Integrand size = 22, antiderivative size = 253 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right )}{329623} \] Output:
-81090/329623/(1+2*x)^(1/2)+1/434*(37+20*x)/(1+2*x)^(1/2)/(5*x^2+3*x+2)^2+ 5/94178*(2329+2080*x)/(1+2*x)^(1/2)/(5*x^2+3*x+2)-15/143056382*(-979586504 708+168143350550*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))+15/143056382*(-979586504708+168143350550* 35^(1/2))^(1/2)*arctan(((20+10*35^(1/2))^(1/2)+10*(1+2*x)^(1/2))/(-20+10*3 5^(1/2))^(1/2))+15/143056382*(979586504708+168143350550*35^(1/2))^(1/2)*ar ctanh((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 = 2.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {-\frac {217 \left (429487+1525635 x+4077245 x^2+4501400 x^3+4054500 x^4\right )}{2 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+15 \sqrt {217 \left (-2257111762+71603149 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+15 \sqrt {217 \left (-2257111762-71603149 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{71528191} \] Input:
Integrate[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^3),x]
Output:
((-217*(429487 + 1525635*x + 4077245*x^2 + 4501400*x^3 + 4054500*x^4))/(2* Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) + 15*Sqrt[217*(-2257111762 + (71603149* I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 15*Sqrt[21 7*(-2257111762 - (71603149*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31] )]*Sqrt[1 + 2*x]])/71528191
Time = 0.67 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.45, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {1165, 27, 1235, 27, 1198, 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.
| \(\displaystyle \int \frac {1}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {1}{434} \int \frac {5 (28 x+69)}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )^2}dx+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{434} \int \frac {28 x+69}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )^2}dx+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {5}{434} \left (\frac {1}{217} \int \frac {3 (2080 x+3743)}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}dx+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \int \frac {2080 x+3743}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}dx+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {1}{7} \int \frac {4577-27030 x}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {2}{7} \int \frac {2 (18092-13515 (2 x+1))}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \int \frac {18092-13515 (2 x+1)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \left (\frac {\int \frac {5 \left (18092 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (18092+2703 \sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (18092+2703 \sqrt {35}\right ) \sqrt {2 x+1}+18092 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \left (\frac {5 \int \frac {18092 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (18092+2703 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (18092+2703 \sqrt {35}\right ) \sqrt {2 x+1}+18092 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \sqrt {3874270750 \sqrt {35}-22571117620} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {3874270750 \sqrt {35}-22571117620} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \sqrt {3874270750 \sqrt {35}-22571117620} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {3874270750 \sqrt {35}-22571117620} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {3874270750 \sqrt {35}-22571117620} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {3874270750 \sqrt {35}-22571117620} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {3874270750 \sqrt {35}-22571117620}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {3874270750 \sqrt {35}-22571117620}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5}{434} \left (\frac {3}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{5} \sqrt {\frac {3874270750 \sqrt {35}-22571117620}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{5} \sqrt {\frac {3874270750 \sqrt {35}-22571117620}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{10} \left (18092+2703 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {10812}{7 \sqrt {2 x+1}}\right )+\frac {2080 x+2329}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^3),x]
Output:
(37 + 20*x)/(434*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) + (5*((2329 + 2080*x)/ (217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (3*(-10812/(7*Sqrt[1 + 2*x]) + (4* ((5*((Sqrt[(-22571117620 + 3874270750*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcT an[(-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] )/5 - ((18092 + 2703*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[(-2 2571117620 + 3874270750*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 + ((18092 + 2703*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))/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), 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_))^(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 = 2.37 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(-\frac {375 \left (\frac {\left (5 x^{2}+3 x +2\right )^{2} \left (\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (88802 \sqrt {5}+58421 \sqrt {7}\right ) \left (\ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )-\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+2243408 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}-\frac {94605}{18092}\right )\right ) \sqrt {1+2 x}}{350}+335172 \left (\frac {815449}{810900} x^{2}+\frac {45014}{40545} x^{3}+x^{4}+\frac {33903}{90100} x +\frac {429487}{4054500}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\right )}{20436626 \sqrt {1+2 x}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (5 x^{2}+3 x +2\right )^{2}}\) | \(295\) |
