\(\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx\) [289]

Optimal result

Integrand size = 27, antiderivative size = 323 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=-\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}+\frac {77 a^2 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {77 a^2 e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {77 a^2 e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d (1+\cos (c+d x)+\sin (c+d x))}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e} \]

[Out]

-1/4*a*(e*cos(d*x+c))^(5/2)*(a+a*sin(d*x+c))^(3/2)/d/e-77/96*a^3*(e*cos(d*x+c))^(5/2)/d/e/(a+a*sin(d*x+c))^(1/
2)-11/24*a^2*(e*cos(d*x+c))^(5/2)*(a+a*sin(d*x+c))^(1/2)/d/e+77/64*a^2*e*(e*cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))
^(1/2)/d-77/64*a^2*e^(3/2)*arcsinh((e*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))+77/64*a^2*e^(3/2)*arctan(sin(d*x+c)*e^(1/2)/(e*cos(d*x+c))^(1/2)/(1+cos(d*x+c))^(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))

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2757, 2764, 2756, 2854, 209, 2912, 65, 221} \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=-\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a \sin (c+d x)+a}}-\frac {77 a^2 e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{64 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {77 a^2 e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {11 a^2 \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{24 d e}+\frac {77 a^2 e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{64 d}-\frac {a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e} \]

[In]

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

[Out]

(-77*a^3*(e*Cos[c + d*x])^(5/2))/(96*d*e*Sqrt[a + a*Sin[c + d*x]]) + (77*a^2*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a
*Sin[c + d*x]])/(64*d) - (11*a^2*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])/(24*d*e) - (77*a^2*e^(3/2)*A
rcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x]
+ Sin[c + d*x])) + (77*a^2*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])
]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*(e*Cos[c + d*
x])^(5/2)*(a + a*Sin[c + d*x])^(3/2))/(4*d*e)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 221

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]

Rule 2756

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)], x_Symbol] :> Dist[a*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] + Dist[b*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[Sin[e + f*x]/(Sqrt[g*Cos[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f,
g}, x] && EqQ[a^2 - b^2, 0]

Rule 2757

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] + Dist[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] && IntegersQ[2*m, 2*p]

Rule 2764

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[g*Sqrt
[g*Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b*f)), x] + Dist[g^2/(2*a), Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Co
s[e + f*x]], x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-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]

Rule 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}+\frac {1}{8} (11 a) \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx \\ & = -\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}+\frac {1}{48} \left (77 a^2\right ) \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}+\frac {1}{64} \left (77 a^3\right ) \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}+\frac {77 a^2 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}+\frac {1}{128} \left (77 a^2 e^2\right ) \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}+\frac {77 a^2 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}+\frac {\left (77 a^3 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{128 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (77 a^3 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{128 (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = -\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}+\frac {77 a^2 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}-\frac {\left (77 a^3 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{128 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (77 a^3 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{64 d (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = -\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}+\frac {77 a^2 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}+\frac {77 a^3 e^{3/2} \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)}}{64 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (77 a^3 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{64 d (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = -\frac {77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt {a+a \sin (c+d x)}}+\frac {77 a^2 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{64 d}-\frac {11 a^2 (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{24 d e}-\frac {a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}}{4 d e}-\frac {77 a^3 e^{3/2} \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)}}{64 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {77 a^3 e^{3/2} \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)}}{64 d (a+a \cos (c+d x)+a \sin (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.24 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=-\frac {16\ 2^{3/4} (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (a (1+\sin (c+d x)))^{5/2}}{5 d e (1+\sin (c+d x))^{15/4}} \]

[In]

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

[Out]

(-16*2^(3/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-11/4, 5/4, 9/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d
*x]))^(5/2))/(5*d*e*(1 + Sin[c + d*x])^(15/4))

Maple [A] (verified)

Time = 5.84 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {\sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, a^{2} e \left (48 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-48 \left (\cos ^{4}\left (d x +c \right )\right )+184 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+136 \left (\cos ^{3}\left (d x +c \right )\right )-154 \cos \left (d x +c \right ) \sin \left (d x +c \right )+338 \left (\cos ^{2}\left (d x +c \right )\right )-231 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-231 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-231 \sin \left (d x +c \right )-77 \cos \left (d x +c \right )-231 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-231 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-231\right )}{192 d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}\)	\(360\)

