Integrand size = 22, antiderivative size = 236 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{10633}-\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{10633}+\frac {5 \sqrt {\frac {2}{217} \left (-12504542+2632525 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right )}{10633} \] Output:
-820/4557/(1+2*x)^(3/2)-4680/10633/(1+2*x)^(1/2)+1/217*(37+20*x)/(1+2*x)^( 3/2)/(5*x^2+3*x+2)+5/2307361*(5426971228+1142515850*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))-5/2307 361*(5426971228+1142515850*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))+5/2307361*(-5426971228+11425158 50*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 = 1.55 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (-\frac {217 \left (34121+112560 x+183140 x^2+140400 x^3\right )}{2 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}-15 \sqrt {217 \left (12504542-1667459 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-15 \sqrt {217 \left (12504542+1667459 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{6922083} \] Input:
Integrate[1/((1 + 2*x)^(5/2)*(2 + 3*x + 5*x^2)^2),x]
Output:
(2*((-217*(34121 + 112560*x + 183140*x^2 + 140400*x^3))/(2*(1 + 2*x)^(3/2) *(2 + 3*x + 5*x^2)) - 15*Sqrt[217*(12504542 - (1667459*I)*Sqrt[31])]*ArcTa n[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - 15*Sqrt[217*(12504542 + (1667 459*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/692 2083
Time = 0.65 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.49, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {1165, 27, 1198, 1198, 25, 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.
| \(\displaystyle \int \frac {1}{(2 x+1)^{5/2} \left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {1}{217} \int \frac {5 (20 x+51)}{(2 x+1)^{5/2} \left (5 x^2+3 x+2\right )}dx+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{217} \int \frac {20 x+51}{(2 x+1)^{5/2} \left (5 x^2+3 x+2\right )}dx+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \int \frac {29-410 x}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}dx-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (\frac {1}{7} \int -\frac {2340 x+1669}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {1}{7} \int \frac {2340 x+1669}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {2}{7} \int \frac {2 (1170 (2 x+1)+499)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \int \frac {1170 (2 x+1)+499}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \left (\frac {\int \frac {5 \left (499 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (499-234 \sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (499-234 \sqrt {35}\right ) \sqrt {2 x+1}+499 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \int \frac {499 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (499-234 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (499-234 \sqrt {35}\right ) \sqrt {2 x+1}+499 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \sqrt {125045420+26325250 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {125045420+26325250 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \sqrt {125045420+26325250 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {125045420+26325250 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {125045420+26325250 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {125045420+26325250 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {125045420+26325250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (499-234 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {125045420+26325250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5}{217} \left (\frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{5} \sqrt {\frac {125045420+26325250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (499-234 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{5} \sqrt {\frac {125045420+26325250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{10} \left (499-234 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {936}{7 \sqrt {2 x+1}}\right )-\frac {164}{21 (2 x+1)^{3/2}}\right )+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}\) |
Input:
Int[1/((1 + 2*x)^(5/2)*(2 + 3*x + 5*x^2)^2),x]
Output:
(37 + 20*x)/(217*(1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)) + (5*(-164/(21*(1 + 2* x)^(3/2)) + (-936/(7*Sqrt[1 + 2*x]) - (4*((5*((Sqrt[(125045420 + 26325250* Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt [1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 - ((499 - 234*Sqrt[35])*Log[Sqrt[3 5] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10))/(2*Sqrt[14 *(2 + Sqrt[35])]) + (5*((Sqrt[(125045420 + 26325250*Sqrt[35])/(10*(-2 + Sq rt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 + ((499 - 234*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])])))/7)/ 7))/217
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), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
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 = 2.27 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.24
| method | result | size |