| derivativedivides | \(-\frac {64}{343 \sqrt {1+2 x}}-\frac {1600 \left (\frac {9793 \left (1+2 x \right )^{\frac {7}{2}}}{30752}-\frac {14343 \left (1+2 x \right )^{\frac {5}{2}}}{19220}+\frac {762223 \left (1+2 x \right )^{\frac {3}{2}}}{768800}-\frac {170877 \sqrt {1+2 x}}{192200}\right )}{343 \left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}-\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}-\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{286112764}+\frac {15 \left (1121704 \sqrt {5}\, \sqrt {7}+\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(444\) |
| default | \(-\frac {64}{343 \sqrt {1+2 x}}-\frac {1600 \left (\frac {9793 \left (1+2 x \right )^{\frac {7}{2}}}{30752}-\frac {14343 \left (1+2 x \right )^{\frac {5}{2}}}{19220}+\frac {762223 \left (1+2 x \right )^{\frac {3}{2}}}{768800}-\frac {170877 \sqrt {1+2 x}}{192200}\right )}{343 \left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}-\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}-\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{286112764}+\frac {15 \left (1121704 \sqrt {5}\, \sqrt {7}+\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(444\) |
| trager | \(\text {Expression too large to display}\) | \(463\) |
| risch | \(-\frac {4054500 x^{4}+4501400 x^{3}+4077245 x^{2}+1525635 x +429487}{659246 \left (5 x^{2}+3 x +2\right )^{2} \sqrt {1+2 x}}-\frac {95145 \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}}{20436626}-\frac {876315 \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}}{286112764}-\frac {475725 \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 )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \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}}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \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}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {95145 \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}}{20436626}+\frac {876315 \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}}{286112764}-\frac {475725 \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 )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \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}}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \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}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(648\) |
Input:
int(1/(1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
-375/20436626/(1+2*x)^(1/2)/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(1/350*(5*x^2+3* x+2)^2*((10*5^(1/2)*7^(1/2)-20)^(1/2)*(88802*5^(1/2)+58421*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)-l n(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)+2243408*(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)-94605/18092))*(1+2*x)^(1/2)+335172*(815449/810900*x^ 2+45014/40545*x^3+x^4+33903/90100*x+429487/4054500)*(10*5^(1/2)*7^(1/2)-20 )^(1/2))/(5*x^2+3*x+2)^2
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (186) = 372\).
Time = 0.09 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/659246*(30*sqrt(1/434)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*sq rt(2711989525*sqrt(5/7) - 2257111762)*arctan(14/71603149*sqrt(1/434)*sqrt( 2*x + 1)*sqrt(2711989525*sqrt(5/7) - 2257111762)*(18092*sqrt(5/7) + 13515) + 1/2219697619*sqrt(2711989525*sqrt(5/7) + 2257111762)*sqrt(2711989525*sq rt(5/7) - 2257111762)*(7*sqrt(5/7) + 2)) - 30*sqrt(1/434)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*sqrt(2711989525*sqrt(5/7) - 2257111762)*arc tan(-14/71603149*sqrt(1/434)*sqrt(2*x + 1)*sqrt(2711989525*sqrt(5/7) - 225 7111762)*(18092*sqrt(5/7) + 13515) + 1/2219697619*sqrt(2711989525*sqrt(5/7 ) + 2257111762)*sqrt(2711989525*sqrt(5/7) - 2257111762)*(7*sqrt(5/7) + 2)) - 15*sqrt(1/434)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*sqrt(2711 989525*sqrt(5/7) + 2257111762)*log(210*sqrt(1/434)*sqrt(2*x + 1)*sqrt(2711 989525*sqrt(5/7) + 2257111762)*(58421*sqrt(5/7) - 63430) + 10740472350*x + 7518330645*sqrt(5/7) + 5370236175) + 15*sqrt(1/434)*(50*x^5 + 85*x^4 + 88 *x^3 + 53*x^2 + 20*x + 4)*sqrt(2711989525*sqrt(5/7) + 2257111762)*log(-210 *sqrt(1/434)*sqrt(2*x + 1)*sqrt(2711989525*sqrt(5/7) + 2257111762)*(58421* sqrt(5/7) - 63430) + 10740472350*x + 7518330645*sqrt(5/7) + 5370236175) - (4054500*x^4 + 4501400*x^3 + 4077245*x^2 + 1525635*x + 429487)*sqrt(2*x + 1))/(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)
\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \] Input:
integrate(1/(1+2*x)**(3/2)/(5*x**2+3*x+2)**3,x)
Output:
Integral(1/((2*x + 1)**(3/2)*(5*x**2 + 3*x + 2)**3), x)
\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)^3*(2*x + 1)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (186) = 372\).