[In]

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

[Out]

-1/192/d*(e*cos(d*x+c))^(1/2)*(a*(1+sin(d*x+c)))^(1/2)*a^2*e/(1+cos(d*x+c)+sin(d*x+c))*(48*cos(d*x+c)^3*sin(d*
x+c)-48*cos(d*x+c)^4+184*cos(d*x+c)^2*sin(d*x+c)+136*cos(d*x+c)^3-154*cos(d*x+c)*sin(d*x+c)+338*cos(d*x+c)^2-2
31*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan((-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))-231*(-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))-231*sin(d*x+c)-77*cos(d*x+c)-2
31*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))-231*sec(d*x+c)*(-c
os(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))-231)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 1020, normalized size of antiderivative = 3.16 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/768*(231*I*(-a^10*e^6/d^4)^(1/4)*d*log(-456533/2*(2*(a^7*e^4*sin(d*x + c) + sqrt(-a^10*e^6/d^4)*(a^2*d^2*e*
cos(d*x + c) + a^2*d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (-a^10*e^6/d^4)^(3/4)*(I*d^3*cos(d*
x + c) + I*d^3 + (2*I*d^3*cos(d*x + c) + I*d^3)*sin(d*x + c)) + (-a^10*e^6/d^4)^(1/4)*(-2*I*a^5*d*e^3*cos(d*x
+ c)^2 - I*a^5*d*e^3*cos(d*x + c) + I*a^5*d*e^3*sin(d*x + c) + I*a^5*d*e^3))/(cos(d*x + c) + sin(d*x + c) + 1)
) - 231*I*(-a^10*e^6/d^4)^(1/4)*d*log(-456533/2*(2*(a^7*e^4*sin(d*x + c) + sqrt(-a^10*e^6/d^4)*(a^2*d^2*e*cos(
d*x + c) + a^2*d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (-a^10*e^6/d^4)^(3/4)*(-I*d^3*cos(d*x +
c) - I*d^3 + (-2*I*d^3*cos(d*x + c) - I*d^3)*sin(d*x + c)) + (-a^10*e^6/d^4)^(1/4)*(2*I*a^5*d*e^3*cos(d*x + c
)^2 + I*a^5*d*e^3*cos(d*x + c) - I*a^5*d*e^3*sin(d*x + c) - I*a^5*d*e^3))/(cos(d*x + c) + sin(d*x + c) + 1)) -
231*(-a^10*e^6/d^4)^(1/4)*d*log(-456533/2*(2*(a^7*e^4*sin(d*x + c) - sqrt(-a^10*e^6/d^4)*(a^2*d^2*e*cos(d*x +
c) + a^2*d^2*e))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (-a^10*e^6/d^4)^(3/4)*(d^3*cos(d*x + c) + d^
3 + (2*d^3*cos(d*x + c) + d^3)*sin(d*x + c)) + (-a^10*e^6/d^4)^(1/4)*(2*a^5*d*e^3*cos(d*x + c)^2 + a^5*d*e^3*c
os(d*x + c) - a^5*d*e^3*sin(d*x + c) - a^5*d*e^3))/(cos(d*x + c) + sin(d*x + c) + 1)) + 231*(-a^10*e^6/d^4)^(1
/4)*d*log(-456533/2*(2*(a^7*e^4*sin(d*x + c) - sqrt(-a^10*e^6/d^4)*(a^2*d^2*e*cos(d*x + c) + a^2*d^2*e))*sqrt(
e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) - (-a^10*e^6/d^4)^(3/4)*(d^3*cos(d*x + c) + d^3 + (2*d^3*cos(d*x + c)
+ d^3)*sin(d*x + c)) - (-a^10*e^6/d^4)^(1/4)*(2*a^5*d*e^3*cos(d*x + c)^2 + a^5*d*e^3*cos(d*x + c) - a^5*d*e^3
*sin(d*x + c) - a^5*d*e^3))/(cos(d*x + c) + sin(d*x + c) + 1)) + 4*(136*a^2*e*cos(d*x + c)^2 - 77*a^2*e + 2*(2
4*a^2*e*cos(d*x + c)^2 - 77*a^2*e)*sin(d*x + c))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a))/d

Sympy [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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