\(\int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx\) [335]

Optimal result

Integrand size = 22, antiderivative size = 187 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {4 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{675 e}+\frac {\operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \] Output:

4/675*(2+34^(1/2))^(1/2)*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(5/3)),1/15 
*(510-30*34^(1/2))^(1/2))/e+1/45*InverseJacobiAM(1/2*d+1/2*e*x-1/2*arctan( 
5/3),1/15*(510-30*34^(1/2))^(1/2))/(2+34^(1/2))^(1/2)/e-1/45*(5*cos(e*x+d) 
-3*sin(e*x+d))/e/(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2)+4/675*(5*cos(e*x+d)-3 
*sin(e*x+d))/e/(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 2.39 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\frac {-24 \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}+\frac {272}{3} \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}+\frac {100 (5+17 \sin (d+e x))}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}-\frac {10 (115+136 \sin (d+e x))}{3 \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+23 \sqrt {\frac {10}{3}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}+\frac {20 \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {\frac {1}{17}+\frac {\cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {34}}}}-\frac {20 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \csc \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}}{6750 e} \] Input:

Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-5/2),x]
 

Output:

(-24*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] + (272*Sqrt[2 + 3*Cos[d 
 + e*x] + 5*Sin[d + e*x]])/3 + (100*(5 + 17*Sin[d + e*x]))/(2 + 3*Cos[d + 
e*x] + 5*Sin[d + e*x])^(3/2) - (10*(115 + 136*Sin[d + e*x]))/(3*Sqrt[2 + 3 
*Cos[d + e*x] + 5*Sin[d + e*x]]) + 23*Sqrt[10/3]*AppellF1[1/2, 1/2, 1/2, 3 
/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] 
 + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcT 
an[3/5]]^2]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + Arc 
Tan[3/5]]] + (20*Sin[d + e*x - ArcTan[5/3]])/Sqrt[1/17 + Cos[d + e*x - Arc 
Tan[5/3]]/Sqrt[34]] - (20*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[3 
4] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d 
 + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Csc[d + e*x - ArcTan[5/3]]*Sqrt[Si 
n[d + e*x - ArcTan[5/3]]^2])/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] 
)/(6750*e)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {3042, 3608, 27, 3042, 3635, 25, 3042, 3628, 3042, 3597, 3042, 3132, 3605, 3042, 3140}

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.

\(\displaystyle \int \frac {1}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}dx\)

\(\Big \downarrow \) 3608

\(\displaystyle \frac {1}{45} \int -\frac {-3 \cos (d+e x)-5 \sin (d+e x)+6}{2 (3 \cos (d+e x)+5 \sin (d+e x)+2)^{3/2}}dx-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{90} \int \frac {-3 \cos (d+e x)-5 \sin (d+e x)+6}{(3 \cos (d+e x)+5 \sin (d+e x)+2)^{3/2}}dx-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{90} \int \frac {-3 \cos (d+e x)-5 \sin (d+e x)+6}{(3 \cos (d+e x)+5 \sin (d+e x)+2)^{3/2}}dx-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3635

\(\displaystyle \frac {1}{90} \left (\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}-\frac {1}{15} \int -\frac {12 \cos (d+e x)+20 \sin (d+e x)+23}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \int \frac {12 \cos (d+e x)+20 \sin (d+e x)+23}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \int \frac {12 \cos (d+e x)+20 \sin (d+e x)+23}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (15 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+4 \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}dx\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (15 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+4 \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}dx\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3597

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (4 \int \sqrt {\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+2}dx+15 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (4 \int \sqrt {\sqrt {34} \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )+\frac {\pi }{2}\right )+2}dx+15 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (15 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+\frac {8 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3605

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (15 \int \frac {1}{\sqrt {\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+2}}dx+\frac {8 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (15 \int \frac {1}{\sqrt {\sqrt {34} \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )+\frac {\pi }{2}\right )+2}}dx+\frac {8 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{90} \left (\frac {1}{15} \left (\frac {30 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}+\frac {8 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )+\frac {8 (5 \cos (d+e x)-3 \sin (d+e x))}{15 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}\right )-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}\)

Input:

Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-5/2),x]
 

Output:

-1/45*(5*Cos[d + e*x] - 3*Sin[d + e*x])/(e*(2 + 3*Cos[d + e*x] + 5*Sin[d + 
 e*x])^(3/2)) + (((8*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/ 
2, (2*(17 - Sqrt[34]))/15])/e + (30*EllipticF[(d + e*x - ArcTan[5/3])/2, ( 
2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e))/15 + (8*(5*Cos[d + e*x] - 
3*Sin[d + e*x]))/(15*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]]))/90
 

