\(\int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx\) [561]

Optimal result

Integrand size = 27, antiderivative size = 378 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\frac {2 \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 (d-a e) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{3 \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac {2 \left (c \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sin (x)\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (x)+c \sin (x)}} \]

[Out]

2/3*(c*(-a*e+d)*cos(x)-b*(-a*e+d)*sin(x))/(a^2-b^2-c^2)/(a+b*cos(x)+c*sin(x))^(3/2)+2/3*(c*(4*a*d-a^2*e-3*(b^2
+c^2)*e)*cos(x)-b*(4*a*d-a^2*e-3*(b^2+c^2)*e)*sin(x))/(a^2-b^2-c^2)^2/(a+b*cos(x)+c*sin(x))^(1/2)+2/3*(4*a*d-a
^2*e-3*(b^2+c^2)*e)*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*arctan(b,c))*EllipticE(sin(1/2*x-1/2*ar
ctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*(a+b*cos(x)+c*sin(x))^(1/2)/(a^2-b^2-c^2)^2/((
a+b*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)-2/3*(-a*e+d)*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/
2*arctan(b,c))*EllipticF(sin(1/2*x-1/2*arctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*((a+b
*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a^2-b^2-c^2)/(a+b*cos(x)+c*sin(x))^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3235, 3228, 3198, 2732, 3206, 2740} \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=-\frac {2 (d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{3 \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac {2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (x)+c \sin (x)}} \]

[In]

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

[Out]

(2*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])
]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(3*(a^2 - b^2 - c^2)^2*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])
- (2*(d - a*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x]
+ c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(3*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[x] + c*Sin[x]]) + (2*(c*(d - a*e)*Cos[
x] - b*(d - a*e)*Sin[x]))/(3*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^(3/2)) + (2*(c*(4*a*d - a^2*e - 3*(b^
2 + c^2)*e)*Cos[x] - b*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*Sin[x]))/(3*(a^2 - b^2 - c^2)^2*Sqrt[a + b*Cos[x] + c
*Sin[x]])

Rule 2732

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]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3206

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3228

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] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(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]

Rule 3235

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_)*((A_.) + cos[(d_.) + (e_.)*(x
_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B -
b*A)*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Dist[
1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C
) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A
, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} \left (a d-\left (b^2+c^2\right ) e\right )+\frac {1}{2} b (d-a e) \cos (x)+\frac {1}{2} c (d-a e) \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx}{3 \left (a^2-b^2-c^2\right )} \\ & = \frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac {2 \left (c \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sin (x)\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (x)+c \sin (x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 a^2 d+\left (b^2+c^2\right ) d-4 a \left (b^2+c^2\right ) e\right )+\frac {1}{4} b \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \cos (x)+\frac {1}{4} c \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx}{3 \left (a^2-b^2-c^2\right )^2} \\ & = \frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac {2 \left (c \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sin (x)\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {(d-a e) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx}{3 \left (a^2-b^2-c^2\right )}+\frac {\left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \int \sqrt {a+b \cos (x)+c \sin (x)} \, dx}{3 \left (a^2-b^2-c^2\right )^2} \\ & = \frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac {2 \left (c \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sin (x)\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (x)+c \sin (x)}}+\frac {\left (\left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sqrt {a+b \cos (x)+c \sin (x)}\right ) \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}} \, dx}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {\left ((d-a e) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}} \, dx}{3 \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}} \\ & = \frac {2 \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 (d-a e) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{3 \left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac {2 \left (c \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (4 a d-a^2 e-3 \left (b^2+c^2\right ) e\right ) \sin (x)\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt {a+b \cos (x)+c \sin (x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.73 (sec) , antiderivative size = 5554, normalized size of antiderivative = 14.69 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4019\) vs. \(2(406)=812\).

