\(\int \frac {(1+2 x)^{9/2}}{(2+3 x+5 x^2)^3} \, dx\) [566]

Optimal result

Integrand size = 22, antiderivative size = 253 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {16}{775} \sqrt {1+2 x}-\frac {(5-4 x) (1+2 x)^{7/2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {\sqrt {1+2 x} (10067+12518 x)}{48050 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{310} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{24025}+\frac {3 \sqrt {\frac {1}{310} \left (250141922+64681225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{24025}-\frac {3 \sqrt {\frac {1}{310} \left (-250141922+64681225 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right )}{24025} \] Output:

-16/775*(1+2*x)^(1/2)-1/62*(5-4*x)*(1+2*x)^(7/2)/(5*x^2+3*x+2)^2-(1+2*x)^( 
1/2)*(10067+12518*x)/(240250*x^2+144150*x+96100)-3/7447750*(77543995820+20 
051179750*35^(1/2))^(1/2)*arctan(((20+10*35^(1/2))^(1/2)-10*(1+2*x)^(1/2)) 
/(-20+10*35^(1/2))^(1/2))+3/7447750*(77543995820+20051179750*35^(1/2))^(1/ 
2)*arctan(((20+10*35^(1/2))^(1/2)+10*(1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2 
))-3/7447750*(-77543995820+20051179750*35^(1/2))^(1/2)*arctanh((20+10*35^( 
1/2))^(1/2)*(1+2*x)^(1/2)/(5+35^(1/2)+10*x))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.57 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {-\frac {155 \sqrt {1+2 x} \left (27977+87291 x+144557 x^2+86150 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {155 \left (250141922-52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {155 \left (250141922+52010281 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{3723875} \] Input:

Integrate[(1 + 2*x)^(9/2)/(2 + 3*x + 5*x^2)^3,x]
 

Output:

((-155*Sqrt[1 + 2*x]*(27977 + 87291*x + 144557*x^2 + 86150*x^3))/(2*(2 + 3 
*x + 5*x^2)^2) + 3*Sqrt[155*(250141922 - (52010281*I)*Sqrt[31])]*ArcTan[Sq 
rt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 3*Sqrt[155*(250141922 + (52010281 
*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/3723875
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.43, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1164, 1233, 27, 1196, 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.

\(\displaystyle \int \frac {(2 x+1)^{9/2}}{\left (5 x^2+3 x+2\right )^3} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle \frac {1}{62} \int \frac {(47-4 x) (2 x+1)^{5/2}}{\left (5 x^2+3 x+2\right )^2}dx-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle \frac {1}{62} \left (-\frac {1}{155} \int -\frac {3 (1423-528 x) \sqrt {2 x+1}}{5 x^2+3 x+2}dx-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \int \frac {(1423-528 x) \sqrt {2 x+1}}{5 x^2+3 x+2}dx-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1196

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {1}{5} \int \frac {14758 x+9227}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {2}{5} \int \frac {2 (7379 (2 x+1)+1848)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {4}{5} \int \frac {7379 (2 x+1)+1848}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1483

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {4}{5} \left (\frac {\int \frac {1848 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (9240-7379 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {\left (9240-7379 \sqrt {35}\right ) \sqrt {2 x+1}+1848 \sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {4}{5} \left (\frac {\frac {1}{2} \sqrt {3501986908+905537150 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{2} \sqrt {3501986908+905537150 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {4}{5} \left (\frac {\frac {1}{2} \sqrt {3501986908+905537150 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{2} \sqrt {3501986908+905537150 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {4}{5} \left (\frac {\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {3501986908+905537150 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {3501986908+905537150 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {4}{5} \left (\frac {\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\sqrt {\frac {3501986908+905537150 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\sqrt {\frac {3501986908+905537150 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{62} \left (\frac {3}{155} \left (\frac {4}{5} \left (\frac {\sqrt {\frac {3501986908+905537150 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\sqrt {\frac {3501986908+905537150 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{10} \left (9240-7379 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {1056}{5} \sqrt {2 x+1}\right )-\frac {(1143-1088 x) (2 x+1)^{3/2}}{155 \left (5 x^2+3 x+2\right )}\right )-\frac {(5-4 x) (2 x+1)^{7/2}}{62 \left (5 x^2+3 x+2\right )^2}\)

Input:

Int[(1 + 2*x)^(9/2)/(2 + 3*x + 5*x^2)^3,x]
 

Output:

