left (\frac {-202713247744 x \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{5}-1452607421494208 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{3} x +3107245926918000 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}-385575076726784 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{3}-2451977189563195620 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right ) x -25592768729287571875 \sqrt {1+2 x}-1049468876066602528 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2} x +7174565 x -3301996}\right )}{6727}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}+1047140227\right ) \ln \left (-\frac {-14479517696 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}+1047140227\right ) \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{4} x -57236247559392 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}+1047140227\right ) x +96324623734458000 \sqrt {1+2 x}\, \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}+27541076909056 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}+1047140227\right )-45826377334291750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}+1047140227\right ) x +1328879814356719851625 \sqrt {1+2 x}+78148905624314000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+188356 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2}+1047140227\right )}{3472 \operatorname {RootOf}\left (1722112 \textit {\_Z}^{4}+9573853504 \textit {\_Z}^{2}+16323986328125\right )^{2} x +12127559 x +3301996}\right )}{1459759}\) | \(458\) |
| risch | \(\frac {\left (8950 x^{3}+8365 x^{2}+7547 x +1849\right ) \sqrt {1+2 x}}{13454 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {451 \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}}{417074}+\frac {7353 \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}}{5839036}-\frac {2255 \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 )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {7353 \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}}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1088 \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}}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {451 \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}}{417074}-\frac {7353 \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}}{5839036}-\frac {2255 \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 )}{208537 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {7353 \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}}{2919518 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1088 \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}}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(643\) |
| derivativedivides | \(\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}+6673227400 \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-214587133600 \sqrt {5}+114637845000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{62696649050 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}+440433008400\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-76332028500 \sqrt {7}+54802482000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{62696649050 \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 {\left (2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-2065235 \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 )}{5839036}+\frac {5 \left (-1315392 \sqrt {35}+4721920+\frac {\left (2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}+6673227400 \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-214587133600 \sqrt {5}+114637845000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{2919518 \left (-41876250+4295000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}+440433008400\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-76332028500 \sqrt {7}+54802482000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2919518 \left (-41876250+4295000 \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 {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{5839036}+\frac {5 \left (-1315392 \sqrt {35}+4721920-\frac {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\) | \(699\) |
| default | \(\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}+6673227400 \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-214587133600 \sqrt {5}+114637845000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{62696649050 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}+440433008400\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}-\frac {\left (-76332028500 \sqrt {7}+54802482000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{62696649050 \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 {\left (2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-2065235 \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 )}{5839036}+\frac {5 \left (-1315392 \sqrt {35}+4721920+\frac {\left (2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}-2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}+\frac {\frac {\sqrt {5}\, \left (-13012793430 \sqrt {5}+6673227400 \sqrt {7}\right ) \left (1+2 x \right )^{\frac {3}{2}}}{6269664905 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-214587133600 \sqrt {5}+114637845000 \sqrt {7}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (1+2 x \right )}{2919518 \left (-41876250+4295000 \sqrt {5}\, \sqrt {7}\right )}+\frac {\left (-141628999400 \sqrt {5}\, \sqrt {7}+440433008400\right ) \sqrt {1+2 x}}{31348324525 \left (-390+40 \sqrt {5}\, \sqrt {7}\right )}+\frac {5 \left (-76332028500 \sqrt {7}+54802482000 \sqrt {5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2919518 \left (-41876250+4295000 \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 {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{5839036}+\frac {5 \left (-1315392 \sqrt {35}+4721920-\frac {\left (-2260650 \sqrt {5}\, \sqrt {2 \sqrt {35}+4}+2065235 \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 )}{1459759 \sqrt {-20+10 \sqrt {35}}}}{20 \sqrt {5}\, \sqrt {7}-195}\) | \(699\) |
Input:
int((1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
-40000/49*(-1253/961*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(5^(1/2)*7^(1/2)-39/4)* (x^3+1673/1790*x^2+7547/8950*x+1849/8950)*(1+2*x)^(1/2)+(x^2+3/5*x+2/5)^2* (-1/476656*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(452130*5^(1/2)-413047*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)-175/4)*(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
