Integrand size = 22, antiderivative size = 185 \[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=\frac {796 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e}+\frac {64 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}-\frac {32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{15 e}-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}{5 e} \] Output:
796/15*(2+34^(1/2))^(1/2)*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(5/3)),1/1 5*(510-30*34^(1/2))^(1/2))/e+64*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-32/15*(5*cos(e*x+d) -3*sin(e*x+d))*(2+3*cos(e*x+d)+5*sin(e*x+d))^(1/2)/e-2/5*(5*cos(e*x+d)-3*s in(e*x+d))*(2+3*cos(e*x+d)+5*sin(e*x+d))^(3/2)/e
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 4.75 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.16 \[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=\frac {-2388 \sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}-2 \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} (550 \cos (d+e x)+3 (-398+75 \cos (2 (d+e x))-110 \sin (d+e x)+40 \sin (2 (d+e x))))+1276 \sqrt {\frac {10}{3}} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}{17+\sqrt {34}}\right ) \sqrt {\cos ^2\left (d+e x+\arctan \left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\arctan \left (\frac {3}{5}\right )\right ) \sqrt {2+\sqrt {34} \sin \left (d+e x+\arctan \left (\frac {3}{5}\right )\right )}+\frac {1990 \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {\frac {1}{17}+\frac {\cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{\sqrt {34}}}}-\frac {1990 \sqrt {30} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{-17+\sqrt {34}},\frac {\sqrt {34}+17 \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}{17+\sqrt {34}}\right ) \csc \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ) \sqrt {\sin ^2\left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )}}}{75 e} \] Input:
Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2),x]
Output:
(-2388*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] - 2*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]]*(550*Cos[d + e*x] + 3*(-398 + 75*Cos[2*(d + e*x)] - 110*Sin[d + e*x] + 40*Sin[2*(d + e*x)])) + 1276*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 + ArcTan[3/5]]^2]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[ d + e*x + ArcTan[3/5]]] + (1990*Sin[d + e*x - ArcTan[5/3]])/Sqrt[1/17 + Co s[d + e*x - ArcTan[5/3]]/Sqrt[34]] - (1990*Sqrt[30]*AppellF1[-1/2, -1/2, - 1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sq rt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Csc[d + e*x - Arc Tan[5/3]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/Sqrt[2 + Sqrt[34]*Cos[d + e* x - ArcTan[5/3]]])/(75*e)
Time = 1.03 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {3042, 3599, 3042, 3625, 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.
| \(\displaystyle \int (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}dx\) |
| \(\Big \downarrow \) 3599 |
| \(\displaystyle \frac {2}{5} \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2} (24 \cos (d+e x)+40 \sin (d+e x)+61)dx-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2} (24 \cos (d+e x)+40 \sin (d+e x)+61)dx-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \int \frac {597 \cos (d+e x)+995 \sin (d+e x)+638}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \int \frac {597 \cos (d+e x)+995 \sin (d+e x)+638}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+199 \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}dx\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+199 \int \sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}dx\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3597 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (199 \int \sqrt {\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+2}dx+240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (199 \int \sqrt {\sqrt {34} \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )+\frac {\pi }{2}\right )+2}dx+240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (240 \int \frac {1}{\sqrt {3 \cos (d+e x)+5 \sin (d+e