-1/62*((5 - 4*x)*(1 + 2*x)^(7/2))/(2 + 3*x + 5*x^2)^2 + (-1/155*((1143 - 1 
088*x)*(1 + 2*x)^(3/2))/(2 + 3*x + 5*x^2) + (3*((-1056*Sqrt[1 + 2*x])/5 + 
(4*((Sqrt[(3501986908 + 905537150*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[( 
-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] - ( 
(9240 - 7379*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[(3501986908 + 90 
5537150*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 
10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] + ((9240 - 7379*Sqrt[35])*Log[ 
Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10)/(2*Sq 
rt[14*(2 + Sqrt[35])])))/5))/155)/62
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 1083

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 1164

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]
 

rule 1196

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]
 

rule 1197

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]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 1483

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]
 

Maple [A] (verified)

Time = 2.38 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(-\frac {26706500 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{3}+\frac {144557}{86150} x^{2}+\frac {87291}{86150} x +\frac {27977}{86150}\right ) \sqrt {1+2 x}+3 \left (5 x^{2}+3 x +2\right )^{2} \left (\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (23998 \sqrt {5}-39535 \sqrt {7}\right ) \left (\ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )-\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+163680 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\frac {7379}{264}+\sqrt {5}\, \sqrt {7}\right )\right )}{14895500 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (5 x^{2}+3 x +2\right )^{2}}\)	\(283\)
derivativedivides	\(\frac {-\frac {3446 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {30664 \left (1+2 x \right )^{\frac {5}{2}}}{24025}-\frac {29386 \left (1+2 x \right )^{\frac {3}{2}}}{24025}-\frac {77616 \sqrt {1+2 x}}{24025}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}+\frac {\left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}-\frac {\left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
default	\(\frac {-\frac {3446 \left (1+2 x \right )^{\frac {7}{2}}}{961}-\frac {30664 \left (1+2 x \right )^{\frac {5}{2}}}{24025}-\frac {29386 \left (1+2 x \right )^{\frac {3}{2}}}{24025}-\frac {77616 \sqrt {1+2 x}}{24025}}{\left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}+\frac {3 \left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}+\frac {\left (-23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3 \left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{14895500}+\frac {3 \left (16368 \sqrt {5}\, \sqrt {7}-\frac {\left (23998 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-39535 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(435\)
trager	\(-\frac {\left (86150 x^{3}+144557 x^{2}+87291 x +27977\right ) \sqrt {1+2 x}}{48050 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) \ln \left (-\frac {-36223395840 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{4} x -3072346456740640 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) x +110841916417883322000 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} \sqrt {1+2 x}+6077383859395584 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right )+217589498184877183750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right ) x +118884731389716793460850625 \sqrt {1+2 x}+794655418120440646000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+96100 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2}[phone]\right )}{2480 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} x [phone] x [phone]}\right )}{3723875}+\frac {6 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right ) \ln \left (\frac {-362233958400 x \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{5}-115421533122664640 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{3} x +3575545690899462000 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} \sqrt {1+2 x}-60773838593955840 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{3}-6367063751256090807636 x \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )-3113705955055050326603275 \sqrt {1+2 x}-4313191619771312411552 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )}{2480 \operatorname {RootOf}\left (1230080 \textit {\_Z}^{4}[phone] \textit {\_Z}^{2}+29285626072504375\right )^{2} x [phone] x -208041124}\right )}{24025}\)	\(458\)
risch	\(-\frac {\left (86150 x^{3}+144557 x^{2}+87291 x +27977\right ) \sqrt {1+2 x}}{48050 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {35997 \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{7447750}+\frac {23721 \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2979100}-\frac {35997 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {23721 \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}}{1489550 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1584 \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}}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {35997 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{7447750}-\frac {23721 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2979100}-\frac {35997 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{744775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {23721 \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}}{1489550 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {1584 \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}}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(643\)

Input:

