Integrand size = 22, antiderivative size = 236 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (5682718+968975 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right ) \] Output:
604/775*(1+2*x)^(1/2)-8/155*(1+2*x)^(3/2)-(5-4*x)*(1+2*x)^(5/2)/(155*x^2+9 3*x+62)+1/120125*(-1761642580+300382250*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))-1/120125*(-1761642 580+300382250*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))-1/120125*(1761642580+300382250*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 = 1.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.58 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (\frac {155 \sqrt {1+2 x} \left (1003+1132 x+2480 x^2\right )}{4+6 x+10 x^2}-\sqrt {155 \left (-5682718+135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {155 \left (-5682718-135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{120125} \] Input:
Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]
Output:
(2*((155*Sqrt[1 + 2*x]*(1003 + 1132*x + 2480*x^2))/(4 + 6*x + 10*x^2) - Sq rt[155*(-5682718 + (135439*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]* Sqrt[1 + 2*x]] - Sqrt[155*(-5682718 - (135439*I)*Sqrt[31])]*ArcTan[Sqrt[(I /7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/120125
Time = 0.68 (sec) , antiderivative size = 345, 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 = {1164, 1196, 1196, 25, 1197, 27, 1483, 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.
| \(\displaystyle \int \frac {(2 x+1)^{7/2}}{\left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle \frac {1}{31} \int \frac {(29-12 x) (2 x+1)^{3/2}}{5 x^2+3 x+2}dx-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \int \frac {\sqrt {2 x+1} (302 x+193)}{5 x^2+3 x+2}dx-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {1}{5} \int -\frac {243-1628 x}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx+\frac {604}{5} \sqrt {2 x+1}\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {1}{5} \int \frac {243-1628 x}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {2}{5} \int \frac {2 (1057-814 (2 x+1))}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {4}{5} \int \frac {1057-814 (2 x+1)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {4}{5} \left (\frac {\int \frac {1057 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (5285+814 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {\left (5285+814 \sqrt {35}\right ) \sqrt {2 x+1}+1057 \sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {4}{5} \left (\frac {\sqrt {\frac {7}{2} \left (968975 \sqrt {35}-5682718\right )} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\sqrt {\frac {7}{2} \left (968975 \sqrt {35}-5682718\right )} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {4}{5} \left (\frac {\sqrt {\frac {7}{2} \left (968975 \sqrt {35}-5682718\right )} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\sqrt {\frac {7}{2} \left (968975 \sqrt {35}-5682718\right )} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {4}{5} \left (\frac {\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {14 \left (968975 \sqrt {35}-5682718\right )} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {14 \left (968975 \sqrt {35}-5682718\right )} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {4}{5} \left (\frac {\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\sqrt {\frac {7 \left (968975 \sqrt {35}-5682718\right )}{5 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\sqrt {\frac {7 \left (968975 \sqrt {35}-5682718\right )}{5 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{31} \left (\frac {1}{5} \left (\frac {604}{5} \sqrt {2 x+1}-\frac {4}{5} \left (\frac {\sqrt {\frac {7 \left (968975 \sqrt {35}-5682718\right )}{5 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\sqrt {\frac {7 \left (968975 \sqrt {35}-5682718\right )}{5 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{10} \left (5285+814 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )\right )-\frac {8}{5} (2 x+1)^{3/2}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}\) |
Input:
Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]
Output:
-1/31*((5 - 4*x)*(1 + 2*x)^(5/2))/(2 + 3*x + 5*x^2) + ((-8*(1 + 2*x)^(3/2) )/5 + ((604*Sqrt[1 + 2*x])/5 - (4*((Sqrt[(7*(-5682718 + 968975*Sqrt[35]))/ (5*(-2 + Sqrt[35]))]*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/ Sqrt[10*(-2 + Sqrt[35])]] - ((5285 + 814*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] )]) + (Sqrt[(7*(-5682718 + 968975*Sqrt[35]))/(5*(-2 + Sqrt[35]))]*ArcTan[( Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] + (( 5285 + 814*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)/5)/31
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)*(d*b - 2*a*e + (2*c*d - b*e)*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 - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
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_)^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 = 2.38 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {768800 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{2}+\frac {283}{620} x +\frac {1003}{2480}\right ) \sqrt {1+2 x}+\left (5 x^{2}+3 x +2\right ) \left (\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (3657 \sqrt {5}+2560 \sqrt {7}\right ) \left (\ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )-\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+93620 \left (-\frac {814}{151}+\sqrt {5}\, \sqrt {7}\right ) \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )\right )}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (1201250 x^{2}+720750 x +480500\right )}\) | \(274\) |