x)+2}}dx+\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3605 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (240 \int \frac {1}{\sqrt {\sqrt {34} \cos \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )+2}}dx+\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (240 \int \frac {1}{\sqrt {\sqrt {34} \sin \left (d+e x-\arctan \left (\frac {5}{3}\right )+\frac {\pi }{2}\right )+2}}dx+\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2}{5} \left (\frac {1}{3} \left (\frac {480 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right ),\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}+\frac {398 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\arctan \left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{e}\right )-\frac {16 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}\right )-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}\) |
Input:
Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2),x]
Output:
(-2*(5*Cos[d + e*x] - 3*Sin[d + e*x])*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x] )^(3/2))/(5*e) + (2*(((398*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[ 5/3])/2, (2*(17 - Sqrt[34]))/15])/e + (480*EllipticF[(d + e*x - ArcTan[5/3 ])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e))/3 - (16*(5*Cos[d + e*x] - 3*Sin[d + e*x])*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])/(3*e)))/ 5
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]
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]
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]
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)), x] + Simp[1/n Int[Simp[n*a^2 + ( n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x ], x]*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]
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]
Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ (n_.)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) ]), x_Symbol] :> Simp[(B*c - b*C - a*C*Cos[d + e*x] + a*B*Sin[d + e*x])*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^n/(a*e*(n + 1))), x] + Simp[1/(a*(n + 1 )) Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)*Simp[a*(b*B + c*C)*n + a^2*A*(n + 1) + (n*(a^2*B - B*c^2 + b*c*C) + a*b*A*(n + 1))*Cos[d + e*x] + (n*(b*B*c + a^2*C - b^2*C) + a*c*A*(n + 1))*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && GtQ[n, 0] && NeQ[a^2 - b^2 - c^2, 0]
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]
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 701, normalized size of antiderivative = 3.79
Input:
int((2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x,method=_RETURNVERBOSE)
Output:
(16/17*((17*sin(e*x+d+arctan(3/5))+34^(1/2))/(34^(1/2)+17))^(1/2)*17^(1/2) *((sin(e*x+d+arctan(3/5))+1)/(-34^(1/2)+17))^(1/2)*(-17*(sin(e*x+d+arctan( 3/5))-1)/(34^(1/2)+17))^(1/2)*EllipticF(((17*sin(e*x+d+arctan(3/5))+34^(1/ 2))/(34^(1/2)+17))^(1/2),I*(1/(-34^(1/2)+17)*(34^(1/2)+17))^(1/2))*34^(1/2 )+16*((17*sin(e*x+d+arctan(3/5))+34^(1/2))/(34^(1/2)+17))^(1/2)*17^(1/2)*( (sin(e*x+d+arctan(3/5))+1)/(-34^(1/2)+17))^(1/2)*(-17*(sin(e*x+d+arctan(3/ 5))-1)/(34^(1/2)+17))^(1/2)*EllipticF(((17*sin(e*x+d+arctan(3/5))+34^(1/2) )/(34^(1/2)+17))^(1/2),I*(1/(-34^(1/2)+17)*(34^(1/2)+17))^(1/2))+1904/15*s in(e*x+d+arctan(3/5))^3-1904/15*sin(e*x+d+arctan(3/5))-116/15*34^(1/2)*sin (e*x+d+arctan(3/5))^2-88/15*34^(1/2)-44*34^(1/2)*(-(17*sin(e*x+d+arctan(3/ 5))+34^(1/2))/(-34^(1/2)+17))^(1/2)*(-17*(sin(e*x+d+arctan(3/5))-1)/(34^(1 /2)+17))^(1/2)*17^(1/2)*((sin(e*x+d+arctan(3/5))+1)/(-34^(1/2)+17))^(1/2)* EllipticF((-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2),I*( (-34^(1/2)+17)/(34^(1/2)+17))^(1/2))-48*17^(1/2)*((sin(e*x+d+arctan(3/5))+ 1)/(-34^(1/2)+17))^(1/2)*(-17*(sin(e*x+d+arctan(3/5))-1)/(34^(1/2)+17))^(1 /2)*(-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2)*EllipticF ((-(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2),I*((-34^(1/2 )+17)/(34^(1/2)+17))^(1/2))+796/17*17^(1/2)*((sin(e*x+d+arctan(3/5))+1)/(- 34^(1/2)+17))^(1/2)*(-17*(sin(e*x+d+arctan(3/5))-1)/(34^(1/2)+17))^(1/2)*( -(17*sin(e*x+d+arctan(3/5))+34^(1/2))/(-34^(1/2)+17))^(1/2)*EllipticE((...