| pseudoelliptic | \(\frac {\frac {5 \left (10 x^{3}+11 x^{2}+7 x +2\right ) \left (\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (6769 \sqrt {5}-9188 \sqrt {7}\right ) \left (\ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )-\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+61876 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}+\frac {8190}{499}\right )\right ) \sqrt {1+2 x}}{4614722}-\frac {46800 \left (\frac {34121}{140400}+\frac {469}{585} x +\frac {9157}{7020} x^{2}+x^{3}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}{10633}}{\left (1+2 x \right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(293\) |
| derivativedivides | \(-\frac {16}{147 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {128}{343 \sqrt {1+2 x}}-\frac {16 \left (\frac {89 \left (1+2 x \right )^{\frac {3}{2}}}{62}+\frac {233 \sqrt {1+2 x}}{620}\right )}{343 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4614722}-\frac {10 \left (30938 \sqrt {5}\, \sqrt {7}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{4614722}+\frac {10 \left (-30938 \sqrt {5}\, \sqrt {7}+\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(433\) |
| default | \(-\frac {16}{147 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {128}{343 \sqrt {1+2 x}}-\frac {16 \left (\frac {89 \left (1+2 x \right )^{\frac {3}{2}}}{62}+\frac {233 \sqrt {1+2 x}}{620}\right )}{343 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{4614722}-\frac {10 \left (30938 \sqrt {5}\, \sqrt {7}-\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{4614722}+\frac {10 \left (-30938 \sqrt {5}\, \sqrt {7}+\frac {\left (33845 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-45940 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(433\) |
| trager | \(-\frac {140400 x^{3}+183140 x^{2}+112560 x +34121}{31899 \left (1+2 x \right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )}+\frac {10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) \ln \left (\frac {1082562656 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{4} x +18274525189608 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) x -16637132201024 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right )-70869658126175580 \sqrt {1+2 x}\, \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+16528108437548820 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right ) x -264894705503873760 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+1356742807\right )-7328924922481506073415 \sqrt {1+2 x}}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} x +17506919 x +6669836}\right )}{2307361}-\frac {10 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right ) \ln \left (-\frac {7577938592 x \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{5}+308754136562536 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{3} x +116459925407168 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{3}-2286118004070180 \sqrt {1+2 x}\, \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2}+2720797097744165060 x \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )+1501216389977029664 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )+170548595197901724125 \sqrt {1+2 x}}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+3101126416 \textit {\_Z}^{2}+34650939378125\right )^{2} x +7502165 x -6669836}\right )}{10633}\) | \(458\) |
| risch | \(-\frac {140400 x^{3}+183140 x^{2}+112560 x +34121}{31899 \left (1+2 x \right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )}+\frac {4835 \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}}{659246}-\frac {22970 \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}}{2307361}+\frac {24175 \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 )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {45940 \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}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {9980 \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}}{74431 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {4835 \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}}{659246}+\frac {22970 \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}}{2307361}+\frac {24175 \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 )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {45940 \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}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {9980 \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}}{74431 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(643\) |
Input:
int(1/(1+2*x)^(5/2)/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
100/74431/(1+2*x)^(3/2)*(1/1240*(10*x^3+11*x^2+7*x+2)*((10*5^(1/2)*7^(1/2) -20)^(1/2)*(6769*5^(1/2)-9188*7^(1/2))*(ln(-(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)-ln(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*5^(1/2)*7^(1/2)+4)^(1/2 )+61876*(arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-10*(1+2*x)^(1/2))/(10 *5^(1/2)*7^(1/2)-20)^(1/2))-arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10 *(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)))*(5^(1/2)*7^(1/2)+8190/499) )*(1+2*x)^(1/2)-3276*(34121/140400+469/585*x+9157/7020*x^2+x^3)*(10*5^(1/2 )*7^(1/2)-20)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)/(5*x^2+3*x+2)
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (169) = 338\).