int((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/14895500/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(26706500*(10*5^(1/2)*7^(1/2)-20 
)^(1/2)*(x^3+144557/86150*x^2+87291/86150*x+27977/86150)*(1+2*x)^(1/2)+3*( 
5*x^2+3*x+2)^2*((10*5^(1/2)*7^(1/2)-20)^(1/2)*(23998*5^(1/2)-39535*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)+163680*(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)))*(7379/264+5^(1/2)*7^(1/2))))/(5*x^2+3*x+2)^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.44 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=-\frac {6 \, \sqrt {\frac {1}{310}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {323406125 \, \sqrt {\frac {7}{5}} + 250141922} \arctan \left (\frac {10}{52010281} \, \sqrt {\frac {1}{310}} {\left (\sqrt {\frac {1}{310}} \sqrt {323406125 \, \sqrt {\frac {7}{5}} - 250141922} {\left (5 \, \sqrt {\frac {7}{5}} + 2\right )} + \sqrt {2 \, x + 1} {\left (1320 \, \sqrt {\frac {7}{5}} - 7379\right )}\right )} \sqrt {323406125 \, \sqrt {\frac {7}{5}} + 250141922}\right ) - 6 \, \sqrt {\frac {1}{310}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {323406125 \, \sqrt {\frac {7}{5}} + 250141922} \arctan \left (\frac {10}{52010281} \, \sqrt {\frac {1}{310}} {\left (\sqrt {\frac {1}{310}} \sqrt {323406125 \, \sqrt {\frac {7}{5}} - 250141922} {\left (5 \, \sqrt {\frac {7}{5}} + 2\right )} - \sqrt {2 \, x + 1} {\left (1320 \, \sqrt {\frac {7}{5}} - 7379\right )}\right )} \sqrt {323406125 \, \sqrt {\frac {7}{5}} + 250141922}\right ) + 3 \, \sqrt {\frac {1}{310}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {323406125 \, \sqrt {\frac {7}{5}} - 250141922} \log \left (6 \, \sqrt {\frac {1}{310}} \sqrt {2 \, x + 1} \sqrt {323406125 \, \sqrt {\frac {7}{5}} - 250141922} {\left (39535 \, \sqrt {\frac {7}{5}} + 23998\right )} + 312061686 \, x + 156030843 \, \sqrt {\frac {7}{5}} + 156030843\right ) - 3 \, \sqrt {\frac {1}{310}} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {323406125 \, \sqrt {\frac {7}{5}} - 250141922} \log \left (-6 \, \sqrt {\frac {1}{310}} \sqrt {2 \, x + 1} \sqrt {323406125 \, \sqrt {\frac {7}{5}} - 250141922} {\left (39535 \, \sqrt {\frac {7}{5}} + 23998\right )} + 312061686 \, x + 156030843 \, \sqrt {\frac {7}{5}} + 156030843\right ) + {\left (86150 \, x^{3} + 144557 \, x^{2} + 87291 \, x + 27977\right )} \sqrt {2 \, x + 1}}{48050 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:

integrate((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
 

Output:

-1/48050*(6*sqrt(1/310)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(3234061 
25*sqrt(7/5) + 250141922)*arctan(10/52010281*sqrt(1/310)*(sqrt(1/310)*sqrt 
(323406125*sqrt(7/5) - 250141922)*(5*sqrt(7/5) + 2) + sqrt(2*x + 1)*(1320* 
sqrt(7/5) - 7379))*sqrt(323406125*sqrt(7/5) + 250141922)) - 6*sqrt(1/310)* 
(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(323406125*sqrt(7/5) + 250141922 
)*arctan(10/52010281*sqrt(1/310)*(sqrt(1/310)*sqrt(323406125*sqrt(7/5) - 2 
50141922)*(5*sqrt(7/5) + 2) - sqrt(2*x + 1)*(1320*sqrt(7/5) - 7379))*sqrt( 
323406125*sqrt(7/5) + 250141922)) + 3*sqrt(1/310)*(25*x^4 + 30*x^3 + 29*x^ 
2 + 12*x + 4)*sqrt(323406125*sqrt(7/5) - 250141922)*log(6*sqrt(1/310)*sqrt 
(2*x + 1)*sqrt(323406125*sqrt(7/5) - 250141922)*(39535*sqrt(7/5) + 23998) 
+ 312061686*x + 156030843*sqrt(7/5) + 156030843) - 3*sqrt(1/310)*(25*x^4 + 
 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(323406125*sqrt(7/5) - 250141922)*log(-6* 
sqrt(1/310)*sqrt(2*x + 1)*sqrt(323406125*sqrt(7/5) - 250141922)*(39535*sqr 
t(7/5) + 23998) + 312061686*x + 156030843*sqrt(7/5) + 156030843) + (86150* 
x^3 + 144557*x^2 + 87291*x + 27977)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x 
^2 + 12*x + 4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((1+2*x)**(9/2)/(5*x**2+3*x+2)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {9}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \,d x } \] Input:

integrate((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
 

Output:

integrate((2*x + 1)^(9/2)/(5*x^2 + 3*x + 2)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (186) = 372\).