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.91 \[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=-\frac {2 \, {\left (51 \, {\left (90 \, \cos \left (e x + d\right )^{2} + 6 \, {\left (8 \, \cos \left (e x + d\right ) - 11\right )} \sin \left (e x + d\right ) + 110 \, \cos \left (e x + d\right ) - 45\right )} \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2} - \left (3354 i + 5590\right ) \, \sqrt {\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right ) + \left (3354 i - 5590\right ) \, \sqrt {-\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right ) + 20298 i \, \sqrt {\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassZeta}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (\frac {860}{289} i + \frac {1376}{867}, -\frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) - i \, \sin \left (e x + d\right ) - \frac {10}{51} i + \frac {2}{17}\right )\right ) - 20298 i \, \sqrt {-\frac {5}{2} i + \frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, {\rm weierstrassPInverse}\left (-\frac {860}{289} i + \frac {1376}{867}, \frac {5480}{132651} i - \frac {12056}{14739}, \cos \left (e x + d\right ) + i \, \sin \left (e x + d\right ) + \frac {10}{51} i + \frac {2}{17}\right )\right )\right )}}{765 \, e} \] Input:
integrate((2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x, algorithm="fricas")
Output:
-2/765*(51*(90*cos(e*x + d)^2 + 6*(8*cos(e*x + d) - 11)*sin(e*x + d) + 110 *cos(e*x + d) - 45)*sqrt(3*cos(e*x + d) + 5*sin(e*x + d) + 2) - (3354*I + 5590)*sqrt(5/2*I + 3/2)*weierstrassPInverse(860/289*I + 1376/867, -5480/13 2651*I - 12056/14739, cos(e*x + d) - I*sin(e*x + d) - 10/51*I + 2/17) + (3 354*I - 5590)*sqrt(-5/2*I + 3/2)*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) + 20298*I*sqrt(5/2*I + 3/2)*weierstrassZeta(860/289*I + 1376/867, -54 80/132651*I - 12056/14739, 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)) - 20298*I*sqrt(-5/2*I + 3/2)*weierstrassZeta(-860/289*I + 1376/867, 5480/1 32651*I - 12056/14739, weierstrassPInverse(-860/289*I + 1376/867, 5480/132 651*I - 12056/14739, cos(e*x + d) + I*sin(e*x + d) + 10/51*I + 2/17)))/e
Timed out. \[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((2+3*cos(e*x+d)+5*sin(e*x+d))**(5/2),x)
Output:
Timed out
\[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=\int { {\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((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)
\[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=\int { {\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((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)
Timed out. \[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=\int {\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{5/2} \,d x \] Input:
int((3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(5/2),x)
Output:
int((3*cos(d + e*x) + 5*sin(d + e*x) + 2)^(5/2), x)
\[ \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx=\frac {\frac {5994 \sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \cos \left (e x +d \right )^{2}}{1075}+\frac {936 \sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \cos \left (e x +d \right ) \sin \left (e x +d \right )}{215}+\frac {30276 \sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \cos \left (e x +d \right )}{46225}+\frac {5166 \sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \sin \left (e x +d \right )^{2}}{215}+\frac {134532 \sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \sin \left (e x +d \right )}{9245}+\frac {127512 \sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}}{46225}+8 \left (\int \frac {\sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}}{3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}d x \right ) e +\frac {39304 \left (\int \frac {\sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \sin \left (e x +d \right )^{3}}{3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}d x \right ) e}{215}+\frac {339864 \left (\int \frac {\sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \sin \left (e x +d \right )^{2}}{3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}d x \right ) e}{1849}+\frac {605064 \left (\int \frac {\sqrt {3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}\, \sin \left (e x +d \right )}{3 \cos \left (e x +d \right )+5 \sin \left (e x +d \right )+2}d x \right ) e}{9245}}{e} \] Input:
int((2+3*cos(e*x+d)+5*sin(e*x+d))^(5/2),x)
Output:
(2*(128871*sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)*cos(d + e*x)**2 + 100 620*sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)*cos(d + e*x)*sin(d + e*x) + 15138*sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)*cos(d + e*x) + 555345*sqrt (3*cos(d + e*x) + 5*sin(d + e*x) + 2)*sin(d + e*x)**2 + 336330*sqrt(3*cos( d + e*x) + 5*sin(d + e*x) + 2)*sin(d + e*x) + 63756*sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2) + 184900*int(sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2) /(3*cos(d + e*x) + 5*sin(d + e*x) + 2),x)*e + 4225180*int((sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)*sin(d + e*x)**3)/(3*cos(d + e*x) + 5*sin(d + e* x) + 2),x)*e + 4248300*int((sqrt(3*cos(d + e*x) + 5*sin(d + e*x) + 2)*sin( d + e*x)**2)/(3*cos(d + e*x) + 5*sin(d + e*x) + 2),x)*e + 1512660*int((sqr t(3*cos(d + e*x) + 5*sin(d + e*x) + 2)*sin(d + e*x))/(3*cos(d + e*x) + 5*s in(d + e*x) + 2),x)*e))/(46225*e)