| derivativedivides | \(\frac {16 \sqrt {1+2 x}}{25}+\frac {-\frac {712 \left (1+2 x \right )^{\frac {3}{2}}}{3875}+\frac {756 \sqrt {1+2 x}}{3875}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}-\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{240250}-\frac {2 \left (9362 \sqrt {5}\, \sqrt {7}+\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{240250}+\frac {2 \left (-9362 \sqrt {5}\, \sqrt {7}-\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(424\) |
| default | \(\frac {16 \sqrt {1+2 x}}{25}+\frac {-\frac {712 \left (1+2 x \right )^{\frac {3}{2}}}{3875}+\frac {756 \sqrt {1+2 x}}{3875}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}-\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{240250}-\frac {2 \left (9362 \sqrt {5}\, \sqrt {7}+\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{240250}+\frac {2 \left (-9362 \sqrt {5}\, \sqrt {7}-\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(424\) |
| trager | \(\frac {\left (2480 x^{2}+1132 x +1003\right ) \sqrt {1+2 x}}{3875 x^{2}+2325 x +1550}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-440410645\right ) \ln \left (\frac {-32135840 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-440410645\right ) \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{4} x +885649429640 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-440410645\right ) x +1081017958534500 \sqrt {1+2 x}\, \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-56160593984 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-440410645\right )-4889233526572500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-440410645\right ) x -20849799689574944375 \sqrt {1+2 x}+1118889208556000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-440410645\right )}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2} x -6089035 x -541756}\right )}{120125}+\frac {2 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right ) \ln \left (-\frac {160679200 x \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{5}-1462685639320 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{3} x +34871547049500 \sqrt {1+2 x}\, \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2}-280802969920 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{3}-2735202958459332 x \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )+33331705773254525 \sqrt {1+2 x}-446948973045024 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+6572387854375\right )^{2} x -5276401 x +541756}\right )}{775}\) | \(453\) |
| risch | \(\frac {\left (2480 x^{2}+1132 x +1003\right ) \sqrt {1+2 x}}{3875 x^{2}+2325 x +1550}+\frac {3657 \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}}{240250}+\frac {256 \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}}{24025}+\frac {3657 \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 )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {512 \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}}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {604 \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}}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3657 \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}}{240250}-\frac {256 \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}}{24025}+\frac {3657 \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 )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {512 \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}}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {604 \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}}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(638\) |
Input:
int((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
1/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(768800*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x^2 +283/620*x+1003/2480)*(1+2*x)^(1/2)+(5*x^2+3*x+2)*((10*5^(1/2)*7^(1/2)-20) ^(1/2)*(3657*5^(1/2)+2560*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)+93 620*(-814/151+5^(1/2)*7^(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))-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)))))/( 1201250*x^2+720750*x+480500)
Time = 0.09 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.26 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {2 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} - \frac {2841359}{155}} \arctan \left (\frac {10}{135439} \, \sqrt {2 \, x + 1} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} - \frac {2841359}{155}} {\left (755 \, \sqrt {\frac {7}{5}} + 814\right )} + \frac {10}{135439} \, \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} + \frac {2841359}{155}} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} - \frac {2841359}{155}} {\left (5 \, \sqrt {\frac {7}{5}} + 2\right )}\right ) - 2 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} - \frac {2841359}{155}} \arctan \left (-\frac {10}{135439} \, \sqrt {2 \, x + 1} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} - \frac {2841359}{155}} {\left (755 \, \sqrt {\frac {7}{5}} + 814\right )} + \frac {10}{135439} \, \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} + \frac {2841359}{155}} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} - \frac {2841359}{155}} {\left (5 \, \sqrt {\frac {7}{5}} + 2\right )}\right ) - {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} + \frac {2841359}{155}} \log \left (2 \, \sqrt {2 \, x + 1} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} + \frac {2841359}{155}} {\left (2560 \, \sqrt {\frac {7}{5}} - 3657\right )} + 270878 \, x + 135439 \, \sqrt {\frac {7}{5}} + 135439\right ) + {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} + \frac {2841359}{155}} \log \left (-2 \, \sqrt {2 \, x + 1} \sqrt {\frac {968975}{62} \, \sqrt {\frac {7}{5}} + \frac {2841359}{155}} {\left (2560 \, \sqrt {\frac {7}{5}} - 3657\right )} + 270878 \, x + 135439 \, \sqrt {\frac {7}{5}} + 135439\right ) - {\left (2480 \, x^{2} + 1132 \, x + 1003\right )} \sqrt {2 \, x + 1}}{775 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \] Input:
integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
-1/775*(2*(5*x^2 + 3*x + 2)*sqrt(968975/62*sqrt(7/5) - 2841359/155)*arctan (10/135439*sqrt(2*x + 1)*sqrt(968975/62*sqrt(7/5) - 2841359/155)*(755*sqrt (7/5) + 814) + 10/135439*sqrt(968975/62*sqrt(7/5) + 2841359/155)*sqrt(9689 75/62*sqrt(7/5) - 2841359/155)*(5*sqrt(7/5) + 2)) - 2*(5*x^2 + 3*x + 2)*sq rt(968975/62*sqrt(7/5) - 2841359/155)*arctan(-10/135439*sqrt(2*x + 1)*sqrt (968975/62*sqrt(7/5) - 2841359/155)*(755*sqrt(7/5) + 814) + 10/135439*sqrt (968975/62*sqrt(7/5) + 2841359/155)*sqrt(968975/62*sqrt(7/5) - 2841359/155 )*(5*sqrt(7/5) + 2)) - (5*x^2 + 3*x + 2)*sqrt(968975/62*sqrt(7/5) + 284135 9/155)*log(2*sqrt(2*x + 1)*sqrt(968975/62*sqrt(7/5) + 2841359/155)*(2560*s qrt(7/5) - 3657) + 270878*x + 135439*sqrt(7/5) + 135439) + (5*x^2 + 3*x + 2)*sqrt(968975/62*sqrt(7/5) + 2841359/155)*log(-2*sqrt(2*x + 1)*sqrt(96897 5/62*sqrt(7/5) + 2841359/155)*(2560*sqrt(7/5) - 3657) + 270878*x + 135439* sqrt(7/5) + 135439) - (2480*x^2 + 1132*x + 1003)*sqrt(2*x + 1))/(5*x^2 + 3 *x + 2)
\[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {7}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \] Input:
integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)
Output:
Integral((2*x + 1)**(7/2)/(5*x**2 + 3*x + 2)**2, x)
\[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \] Input:
integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (169) = 338\).
Time = 0.78 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.68 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")
Output:
1/14421006250*sqrt(31)*(85470*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt( -140*sqrt(35) + 2450) - 407*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3 /2) + 814*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 170940*(7/5)^(3/4)*sqr t(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2589650*sqrt(31)*(7/5)^(1/4)*sq rt(-140*sqrt(35) + 2450) - 5179300*(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)) + 1/14421006250*sqrt(31)*(85470*sqrt(31)* (7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 407*sqrt(31)*(7 /5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 814*(7/5)^(3/4)*(140*sqrt(35) + 2 450)^(3/2) + 170940*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35 ) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 5179300*(7/5 )^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*sq rt(1/35*sqrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 1/2 8842012500*sqrt(31)*(407*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 85470*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 170940*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 814*(7/ 5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt( 140*sqrt(35) + 2450) + 5179300*(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) - 1/28842012500*sqrt(31)*(407*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + ...
Time = 0.09 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.92 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {16\,\sqrt {2\,x+1}}{25}+\frac {\frac {756\,\sqrt {2\,x+1}}{3875}-\frac {712\,{\left (2\,x+1\right )}^{3/2}}{3875}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}+\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}-\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125} \] Input:
int((2*x + 1)^(7/2)/(3*x + 5*x^2 + 2)^2,x)
Output:
(16*(2*x + 1)^(1/2))/25 + ((756*(2*x + 1)^(1/2))/3875 - (712*(2*x + 1)^(3/ 2))/3875)/((2*x + 1)^2 - (8*x)/5 + 3/5) - (155^(1/2)*atan((155^(1/2)*(5682 718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^(1/2)*559232i)/(46923828125*((31^( 1/2)*591108224i)/9384765625 - 2004287488/9384765625)) + (1118464*31^(1/2)* 155^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^(1/2))/(14546386718 75*((31^(1/2)*591108224i)/9384765625 - 2004287488/9384765625)))*(5682718 - 31^(1/2)*135439i)^(1/2)*2i)/120125 + (155^(1/2)*atan((155^(1/2)*(31^(1/2) *135439i + 5682718)^(1/2)*(2*x + 1)^(1/2)*559232i)/(46923828125*((31^(1/2) *591108224i)/9384765625 + 2004287488/9384765625)) - (1118464*31^(1/2)*155^ (1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x + 1)^(1/2))/(1454638671875*( (31^(1/2)*591108224i)/9384765625 + 2004287488/9384765625)))*(31^(1/2)*1354 39i + 5682718)^(1/2)*2i)/120125
Time = 0.23 (sec) , antiderivative size = 980, normalized size of antiderivative = 4.15 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x)
Output:
( - 25600*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 - 15360*sqrt(sq rt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sq rt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x - 10240*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))) + 36570*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 + 21942*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 + 14628*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))) + 25600*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 + 15360*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 + 102 40*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))) - 36570*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 - 21942*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 - 14628*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*...