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3597

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Int[Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]] 
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[b^2 
+ c^2], 0]
 

rule 3605

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( 
x_)]], x_Symbol] :> Int[1/Sqrt[a + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, 
c]]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 + c^2, 0] && GtQ[a + Sqrt[ 
b^2 + c^2], 0]
 

rule 3608

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_), x_Symbol] :> Simp[((-c)*Cos[d + e*x] + b*Sin[d + e*x])*((a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 
1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c 
*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x 
] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && 
NeQ[n, -3/2]
 

rule 3628

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) 
/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]] 
, x_Symbol] :> Simp[B/b   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] 
, x] + Simp[(A*b - a*B)/b   Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]] 
, x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[ 
A*b - a*B, 0]
 

rule 3635

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) 
^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) 
*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( 
a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a + b*Co 
s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 
2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] 
/; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 
 0] && NeQ[n, -2]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.13

method	result	size
default	\(\frac {17 \sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}\, \left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}+2 \sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2\right ) \left (-\frac {\sqrt {34}\, \sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}}{765 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\frac {\sqrt {34}}{17}\right )^{2}}+\frac {136 \sqrt {34}\, \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}{675 \sqrt {-\cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2} \sqrt {34}\, \left (-289 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-17 \sqrt {34}\right )}}+\frac {46 \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17+17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{-\sqrt {34}+17}}\, \operatorname {EllipticF}\left (\sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right )}{675 \sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}}+\frac {8 \sqrt {34}\, \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17+17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{-\sqrt {34}+17}}\, \left (\left (-\frac {\sqrt {34}}{17}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right )\right )}{675 \sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )^{2}}}\right )}{2 \left (17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}\right )^{2} \cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right ) \sqrt {\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2}\, e}\)	\(586\)

Input:

int(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

17/2*(-(-34^(1/2)*sin(e*x+d+arctan(3/5))-2)*cos(e*x+d+arctan(3/5))^2)^(1/2 
)/(17*sin(e*x+d+arctan(3/5))+34^(1/2))^2*(17*sin(e*x+d+arctan(3/5))^2+2*34 
^(1/2)*sin(e*x+d+arctan(3/5))+2)*(-1/765*34^(1/2)*(-(-34^(1/2)*sin(e*x+d+a 
rctan(3/5))-2)*cos(e*x+d+arctan(3/5))^2)^(1/2)/(sin(e*x+d+arctan(3/5))+1/1 
7*34^(1/2))^2+136/675*34^(1/2)*cos(e*x+d+arctan(3/5))^2/(-cos(e*x+d+arctan 
(3/5))^2*34^(1/2)*(-289*sin(e*x+d+arctan(3/5))-17*34^(1/2)))^(1/2)+46/675* 
(-1+1/17*34^(1/2))*(1/(-34^(1/2)+17)*(-17*sin(e*x+d+arctan(3/5))-34^(1/2)) 
)^(1/2)*((-17*sin(e*x+d+arctan(3/5))+17)/(34^(1/2)+17))^(1/2)*(1/(-34^(1/2 
)+17)*(17+17*sin(e*x+d+arctan(3/5))))^(1/2)/(-(-34^(1/2)*sin(e*x+d+arctan( 
3/5))-2)*cos(e*x+d+arctan(3/5))^2)^(1/2)*EllipticF((1/(-34^(1/2)+17)*(-17* 
sin(e*x+d+arctan(3/5))-34^(1/2)))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^( 
1/2))+8/675*34^(1/2)*(-1+1/17*34^(1/2))*(1/(-34^(1/2)+17)*(-17*sin(e*x+d+a 
rctan(3/5))-34^(1/2)))^(1/2)*((-17*sin(e*x+d+arctan(3/5))+17)/(34^(1/2)+17 
))^(1/2)*(1/(-34^(1/2)+17)*(17+17*sin(e*x+d+arctan(3/5))))^(1/2)/(-(-34^(1 
/2)*sin(e*x+d+arctan(3/5))-2)*cos(e*x+d+arctan(3/5))^2)^(1/2)*((-1/17*34^( 
1/2)-1)*EllipticE((1/(-34^(1/2)+17)*(-17*sin(e*x+d+arctan(3/5))-34^(1/2))) 
^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+17))^(1/2))+EllipticF((1/(-34^(1/2)+17) 
*(-17*sin(e*x+d+arctan(3/5))-34^(1/2)))^(1/2),I*((-34^(1/2)+17)/(34^(1/2)+ 
17))^(1/2))))/cos(e*x+d+arctan(3/5))/(34^(1/2)*sin(e*x+d+arctan(3/5))+2)^( 
1/2)/e
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x, algorithm="fricas")
 