Time = 142.43 (sec) , antiderivative size = 4020, normalized size of antiderivative = 10.63

method	result	size
default	\(\text {Expression too large to display}\)	\(4020\)
parts	\(\text {Expression too large to display}\)	\(1032654\)

[In]

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

[Out]

-(-(-b^2*sin(x-arctan(-b,c))-c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))*cos(x-arctan(-b,c))^2/(b^2+c^2)^(1/2))
^(1/2)/(b^2+c^2)^(1/2)*(b^4*sin(x-arctan(-b,c))^4+2*b^2*c^2*sin(x-arctan(-b,c))^4+c^4*sin(x-arctan(-b,c))^4-2*
a^2*b^2*sin(x-arctan(-b,c))^2-2*a^2*c^2*sin(x-arctan(-b,c))^2+a^4)*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin
(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*(sin(x-arctan(-b,c))^2*b^2*e+sin(x-arctan(-b,c))^2*c^2*e+e*a*sin(x-arctan
(-b,c))*(b^2+c^2)^(1/2)+d*sin(x-arctan(-b,c))*(b^2+c^2)^(1/2)+a*d)/(2*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)
*cos(x-arctan(-b,c))^2)^(1/2)*sin(x-arctan(-b,c))^2*a*b^2*e+2*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-a
rctan(-b,c))^2)^(1/2)*sin(x-arctan(-b,c))^2*a*c^2*e-(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c)
)+a)*(b^2+c^2))^(1/2)*sin(x-arctan(-b,c))^3*b^2*e-(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+
a)*(b^2+c^2))^(1/2)*sin(x-arctan(-b,c))^3*c^2*e-(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2
)^(1/2)*sin(x-arctan(-b,c))^2*b^2*d-(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*sin(
x-arctan(-b,c))^2*c^2*d-e*a^2*sin(x-arctan(-b,c))*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+
a)*(b^2+c^2))^(1/2)-d*a^2*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)+2*d*a*sin(x-ar
ctan(-b,c))*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2))/(b^4*sin(x-arctan
(-b,c))^3+2*b^2*c^2*sin(x-arctan(-b,c))^3+c^4*sin(x-arctan(-b,c))^3+3*b^2*sin(x-arctan(-b,c))^2*a*(b^2+c^2)^(1
/2)+3*c^2*sin(x-arctan(-b,c))^2*a*(b^2+c^2)^(1/2)+3*a^2*b^2*sin(x-arctan(-b,c))+3*a^2*c^2*sin(x-arctan(-b,c))+
(b^2+c^2)^(1/2)*a^3)*(-1/4*(a*b^2*e+a*c^2*e-b^2*d-c^2*d)/a/(a^2-b^2-c^2)*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/
2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)/(b^2*sin(x-arctan(-b,c))+c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))
-1/3*(a*e-d)/(a^2-b^2-c^2)/(b^2+c^2)^(1/2)*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2
+c^2))^(1/2)/(sin(x-arctan(-b,c))+1/(b^2+c^2)^(1/2)*a)^2+1/3*(b^2+c^2)^(1/2)*(-b^2-c^2)*cos(x-arctan(-b,c))^2/
(a^2-b^2-c^2)^2*(a^2*e+3*b^2*e+3*c^2*e-4*a*d)/(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(
b^2+c^2))^(1/2)+2*(7/24*(a*b^2*e+a*c^2*e-b^2*d-c^2*d)/(a^2-b^2-c^2)-1/6*a*(b^2+c^2)*(a^2*e+3*b^2*e+3*c^2*e-4*a
*d)/(a^2-b^2-c^2)^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/
2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1
/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)
*EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+
c^2)^(1/2)))^(1/2))+2*(-1/8*(b^2+c^2)^(1/2)*(a*b^2*e+a*c^2*e-b^2*d-c^2*d)/a/(a^2-b^2-c^2)+1/6*(b^2+c^2)^(3/2)*
(a^2*e+3*b^2*e+3*c^2*e-4*a*d)/(a^2-b^2-c^2)^2-1/6*(b^2+c^2)^(1/2)*(2*b^2+2*c^2)/(a^2-b^2-c^2)^2*(a^2*e+3*b^2*e
+3*c^2*e-4*a*d))*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*(
(sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/
(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*((-
1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2
+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2
)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2)))-1/8*(a^3*b^2*e+a^3*c^2*e+3*a*b^4*e+6*a*b^2*c^2*e
+3*a*c^4*e-5*a^2*b^2*d-5*a^2*c^2*d+b^4*d+2*b^2*c^2*d+c^4*d)/a^2/(a^2-b^2-c^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c
^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+
c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(x-arctan(-b,c))^2
*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*EllipticPi((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/
(a+(b^2+c^2)^(1/2)))^(1/2),-1/2*(-1/(b^2+c^2)^(1/2)*a-1)*(b^2+c^2)^(1/2)/a,((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)
^(1/2)))^(1/2))+1/4*(b^2+c^2)^(3/2)*(a*e-d)/a/(a^2-b^2-c^2)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arc
tan(-b,c))^2)^(1/2)/(b^2*sin(x-arctan(-b,c))+c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))-1/3*(a*e-d)/(a^2-b^2-c
^2)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)/(sin(x-arctan(-b,c))+1/(b^2+c^2)^(1/
2)*a)^2-1/3*(b^2+c^2)*cos(x-arctan(-b,c))^2/(a^2-b^2-c^2)^2*(a^2*e+3*b^2*e+3*c^2*e-4*a*d)/(((b^2+c^2)^(1/2)*si
n(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)+2*(1/24*(b^2+c^2)^(1/2)*(a*e-d)/(a^2-b^2-c^2)-1/6*a*(b^2+c^2
)^(1/2)*(a^2*e+3*b^2*e+3*c^2*e-4*a*d)/(a^2-b^2-c^2)^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(
-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-s
in(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(
x-arctan(-b,c))^2)^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b
^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+2*(1/8*(a*b^2*e+a*c^2*e-b^2*d-c^2*d)/a/(a^2-b^2-c^2)-1/6*(b^2+c^2)
*(a^2*e+3*b^2*e+3*c^2*e-4*a*d)/(a^2-b^2-c^2)^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+
a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-ar
ctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arcta
n(-b,c))^2)^(1/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1
/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))
+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2)))+1/8*(a^3*b^2*e+a^3*c^2*e+3*
a*b^4*e+6*a*b^2*c^2*e+3*a*c^4*e-5*a^2*b^2*d-5*a^2*c^2*d+b^4*d+2*b^2*c^2*d+c^4*d)/a^2/(a^2-b^2-c^2)/(b^2+c^2)^(
1/2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arcta
n(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)