Time = 0.82 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.54 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x, algorithm="giac")
 

Output:

3/1788204775000*sqrt(31)*(1549590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*s 
qrt(-140*sqrt(35) + 2450) - 7379*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 245 
0)^(3/2) + 14758*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 3099180*(7/5)^( 
3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 9055200*sqrt(31)*(7/5)^ 
(1/4)*sqrt(-140*sqrt(35) + 2450) + 18110400*(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) + s 
qrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) + 3/1788204775000*sqrt(31)*(1549 
590*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 73 
79*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 14758*(7/5)^(3/4)*( 
140*sqrt(35) + 2450)^(3/2) + 3099180*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450) 
*(2*sqrt(35) - 35) + 9055200*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 245 
0) + 18110400*(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)) + 3/3576409550000*sqrt(31)*(7379*sqrt(31)*(7/5)^(3/4)*(140*sq 
rt(35) + 2450)^(3/2) + 1549590*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 24 
50)*(2*sqrt(35) - 35) - 3099180*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sq 
rt(35) + 2450) + 14758*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9055200* 
sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 18110400*(7/5)^(1/4)*sqrt 
(-140*sqrt(35) + 2450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) 
 + 1/2) + 2*x + sqrt(7/5) + 1) - 3/3576409550000*sqrt(31)*(7379*sqrt(31...
 

Mupad [B] (verification not implemented)

Time = 5.71 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.97 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {77616\,\sqrt {2\,x+1}}{600625}+\frac {29386\,{\left (2\,x+1\right )}^{3/2}}{600625}+\frac {30664\,{\left (2\,x+1\right )}^{5/2}}{600625}+\frac {3446\,{\left (2\,x+1\right )}^{7/2}}{24025}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{45093798828125\,\left (-\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}-\frac {46760544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{1397907763671875\,\left (-\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}\right )\,\sqrt {-250141922-\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{3723875}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}\,23380272{}\mathrm {i}}{45093798828125\,\left (\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}+\frac {46760544\,\sqrt {31}\,\sqrt {155}\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,\sqrt {2\,x+1}}{1397907763671875\,\left (\frac {1294074674928}{9018759765625}+\frac {\sqrt {31}\,43206742656{}\mathrm {i}}{9018759765625}\right )}\right )\,\sqrt {-250141922+\sqrt {31}\,52010281{}\mathrm {i}}\,3{}\mathrm {i}}{3723875} \] Input:

int((2*x + 1)^(9/2)/(3*x + 5*x^2 + 2)^3,x)
 

Output:

((77616*(2*x + 1)^(1/2))/600625 + (29386*(2*x + 1)^(3/2))/600625 + (30664* 
(2*x + 1)^(5/2))/600625 + (3446*(2*x + 1)^(7/2))/24025)/((112*x)/25 - (86* 
(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) - (155^(1/2)*ata 
n((155^(1/2)*(- 31^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*2338 
0272i)/(45093798828125*((31^(1/2)*43206742656i)/9018759765625 - 1294074674 
928/9018759765625)) - (46760544*31^(1/2)*155^(1/2)*(- 31^(1/2)*52010281i - 
 250141922)^(1/2)*(2*x + 1)^(1/2))/(1397907763671875*((31^(1/2)*4320674265 
6i)/9018759765625 - 1294074674928/9018759765625)))*(- 31^(1/2)*52010281i - 
 250141922)^(1/2)*3i)/3723875 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)*52010 
281i - 250141922)^(1/2)*(2*x + 1)^(1/2)*23380272i)/(45093798828125*((31^(1 
/2)*43206742656i)/9018759765625 + 1294074674928/9018759765625)) + (4676054 
4*31^(1/2)*155^(1/2)*(31^(1/2)*52010281i - 250141922)^(1/2)*(2*x + 1)^(1/2 
))/(1397907763671875*((31^(1/2)*43206742656i)/9018759765625 + 129407467492 
8/9018759765625)))*(31^(1/2)*52010281i - 250141922)^(1/2)*3i)/3723875
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1653, normalized size of antiderivative = 6.53 \[ \int \frac {(1+2 x)^{9/2}}{\left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((1+2*x)^(9/2)/(5*x^2+3*x+2)^3,x)
 

Output:

( - 5930250*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 - 7116300*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 - 6879090*sqrt(sqrt(35) - 2) 
*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqr 
t(sqrt(35) - 2)*sqrt(2)))*x**2 - 2846520*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 - 948840*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))) - 359970 
0*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2* 
x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**4 - 4319640*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 - 4175652*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 - 1727856*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 - 575952*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))) + 5930250*sqrt(sqr 
t(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1)*sqr 
t(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**4 + 7116300*sqrt(sqrt(35) - 2)*s...