Time = 0.61 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.57 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
Output:
-3/981366780520*sqrt(31)*(567630*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq rt(-140*sqrt(35) + 2450) - 2703*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450 )^(3/2) + 5406*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 1135260*(7/5)^(3/ 4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 17730160*sqrt(31)*(7/5)^( 1/4)*sqrt(-140*sqrt(35) + 2450) - 35460320*(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) + sq rt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 3/981366780520*sqrt(31)*(567630 *sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 2703* sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 5406*(7/5)^(3/4)*(140* sqrt(35) + 2450)^(3/2) + 1135260*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2* sqrt(35) - 35) - 17730160*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 35460320*(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)) - 3/1962733561040*sqrt(31)*(2703*sqrt(31)*(7/5)^(3/4)*(140*sqrt( 35) + 2450)^(3/2) + 567630*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)* (2*sqrt(35) - 35) - 1135260*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(3 5) + 2450) + 5406*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 17730160*sqrt (31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 35460320*(7/5)^(1/4)*sqrt(-14 0*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) + 3/1962733561040*sqrt(31)*(2703*sqrt(31)*(7...
Time = 0.11 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {274928\,x}{235445}-\frac {1362758\,{\left (2\,x+1\right )}^2}{1648115}+\frac {144304\,{\left (2\,x+1\right )}^3}{329623}-\frac {81090\,{\left (2\,x+1\right )}^4}{329623}+\frac {256792}{1177225}}{\frac {49\,\sqrt {2\,x+1}}{25}-\frac {56\,{\left (2\,x+1\right )}^{3/2}}{25}+\frac {86\,{\left (2\,x+1\right )}^{5/2}}{25}-\frac {8\,{\left (2\,x+1\right )}^{7/2}}{5}+{\left (2\,x+1\right )}^{9/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}+\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}-\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191} \] Input:
int(1/((2*x + 1)^(3/2)*(3*x + 5*x^2 + 2)^3),x)
Output:
((274928*x)/235445 - (1362758*(2*x + 1)^2)/1648115 + (144304*(2*x + 1)^3)/ 329623 - (81090*(2*x + 1)^4)/329623 + 256792/1177225)/((49*(2*x + 1)^(1/2) )/25 - (56*(2*x + 1)^(3/2))/25 + (86*(2*x + 1)^(5/2))/25 - (8*(2*x + 1)^(7 /2))/5 + (2*x + 1)^(9/2)) + (217^(1/2)*atan((217^(1/2)*(2*x + 1)^(1/2)*(22 57111762 - 31^(1/2)*71603149i)^(1/2)*32187888i)/(1826102771022103*((31^(1/ 2)*582343269696i)/260871824431729 - 1880448604848/260871824431729)) + (643 75776*31^(1/2)*217^(1/2)*(2*x + 1)^(1/2)*(2257111762 - 31^(1/2)*71603149i) ^(1/2))/(56609185901685193*((31^(1/2)*582343269696i)/260871824431729 - 188 0448604848/260871824431729)))*(2257111762 - 31^(1/2)*71603149i)^(1/2)*15i) /71528191 - (217^(1/2)*atan((217^(1/2)*(2*x + 1)^(1/2)*(31^(1/2)*71603149i + 2257111762)^(1/2)*32187888i)/(1826102771022103*((31^(1/2)*582343269696i )/260871824431729 + 1880448604848/260871824431729)) - (64375776*31^(1/2)*2 17^(1/2)*(2*x + 1)^(1/2)*(31^(1/2)*71603149i + 2257111762)^(1/2))/(5660918 5901685193*((31^(1/2)*582343269696i)/260871824431729 + 1880448604848/26087 1824431729)))*(31^(1/2)*71603149i + 2257111762)^(1/2)*15i)/71528191
Time = 0.20 (sec) , antiderivative size = 1882, normalized size of antiderivative = 7.44 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x)
Output:
(43815750*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**4 + 52578900*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 + 50826270*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 + 21031560*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 + 70 10520*sqrt(2*x + 1)*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*s qrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2))) - 66601500 *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**4 - 79921800 *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 - 77257740 *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 - 31968720 *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 - 10656240*sq rt(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))) - 43815750*sqrt...