Time = 0.09 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {2 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} + \frac {4825531}{217}} \arctan \left (\frac {14}{825499} \, {\left (\sqrt {2 \, x + 1} {\left (544 \, \sqrt {\frac {5}{7}} - 895\right )} + \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} - \frac {4825531}{217}} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )}\right )} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} + \frac {4825531}{217}}\right ) - 2 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} + \frac {4825531}{217}} \arctan \left (-\frac {14}{825499} \, {\left (\sqrt {2 \, x + 1} {\left (544 \, \sqrt {\frac {5}{7}} - 895\right )} - \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} - \frac {4825531}{217}} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )}\right )} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} + \frac {4825531}{217}}\right ) + {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} - \frac {4825531}{217}} \log \left (14 \, \sqrt {2 \, x + 1} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} - \frac {4825531}{217}} {\left (7353 \, \sqrt {\frac {5}{7}} + 4510\right )} + 8254990 \, x + 5778493 \, \sqrt {\frac {5}{7}} + 4127495\right ) - {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} - \frac {4825531}{217}} \log \left (-14 \, \sqrt {2 \, x + 1} \sqrt {\frac {1806875}{62} \, \sqrt {\frac {5}{7}} - \frac {4825531}{217}} {\left (7353 \, \sqrt {\frac {5}{7}} + 4510\right )} + 8254990 \, x + 5778493 \, \sqrt {\frac {5}{7}} + 4127495\right ) - {\left (8950 \, x^{3} + 8365 \, x^{2} + 7547 \, x + 1849\right )} \sqrt {2 \, x + 1}}{13454 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
-1/13454*(2*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(1806875/62*sqrt(5/7 ) + 4825531/217)*arctan(14/825499*(sqrt(2*x + 1)*(544*sqrt(5/7) - 895) + s qrt(1806875/62*sqrt(5/7) - 4825531/217)*(7*sqrt(5/7) + 2))*sqrt(1806875/62 *sqrt(5/7) + 4825531/217)) - 2*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt( 1806875/62*sqrt(5/7) + 4825531/217)*arctan(-14/825499*(sqrt(2*x + 1)*(544* sqrt(5/7) - 895) - sqrt(1806875/62*sqrt(5/7) - 4825531/217)*(7*sqrt(5/7) + 2))*sqrt(1806875/62*sqrt(5/7) + 4825531/217)) + (25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(1806875/62*sqrt(5/7) - 4825531/217)*log(14*sqrt(2*x + 1) *sqrt(1806875/62*sqrt(5/7) - 4825531/217)*(7353*sqrt(5/7) + 4510) + 825499 0*x + 5778493*sqrt(5/7) + 4127495) - (25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) *sqrt(1806875/62*sqrt(5/7) - 4825531/217)*log(-14*sqrt(2*x + 1)*sqrt(18068 75/62*sqrt(5/7) - 4825531/217)*(7353*sqrt(5/7) + 4510) + 8254990*x + 57784 93*sqrt(5/7) + 4127495) - (8950*x^3 + 8365*x^2 + 7547*x + 1849)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
\[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {\sqrt {2 x + 1}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \] Input:
integrate((1+2*x)**(1/2)/(5*x**2+3*x+2)**3,x)
Output:
Integral(sqrt(2*x + 1)/(5*x**2 + 3*x + 2)**3, x)
\[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {\sqrt {2 \, x + 1}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \] Input:
integrate((1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
integrate(sqrt(2*x + 1)/(5*x^2 + 3*x + 2)^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (177) = 354\).
Time = 0.80 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.68 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
Output:
1/100139467400*sqrt(31)*(37590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt (-140*sqrt(35) + 2450) - 179*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^( 3/2) + 358*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 75180*(7/5)^(3/4)*sqr t(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 533120*sqrt(31)*(7/5)^(1/4)*sqr t(-140*sqrt(35) + 2450) + 1066240*(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)) + 1/100139467400*sqrt(31)*(37590*sqrt(31)* (7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 179*sqrt(31)*(7 /5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 358*(7/5)^(3/4)*(140*sqrt(35) + 2 450)^(3/2) + 75180*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 1066240*(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)) + 1/200 278934800*sqrt(31)*(179*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 37590*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 75180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 358*(7/5) ^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 533120*sqrt(31)*(7/5)^(1/4)*sqrt(140 *sqrt(35) + 2450) - 1066240*(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) - 1/200278934800*sqrt(31)*(179*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 24...
Time = 0.09 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {1088\,\sqrt {2\,x+1}}{24025}-\frac {23578\,{\left (2\,x+1\right )}^{3/2}}{168175}+\frac {2024\,{\left (2\,x+1\right )}^{5/2}}{33635}-\frac {358\,{\left (2\,x+1\right )}^{7/2}}{6727}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}-\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (-\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062-\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}\,13744{}\mathrm {i}}{1940202180875\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}+\frac {27488\,\sqrt {31}\,\sqrt {217}\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,\sqrt {2\,x+1}}{60146267607125\,\left (\frac {101059632}{277171740125}+\frac {\sqrt {31}\,7476736{}\mathrm {i}}{277171740125}\right )}\right )\,\sqrt {-9651062+\sqrt {31}\,825499{}\mathrm {i}}\,1{}\mathrm {i}}{1459759} \] Input:
int((2*x + 1)^(1/2)/(3*x + 5*x^2 + 2)^3,x)
Output:
((1088*(2*x + 1)^(1/2))/24025 - (23578*(2*x + 1)^(3/2))/168175 + (2024*(2* x + 1)^(5/2))/33635 - (358*(2*x + 1)^(7/2))/6727)/((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)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744i)/(19402 02180875*((31^(1/2)*7476736i)/277171740125 - 101059632/277171740125)) - (2 7488*31^(1/2)*217^(1/2)*(- 31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/ 2))/(60146267607125*((31^(1/2)*7476736i)/277171740125 - 101059632/27717174 0125)))*(- 31^(1/2)*825499i - 9651062)^(1/2)*1i)/1459759 + (217^(1/2)*atan ((217^(1/2)*(31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1/2)*13744i)/(19 40202180875*((31^(1/2)*7476736i)/277171740125 + 101059632/277171740125)) + (27488*31^(1/2)*217^(1/2)*(31^(1/2)*825499i - 9651062)^(1/2)*(2*x + 1)^(1 /2))/(60146267607125*((31^(1/2)*7476736i)/277171740125 + 101059632/2771717 40125)))*(31^(1/2)*825499i - 9651062)^(1/2)*1i)/1459759
Time = 0.20 (sec) , antiderivative size = 1653, normalized size of antiderivative = 6.89 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int((1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x)
Output:
( - 367650*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 - 441180*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**3 - 426474*sqrt(sqrt(35) - 2)*sq rt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(s qrt(35) - 2)*sqrt(2)))*x**2 - 176472*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqr t(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqr t(2)))*x - 58824*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))) - 315700*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 - 378840*sqrt(sqrt(35) - 2)*s qrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt( sqrt(35) - 2)*sqrt(2)))*x**3 - 366212*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sq rt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sq rt(2)))*x**2 - 151536*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 - 5051 2*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))) + 367650*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 + 441180*sqrt(sqrt(35) - 2)*sqrt(14)*atan...