Time = 0.09 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {30 \, \sqrt {\frac {1}{434}} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {18427675 \, \sqrt {\frac {5}{7}} + 12504542} \arctan \left (\frac {14}{1667459} \, \sqrt {\frac {1}{434}} {\left (\sqrt {\frac {1}{434}} \sqrt {18427675 \, \sqrt {\frac {5}{7}} - 12504542} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} + \sqrt {2 \, x + 1} {\left (499 \, \sqrt {\frac {5}{7}} - 1170\right )}\right )} \sqrt {18427675 \, \sqrt {\frac {5}{7}} + 12504542}\right ) - 30 \, \sqrt {\frac {1}{434}} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {18427675 \, \sqrt {\frac {5}{7}} + 12504542} \arctan \left (\frac {14}{1667459} \, \sqrt {\frac {1}{434}} {\left (\sqrt {\frac {1}{434}} \sqrt {18427675 \, \sqrt {\frac {5}{7}} - 12504542} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} - \sqrt {2 \, x + 1} {\left (499 \, \sqrt {\frac {5}{7}} - 1170\right )}\right )} \sqrt {18427675 \, \sqrt {\frac {5}{7}} + 12504542}\right ) + 15 \, \sqrt {\frac {1}{434}} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {18427675 \, \sqrt {\frac {5}{7}} - 12504542} \log \left (70 \, \sqrt {\frac {1}{434}} \sqrt {2 \, x + 1} \sqrt {18427675 \, \sqrt {\frac {5}{7}} - 12504542} {\left (9188 \, \sqrt {\frac {5}{7}} + 4835\right )} + 83372950 \, x + 58361065 \, \sqrt {\frac {5}{7}} + 41686475\right ) - 15 \, \sqrt {\frac {1}{434}} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {18427675 \, \sqrt {\frac {5}{7}} - 12504542} \log \left (-70 \, \sqrt {\frac {1}{434}} \sqrt {2 \, x + 1} \sqrt {18427675 \, \sqrt {\frac {5}{7}} - 12504542} {\left (9188 \, \sqrt {\frac {5}{7}} + 4835\right )} + 83372950 \, x + 58361065 \, \sqrt {\frac {5}{7}} + 41686475\right ) - {\left (140400 \, x^{3} + 183140 \, x^{2} + 112560 \, x + 34121\right )} \sqrt {2 \, x + 1}}{31899 \, {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )}} \] Input:
integrate(1/(1+2*x)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/31899*(30*sqrt(1/434)*(20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2)*sqrt(1842767 5*sqrt(5/7) + 12504542)*arctan(14/1667459*sqrt(1/434)*(sqrt(1/434)*sqrt(18 427675*sqrt(5/7) - 12504542)*(7*sqrt(5/7) + 2) + sqrt(2*x + 1)*(499*sqrt(5 /7) - 1170))*sqrt(18427675*sqrt(5/7) + 12504542)) - 30*sqrt(1/434)*(20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2)*sqrt(18427675*sqrt(5/7) + 12504542)*arctan( 14/1667459*sqrt(1/434)*(sqrt(1/434)*sqrt(18427675*sqrt(5/7) - 12504542)*(7 *sqrt(5/7) + 2) - sqrt(2*x + 1)*(499*sqrt(5/7) - 1170))*sqrt(18427675*sqrt (5/7) + 12504542)) + 15*sqrt(1/434)*(20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2)* sqrt(18427675*sqrt(5/7) - 12504542)*log(70*sqrt(1/434)*sqrt(2*x + 1)*sqrt( 18427675*sqrt(5/7) - 12504542)*(9188*sqrt(5/7) + 4835) + 83372950*x + 5836 1065*sqrt(5/7) + 41686475) - 15*sqrt(1/434)*(20*x^4 + 32*x^3 + 25*x^2 + 11 *x + 2)*sqrt(18427675*sqrt(5/7) - 12504542)*log(-70*sqrt(1/434)*sqrt(2*x + 1)*sqrt(18427675*sqrt(5/7) - 12504542)*(9188*sqrt(5/7) + 4835) + 83372950 *x + 58361065*sqrt(5/7) + 41686475) - (140400*x^3 + 183140*x^2 + 112560*x + 34121)*sqrt(2*x + 1))/(20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2)
\[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \] Input:
integrate(1/(1+2*x)**(5/2)/(5*x**2+3*x+2)**2,x)
Output:
Integral(1/((2*x + 1)**(5/2)*(5*x**2 + 3*x + 2)**2), x)
\[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(1+2*x)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)^2*(2*x + 1)^(5/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (169) = 338\).
