Integrand size = 25, antiderivative size = 202 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx=\frac {2 a \left (3 a^2+2 b^2\right ) e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {2 a \left (3 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{3/2}}{15 d}+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{231 d e}+\frac {10 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{33 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}}{11 d e} \]
-2/15*a*(3*a^2+2*b^2)*e*cos(d*x+c)*(e*sin(d*x+c))^(3/2)/d+2/231*b*(43*a^2+ 12*b^2)*(e*sin(d*x+c))^(7/2)/d/e+10/33*a*b*(a+b*cos(d*x+c))*(e*sin(d*x+c)) ^(7/2)/d/e+2/11*b*(a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(7/2)/d/e-2/5*a*(3*a^2 +2*b^2)*e^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)* EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/d/sin(d* x+c)^(1/2)
Time = 2.24 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.74 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx=-\frac {(e \sin (c+d x))^{5/2} \left (1848 \left (3 a^3+2 a b^2\right ) E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+\left (462 a \left (4 a^2+b^2\right ) \cos (c+d x)+5 b \left (-396 a^2-69 b^2+12 \left (33 a^2+4 b^2\right ) \cos (2 (c+d x))+154 a b \cos (3 (c+d x))+21 b^2 \cos (4 (c+d x))\right )\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{4620 d \sin ^{\frac {5}{2}}(c+d x)} \]
-1/4620*((e*Sin[c + d*x])^(5/2)*(1848*(3*a^3 + 2*a*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + (462*a*(4*a^2 + b^2)*Cos[c + d*x] + 5*b*(-396*a^2 - 69 *b^2 + 12*(33*a^2 + 4*b^2)*Cos[2*(c + d*x)] + 154*a*b*Cos[3*(c + d*x)] + 2 1*b^2*Cos[4*(c + d*x)]))*Sin[c + d*x]^(3/2)))/(d*Sin[c + d*x]^(5/2))
Time = 0.95 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3171, 27, 3042, 3341, 27, 3042, 3148, 3042, 3115, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} (a+b \cos (c+d x)) \left (11 a^2+15 b \cos (c+d x) a+4 b^2\right ) (e \sin (c+d x))^{5/2}dx+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int (a+b \cos (c+d x)) \left (11 a^2+15 b \cos (c+d x) a+4 b^2\right ) (e \sin (c+d x))^{5/2}dx+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (11 a^2+15 b \sin \left (c+d x+\frac {\pi }{2}\right ) a+4 b^2\right )dx+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {3}{2} \left (11 a \left (3 a^2+2 b^2\right )+b \left (43 a^2+12 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{5/2}dx+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int \left (11 a \left (3 a^2+2 b^2\right )+b \left (43 a^2+12 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{5/2}dx+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2} \left (11 a \left (3 a^2+2 b^2\right )-b \left (43 a^2+12 b^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )\right )dx+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \int (e \sin (c+d x))^{5/2}dx+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \int (e \sin (c+d x))^{5/2}dx+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \sin (c+d x)}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3}{5} e^2 \int \sqrt {e \sin (c+d x)}dx-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{5 \sqrt {\sin (c+d x)}}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {3 e^2 \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{5 \sqrt {\sin (c+d x)}}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{3} \left (11 a \left (3 a^2+2 b^2\right ) \left (\frac {6 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}\right )+\frac {2 b \left (43 a^2+12 b^2\right ) (e \sin (c+d x))^{7/2}}{7 d e}\right )+\frac {10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{3 d e}\right )+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}\) |
(2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2))/(11*d*e) + ((10*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(3*d*e) + ((2*b*(43*a^2 + 12*b^ 2)*(e*Sin[c + d*x])^(7/2))/(7*d*e) + 11*a*(3*a^2 + 2*b^2)*((6*e^2*Elliptic E[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d)))/3)/11
3.1.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Time = 40.31 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (7 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+33 a^{2}+4 b^{2}\right )}{77 e}-\frac {e^{3} a \left (10 \left (\sin ^{6}\left (d x +c \right )\right ) b^{2}+18 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+12 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-9 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-6 a^{2} \left (\sin ^{4}\left (d x +c \right )\right )-14 \left (\sin ^{4}\left (d x +c \right )\right ) b^{2}+6 a^{2} \left (\sin ^{2}\left (d x +c \right )\right )+4 b^{2} \left (\sin ^{2}\left (d x +c \right )\right )\right )}{15 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(356\) |
| parts | \(-\frac {a^{3} e^{3} \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{4}\left (d x +c \right )\right )+2 \left (\sin ^{2}\left (d x +c \right )\right )\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{3} \left (\frac {\left (e \sin \left (d x +c \right )\right )^{\frac {11}{2}}}{11}-\frac {e^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}\right )}{d \,e^{3}}+\frac {6 a^{2} b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 e d}-\frac {2 a \,b^{2} e^{3} \left (5 \left (\sin ^{6}\left (d x +c \right )\right )+6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-7 \left (\sin ^{4}\left (d x +c \right )\right )+2 \left (\sin ^{2}\left (d x +c \right )\right )\right )}{15 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(382\) |
(2/77/e*b*(e*sin(d*x+c))^(7/2)*(7*b^2*cos(d*x+c)^2+33*a^2+4*b^2)-1/15*e^3* a*(10*sin(d*x+c)^6*b^2+18*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin( d*x+c)^(1/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2+12*(1-sin(d*x +c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticE((1-sin(d*x+c) )^(1/2),1/2*2^(1/2))*b^2-9*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin (d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))*a^2-6*(1-sin(d*x +c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c) )^(1/2),1/2*2^(1/2))*b^2-6*a^2*sin(d*x+c)^4-14*sin(d*x+c)^4*b^2+6*a^2*sin( d*x+c)^2+4*b^2*sin(d*x+c)^2)/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.05 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx=\frac {231 i \, \sqrt {2} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} \sqrt {-i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} \sqrt {i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (105 \, b^{3} e^{2} \cos \left (d x + c\right )^{4} + 385 \, a b^{2} e^{2} \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, a^{2} b - b^{3}\right )} e^{2} \cos \left (d x + c\right )^{2} + 231 \, {\left (a^{3} - a b^{2}\right )} e^{2} \cos \left (d x + c\right ) - 15 \, {\left (33 \, a^{2} b + 4 \, b^{3}\right )} e^{2}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right )}{1155 \, d} \]
1/1155*(231*I*sqrt(2)*(3*a^3 + 2*a*b^2)*sqrt(-I*e)*e^2*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*I*sqrt( 2)*(3*a^3 + 2*a*b^2)*sqrt(I*e)*e^2*weierstrassZeta(4, 0, weierstrassPInver se(4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(105*b^3*e^2*cos(d*x + c)^4 + 385*a*b^2*e^2*cos(d*x + c)^3 + 45*(11*a^2*b - b^3)*e^2*cos(d*x + c)^2 + 2 31*(a^3 - a*b^2)*e^2*cos(d*x + c) - 15*(33*a^2*b + 4*b^3)*e^2)*sqrt(e*sin( d*x + c))*sin(d*x + c))/d
Timed out. \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]