Output:

1/68850*(53*sqrt(5/2*I + 3/2)*((48*I + 80)*cos(e*x + d)^2 + 10*(-(9*I + 15 
)*cos(e*x + d) - 6*I - 10)*sin(e*x + d) - (36*I + 60)*cos(e*x + d) - 87*I 
- 145)*weierstrassPInverse(860/289*I + 1376/867, -5480/132651*I - 12056/14 
739, cos(e*x + d) - I*sin(e*x + d) - 10/51*I + 2/17) + 53*sqrt(-5/2*I + 3/ 
2)*(-(48*I - 80)*cos(e*x + d)^2 + 10*((9*I - 15)*cos(e*x + d) + 6*I - 10)* 
sin(e*x + d) + (36*I - 60)*cos(e*x + d) + 87*I - 145)*weierstrassPInverse( 
-860/289*I + 1376/867, 5480/132651*I - 12056/14739, cos(e*x + d) + I*sin(e 
*x + d) + 10/51*I + 2/17) + 408*sqrt(5/2*I + 3/2)*(-16*I*cos(e*x + d)^2 + 
10*(3*I*cos(e*x + d) + 2*I)*sin(e*x + d) + 12*I*cos(e*x + d) + 29*I)*weier 
strassZeta(860/289*I + 1376/867, -5480/132651*I - 12056/14739, weierstrass 
PInverse(860/289*I + 1376/867, -5480/132651*I - 12056/14739, cos(e*x + d) 
- I*sin(e*x + d) - 10/51*I + 2/17)) + 408*sqrt(-5/2*I + 3/2)*(16*I*cos(e*x 
 + d)^2 + 10*(-3*I*cos(e*x + d) - 2*I)*sin(e*x + d) - 12*I*cos(e*x + d) - 
29*I)*weierstrassZeta(-860/289*I + 1376/867, 5480/132651*I - 12056/14739, 
weierstrassPInverse(-860/289*I + 1376/867, 5480/132651*I - 12056/14739, co 
s(e*x + d) + I*sin(e*x + d) + 10/51*I + 2/17)) - 102*(120*cos(e*x + d)^2 + 
 (64*cos(e*x + d) + 21)*sin(e*x + d) - 35*cos(e*x + d) - 60)*sqrt(3*cos(e* 
x + d) + 5*sin(e*x + d) + 2))/(16*e*cos(e*x + d)^2 - 12*e*cos(e*x + d) - 1 
0*(3*e*cos(e*x + d) + 2*e)*sin(e*x + d) - 29*e)
 

Sympy [F]

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

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))**(5/2),x)
 

Output:

Integral((5*sin(d + e*x) + 3*cos(d + e*x) + 2)**(-5/2), x)
 

Maxima [F]

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

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x, algorithm="maxima")
 

Output:

integrate((3*cos(e*x + d) + 5*sin(e*x + d) + 2)^(-5/2), x)
 

Giac [F]

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

integrate(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x, algorithm="giac")
 

Output:

integrate((3*cos(e*x + d) + 5*sin(e*x + d) + 2)^(-5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int(1/(3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(5/2),x)
 

Output:

int(1/(3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx=\int \frac {\sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}}{27 \cos \left (e x +d \right )^{3}+135 \cos \left (e x +d \right )^{2} \sin \left (e x +d \right )+54 \cos \left (e x +d \right )^{2}+225 \cos \left (e x +d \right ) \sin \left (e x +d \right )^{2}+180 \cos \left (e x +d \right ) \sin \left (e x +d \right )+36 \cos \left (e x +d \right )+125 \sin \left (e x +d \right )^{3}+150 \sin \left (e x +d \right )^{2}+60 \sin \left (e x +d \right )+8}d x \] Input:

int(1/(2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x)
 

Output:

int(sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)/(27*cos(d + e*x)**3 + 135*co 
s(d + e*x)**2*sin(d + e*x) + 54*cos(d + e*x)**2 + 225*cos(d + e*x)*sin(d + 
 e*x)**2 + 180*cos(d + e*x)*sin(d + e*x) + 36*cos(d + e*x) + 125*sin(d + e 
*x)**3 + 150*sin(d + e*x)**2 + 60*sin(d + e*x) + 8),x)