Integrand size = 27, antiderivative size = 243 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {5 a \sqrt {e} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {5 a \sqrt {e} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (1+\cos (c+d x)+\sin (c+d x))} \]
-5/4*a^2*(e*cos(d*x+c))^(3/2)/d/e/(a+a*sin(d*x+c))^(1/2)-1/2*a*(e*cos(d*x+ c))^(3/2)*(a+a*sin(d*x+c))^(1/2)/d/e+5/4*a*arcsinh((e*cos(d*x+c))^(1/2)/e^ (1/2))*e^(1/2)*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(1+cos(d*x+c) +sin(d*x+c))+5/4*a*arctan(sin(d*x+c)*e^(1/2)/(e*cos(d*x+c))^(1/2)/(1+cos(d *x+c))^(1/2))*e^(1/2)*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(1+cos (d*x+c)+sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.32 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx=-\frac {8 \sqrt [4]{2} (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{3/2}}{3 d e (1+\sin (c+d x))^{9/4}} \]
(-8*2^(1/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(3*d*e*(1 + Sin[c + d*x])^( 9/4))
Time = 0.99 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3157, 3042, 3157, 3042, 3163, 3042, 25, 3254, 216, 3312, 63, 222}
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.
| \(\displaystyle \int (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {5}{4} a \int \sqrt {e \cos (c+d x)} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{4} a \int \sqrt {e \cos (c+d x)} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 3163 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}+\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 3254 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}-\frac {2 e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{a \sin (c+d x)+a \cos (c+d x)+a}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {e \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {5}{4} a \left (\frac {1}{2} a \left (\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}\right )-\frac {a (e \cos (c+d x))^{3/2}}{d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}\) |
-1/2*(a*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]])/(d*e) + (5*a*(-(( a*(e*Cos[c + d*x])^(3/2))/(d*e*Sqrt[a + a*Sin[c + d*x]])) + (a*((2*Sqrt[e] *ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*S in[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) + (2*Sqrt[e]*ArcTa n[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sq rt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a* Sin[c + d*x]))))/2))/4
3.3.82.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
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[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[g*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x ]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x])) Int[Sqrt[1 + Cos[e + f*x]]/Sqrt [g*Cos[e + f*x]], x], x] - Simp[g*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b + b*Cos[e + f*x] + a*Sin[e + f*x])) Int[Sin[e + f*x]/(Sqrt[g*C os[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] & & EqQ[a^2 - b^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))], x ] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 6.50 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.26
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {e \cos \left (d x +c \right )}\, a \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )+5 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-5 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+7 \cos \left (d x +c \right )-5 \sin \left (d x +c \right )+5 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-5 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+5\right )}{4 d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}\) | \(305\) |
-1/4/d*(a*(1+sin(d*x+c)))^(1/2)*(e*cos(d*x+c))^(1/2)*a/(1+cos(d*x+c)+sin(d *x+c))*(2*cos(d*x+c)^2+2*cos(d*x+c)*sin(d*x+c)+5*(-cos(d*x+c)/(1+cos(d*x+c )))^(1/2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-5*(-cos(d*x+c)/(1+cos (d*x+c)))^(1/2)*arctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c)/(1+cos(d*x+ c)))^(1/2))+7*cos(d*x+c)-5*sin(d*x+c)+5*sec(d*x+c)*(-cos(d*x+c)/(1+cos(d*x +c)))^(1/2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-5*sec(d*x+c)*(-cos( d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(sin(d*x+c)/(1+cos(d*x+c))/(-cos(d*x+c )/(1+cos(d*x+c)))^(1/2))+5)
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 1051, normalized size of antiderivative = 4.33 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx=\text {Too large to display} \]
1/16*(5*(-a^6*e^2/d^4)^(1/4)*(d*cos(d*x + c) + d*sin(d*x + c) + d)*log(125 /2*(2*(a^4*e*sin(d*x + c) + sqrt(-a^6*e^2/d^4)*(a*d^2*cos(d*x + c) + a*d^2 ))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (2*d^3*cos(d*x + c)^2 + d^3*cos(d*x + c) - d^3*sin(d*x + c) - d^3)*(-a^6*e^2/d^4)^(3/4) + (a^3*d* e*cos(d*x + c) + a^3*d*e + (2*a^3*d*e*cos(d*x + c) + a^3*d*e)*sin(d*x + c) )*(-a^6*e^2/d^4)^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 5*(-a^6*e^2/d ^4)^(1/4)*(d*cos(d*x + c) + d*sin(d*x + c) + d)*log(125/2*(2*(a^4*e*sin(d* x + c) + sqrt(-a^6*e^2/d^4)*(a*d^2*cos(d*x + c) + a*d^2))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) - (2*d^3*cos(d*x + c)^2 + d^3*cos(d*x + c) - d^3*sin(d*x + c) - d^3)*(-a^6*e^2/d^4)^(3/4) - (a^3*d*e*cos(d*x + c) + a^ 3*d*e + (2*a^3*d*e*cos(d*x + c) + a^3*d*e)*sin(d*x + c))*(-a^6*e^2/d^4)^(1 /4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 5*(-a^6*e^2/d^4)^(1/4)*(-I*d*cos (d*x + c) - I*d*sin(d*x + c) - I*d)*log(125/2*(2*(a^4*e*sin(d*x + c) - sqr t(-a^6*e^2/d^4)*(a*d^2*cos(d*x + c) + a*d^2))*sqrt(e*cos(d*x + c))*sqrt(a* sin(d*x + c) + a) - (2*I*d^3*cos(d*x + c)^2 + I*d^3*cos(d*x + c) - I*d^3*s in(d*x + c) - I*d^3)*(-a^6*e^2/d^4)^(3/4) - (-I*a^3*d*e*cos(d*x + c) - I*a ^3*d*e + (-2*I*a^3*d*e*cos(d*x + c) - I*a^3*d*e)*sin(d*x + c))*(-a^6*e^2/d ^4)^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 5*(-a^6*e^2/d^4)^(1/4)*(I* d*cos(d*x + c) + I*d*sin(d*x + c) + I*d)*log(125/2*(2*(a^4*e*sin(d*x + c) - sqrt(-a^6*e^2/d^4)*(a*d^2*cos(d*x + c) + a*d^2))*sqrt(e*cos(d*x + c))...
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {e \cos {\left (c + d x \right )}}\, dx \]
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]