\(\int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx\) [96]

Optimal result

Integrand size = 42, antiderivative size = 292 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {2 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{65 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

4/13*a*(g*cos(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(1/2)-2/39*a*(g*cos(f*x+e))^(5/2)/c/f/
g/(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2)-2/65*a*(g*cos(f*x+e))^(5/2)/c^2/f/g/(c-c*sin(f*x+e))^(5/2)/(a+
a*sin(f*x+e))^(1/2)-2/65*a*(g*cos(f*x+e))^(5/2)/c^3/f/g/(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)+2/65*a*g
*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(g*cos
(f*x+e))^(1/2)/c^4/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{65 c^4 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}} \]

[In]

Int[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2))/(13*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (2*a*(g*Cos[e + f*
x])^(5/2))/(39*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(65*c
^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (2*a*(g*Cos[e + f*x])^(5/2))/(65*c^3*f*g*Sqrt[a
+ a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (2*a*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e +
f*x)/2, 2])/(65*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 2719

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]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2921

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

Rule 2929

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e
+ f*x])^n/(f*g*(2*n + p + 1))), x] - Dist[b*((2*m + p - 1)/(d*(2*n + p + 1))), Int[(g*Cos[e + f*x])^p*(a + b*
Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c +
a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 2931

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^
n/(a*f*g*(2*m + p + 1))), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f
*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && E
qQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] &&  !LtQ[m, n, -1] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {(3 a) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{13 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{13 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{65 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{65 c^4} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {(a g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{65 c^4 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{65 c^4 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {2 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{65 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.01 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 e^{3 i (e+f x)} \left (e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right ) g\right )^{3/2} \left (\sqrt {1+e^{2 i (e+f x)}} \left (-1+149 i e^{i (e+f x)}+44 e^{2 i (e+f x)}-64 i e^{3 i (e+f x)}+21 e^{4 i (e+f x)}+3 i e^{5 i (e+f x)}\right )-i \left (-i+e^{i (e+f x)}\right )^7 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{195 c^4 \left (1-i e^{i (e+f x)}\right ) \left (-i+e^{i (e+f x)}\right )^6 \sqrt {i c e^{-i (e+f x)} \left (-i+e^{i (e+f x)}\right )^2} \left (1+e^{2 i (e+f x)}\right )^{3/2} f} \]

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(4*E^((3*I)*(e + f*x))*(((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))]*(-
1 + (149*I)*E^(I*(e + f*x)) + 44*E^((2*I)*(e + f*x)) - (64*I)*E^((3*I)*(e + f*x)) + 21*E^((4*I)*(e + f*x)) + (
3*I)*E^((5*I)*(e + f*x))) - I*(-I + E^(I*(e + f*x)))^7*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])
*Sqrt[a*(1 + Sin[e + f*x])])/(195*c^4*(1 - I*E^(I*(e + f*x)))*(-I + E^(I*(e + f*x)))^6*Sqrt[(I*c*(-I + E^(I*(e
+ f*x)))^2)/E^(I*(e + f*x))]*(1 + E^((2*I)*(e + f*x)))^(3/2)*f)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.37 (sec) , antiderivative size = 1388, normalized size of antiderivative = 4.75

method	result	size
default	\(\text {Expression too large to display}\)	\(1388\)

[In]

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

2/195*I/f*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g/(1+cos(f*x+e))/(cos(f*x+e)^2+2*sin(f*x+e)-2)/(-c*(si
n(f*x+e)-1))^(1/2)/c^4*(-3*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-
cot(f*x+e)),I)+3*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)
),I)+30*I*sec(f*x+e)^2+9*I*sin(f*x+e)+18*I*sec(f*x+e)+42*I*tan(f*x+e)+3*I*cos(f*x+e)^2-14*I-3*sin(f*x+e)*(cos(
f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+3*si
n(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos
(f*x+e)^2-6*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*
cos(f*x+e)*sin(f*x+e)+6*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x
+e)))^(1/2)*cos(f*x+e)*sin(f*x+e)+I*cos(f*x+e)+30*I*tan(f*x+e)*sec(f*x+e)+9*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*
EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-9*(cos(f*x+e)/(1+cos(f*x+e)))^(1/
2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+18*(cos(f*x+e)/(1+cos(f*x+e)))
^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)+9*sin(f*x+e)*(cos(f*x+e)/(1+
cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)-18*(cos(f*x+e)/(1+cos(f*x+e
)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)-9*sin(f*x+e)*(cos(f*x+e)/
(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)+24*(1/(1+cos(f*x+e)))^(1
/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*tan(f*x+e)-24*(1/(1+cos(f*x+e)))^
(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*tan(f*x+e)-24*(1/(1+cos(f*x+e))
)^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)+24*(1/(1+cos(f*x+e
)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)-12*(1/(1+cos(f*x
+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)^2+12*(1/(1+cos
(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)^2+12*(1/(1
+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*tan(f*x+e)*sec(f*
x+e)-12*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*tan(
f*x+e)*sec(f*x+e))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.14 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.12 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {2 \, {\left (12 \, g \cos \left (f x + e\right )^{2} - {\left (3 \, g \cos \left (f x + e\right )^{2} - 23 \, g\right )} \sin \left (f x + e\right ) + 7 \, g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 3 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{4} - 8 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 4 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) + 8 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 3 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{4} + 8 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 4 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} g\right )} \sin \left (f x + e\right ) - 8 i \, \sqrt {2} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{195 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

1/195*(2*(12*g*cos(f*x + e)^2 - (3*g*cos(f*x + e)^2 - 23*g)*sin(f*x + e) + 7*g)*sqrt(g*cos(f*x + e))*sqrt(a*si
n(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) - 3*(I*sqrt(2)*g*cos(f*x + e)^4 - 8*I*sqrt(2)*g*cos(f*x + e)^2 + 4*(
I*sqrt(2)*g*cos(f*x + e)^2 - 2*I*sqrt(2)*g)*sin(f*x + e) + 8*I*sqrt(2)*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, w
eierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 3*(-I*sqrt(2)*g*cos(f*x + e)^4 + 8*I*sqrt(2)*g*cos
(f*x + e)^2 + 4*(-I*sqrt(2)*g*cos(f*x + e)^2 + 2*I*sqrt(2)*g)*sin(f*x + e) - 8*I*sqrt(2)*g)*sqrt(a*c*g)*weiers
trassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(c^5*f*cos(f*x + e)^4 - 8*c^5*f*c
os(f*x + e)^2 + 8*c^5*f + 4*(c^5*f*cos(f*x + e)^2 - 2*c^5*f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(1/2)/(c-c*sin(f*x+e))**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*sqrt(a*sin(f*x + e) + a)/(-c*sin(f*x + e) + c)^(9/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]

[In]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(1/2))/(c - c*sin(e + f*x))^(9/2),x)

[Out]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(1/2))/(c - c*sin(e + f*x))^(9/2), x)