Time = 0.58 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(1+2*x)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")
Output:
-1/7914248230*sqrt(31)*(24570*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt( -140*sqrt(35) + 2450) - 117*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3 /2) - 234*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) - 49140*(7/5)^(3/4)*sqrt (140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 244510*sqrt(31)*(7/5)^(1/4)*sqrt (-140*sqrt(35) + 2450) - 489020*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arc tan(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)) - 1/7914248230*sqrt(31)*(24570*sqrt(31)*(7/5 )^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 117*sqrt(31)*(7/5)^ (3/4)*(-140*sqrt(35) + 2450)^(3/2) - 234*(7/5)^(3/4)*(140*sqrt(35) + 2450) ^(3/2) - 49140*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 2 44510*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 489020*(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)) - 1/15828496 460*sqrt(31)*(117*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 24570 *sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 49140* (7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 234*(7/5)^(3/4) *(-140*sqrt(35) + 2450)^(3/2) + 244510*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt( 35) + 2450) + 489020*(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) + 1/15 828496460*sqrt(31)*(117*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2...
Time = 0.10 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {128\,x}{147}-\frac {5492\,{\left (2\,x+1\right )}^2}{31899}+\frac {4680\,{\left (2\,x+1\right )}^3}{10633}+\frac {144}{245}}{\frac {7\,{\left (2\,x+1\right )}^{3/2}}{5}-\frac {4\,{\left (2\,x+1\right )}^{5/2}}{5}+{\left (2\,x+1\right )}^{7/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}\,6884992{}\mathrm {i}}{1900211000023\,\left (-\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}-\frac {13769984\,\sqrt {31}\,\sqrt {217}\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}}{58906541000713\,\left (-\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}\right )\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,10{}\mathrm {i}}{2307361}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}\,6884992{}\mathrm {i}}{1900211000023\,\left (\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}+\frac {13769984\,\sqrt {31}\,\sqrt {217}\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}}{58906541000713\,\left (\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}\right )\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,10{}\mathrm {i}}{2307361} \] Input:
int(1/((2*x + 1)^(5/2)*(3*x + 5*x^2 + 2)^2),x)
Output:
(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*1667459i - 12504542)^(1/2)*(2*x + 1 )^(1/2)*6884992i)/(1900211000023*((31^(1/2)*3435611008i)/271458714289 - 63 259306496/271458714289)) - (13769984*31^(1/2)*217^(1/2)*(- 31^(1/2)*166745 9i - 12504542)^(1/2)*(2*x + 1)^(1/2))/(58906541000713*((31^(1/2)*343561100 8i)/271458714289 - 63259306496/271458714289)))*(- 31^(1/2)*1667459i - 1250 4542)^(1/2)*10i)/2307361 - ((128*x)/147 - (5492*(2*x + 1)^2)/31899 + (4680 *(2*x + 1)^3)/10633 + 144/245)/((7*(2*x + 1)^(3/2))/5 - (4*(2*x + 1)^(5/2) )/5 + (2*x + 1)^(7/2)) - (217^(1/2)*atan((217^(1/2)*(31^(1/2)*1667459i - 1 2504542)^(1/2)*(2*x + 1)^(1/2)*6884992i)/(1900211000023*((31^(1/2)*3435611 008i)/271458714289 + 63259306496/271458714289)) + (13769984*31^(1/2)*217^( 1/2)*(31^(1/2)*1667459i - 12504542)^(1/2)*(2*x + 1)^(1/2))/(58906541000713 *((31^(1/2)*3435611008i)/271458714289 + 63259306496/271458714289)))*(31^(1 /2)*1667459i - 12504542)^(1/2)*10i)/2307361
Time = 0.20 (sec) , antiderivative size = 1498, normalized size of antiderivative = 6.35 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(1+2*x)^(5/2)/(5*x^2+3*x+2)^2,x)
Output:
(2756400*sqrt(2*x + 1)*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 + 3032040*sqrt(2*x + 1)*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**2 + 1 929480*sqrt(2*x + 1)*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 + 55128 0*sqrt(2*x + 1)*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))) + 2030700*sqrt (2*x + 1)*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 + 2233770*sqrt( 2*x + 1)*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 + 1421490*sqrt(2 *x + 1)*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*s qrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x + 406140*sqrt(2*x + 1)*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))) - 2756400*sqrt(2*x + 1)*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 - 3032040*sqrt(2*x + 1)*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**2 - 1929480*sqrt(2*x + 1)*sq...