^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*EllipticPi((((b^2+c^2)^(1
/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),-1/2*(-1/(b^2+c^2)^(1/2)*a-1)*(b^2+c^2)^(1/2)/a,((-a-(b^
2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2)))/cos(x-arctan(-b,c))/((b^2*sin(x-arctan(-b,c))+c^2*sin(x-arctan(-b,
c))+a*(b^2+c^2)^(1/2))/(b^2+c^2)^(1/2))^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.37 (sec) , antiderivative size = 4057, normalized size of antiderivative = 10.73 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/9*((sqrt(2)*(I*(a^2*b^3 + 3*b^5 - a^2*b*c^2 - 3*b*c^4)*d - (a^2*c^3 + 3*c^5 - (a^2*b^2 + 3*b^4)*c)*d + 2*I*(
a^3*b^3 - 3*a*b^5 - a^3*b*c^2 + 3*a*b*c^4)*e - 2*(a^3*c^3 - 3*a*c^5 - (a^3*b^2 - 3*a*b^4)*c)*e)*cos(x)^2 - 2*s
qrt(2)*(-I*(a^3*b^2 + 3*a*b^4 + 3*a*b^2*c^2)*d - (3*a*b*c^3 + (a^3*b + 3*a*b^3)*c)*d - 2*I*(a^4*b^2 - 3*a^2*b^
4 - 3*a^2*b^2*c^2)*e + 2*(3*a^2*b*c^3 - (a^4*b - 3*a^2*b^3)*c)*e)*cos(x) - 2*(sqrt(2)*(-I*(3*b^2*c^3 + (a^2*b^
2 + 3*b^4)*c)*d - (3*b*c^4 + (a^2*b + 3*b^3)*c^2)*d + 2*I*(3*a*b^2*c^3 - (a^3*b^2 - 3*a*b^4)*c)*e + 2*(3*a*b*c
^4 - (a^3*b - 3*a*b^3)*c^2)*e)*cos(x) + sqrt(2)*(-I*(3*a*b*c^3 + (a^3*b + 3*a*b^3)*c)*d - (3*a*c^4 + (a^3 + 3*
a*b^2)*c^2)*d + 2*I*(3*a^2*b*c^3 - (a^4*b - 3*a^2*b^3)*c)*e + 2*(3*a^2*c^4 - (a^4 - 3*a^2*b^2)*c^2)*e))*sin(x)
+ sqrt(2)*(I*(a^4*b + 3*a^2*b^3 + 3*b*c^4 + (4*a^2*b + 3*b^3)*c^2)*d + (3*c^5 + (4*a^2 + 3*b^2)*c^3 + (a^4 +
3*a^2*b^2)*c)*d + 2*I*(a^5*b - 3*a^3*b^3 - 3*a*b*c^4 - (2*a^3*b + 3*a*b^3)*c^2)*e - 2*(3*a*c^5 + (2*a^3 + 3*a*
b^2)*c^3 - (a^5 - 3*a^3*b^2)*c)*e))*sqrt(b + I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I
*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9
*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c
^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(I*b^2 + I*c^2)*sin(x))/(b^2 + c^2)) +
(sqrt(2)*(-I*(a^2*b^3 + 3*b^5 - a^2*b*c^2 - 3*b*c^4)*d - (a^2*c^3 + 3*c^5 - (a^2*b^2 + 3*b^4)*c)*d - 2*I*(a^3*
b^3 - 3*a*b^5 - a^3*b*c^2 + 3*a*b*c^4)*e - 2*(a^3*c^3 - 3*a*c^5 - (a^3*b^2 - 3*a*b^4)*c)*e)*cos(x)^2 - 2*sqrt(
2)*(I*(a^3*b^2 + 3*a*b^4 + 3*a*b^2*c^2)*d - (3*a*b*c^3 + (a^3*b + 3*a*b^3)*c)*d + 2*I*(a^4*b^2 - 3*a^2*b^4 - 3
*a^2*b^2*c^2)*e + 2*(3*a^2*b*c^3 - (a^4*b - 3*a^2*b^3)*c)*e)*cos(x) - 2*(sqrt(2)*(I*(3*b^2*c^3 + (a^2*b^2 + 3*
b^4)*c)*d - (3*b*c^4 + (a^2*b + 3*b^3)*c^2)*d - 2*I*(3*a*b^2*c^3 - (a^3*b^2 - 3*a*b^4)*c)*e + 2*(3*a*b*c^4 - (
a^3*b - 3*a*b^3)*c^2)*e)*cos(x) + sqrt(2)*(I*(3*a*b*c^3 + (a^3*b + 3*a*b^3)*c)*d - (3*a*c^4 + (a^3 + 3*a*b^2)*
c^2)*d - 2*I*(3*a^2*b*c^3 - (a^4*b - 3*a^2*b^3)*c)*e + 2*(3*a^2*c^4 - (a^4 - 3*a^2*b^2)*c^2)*e))*sin(x) + sqrt
(2)*(-I*(a^4*b + 3*a^2*b^3 + 3*b*c^4 + (4*a^2*b + 3*b^3)*c^2)*d + (3*c^5 + (4*a^2 + 3*b^2)*c^3 + (a^4 + 3*a^2*
b^2)*c)*d - 2*I*(a^5*b - 3*a^3*b^3 - 3*a*b*c^4 - (2*a^3*b + 3*a*b^3)*c^2)*e - 2*(3*a*c^5 + (2*a^3 + 3*a*b^2)*c
^3 - (a^5 - 3*a^3*b^2)*c)*e))*sqrt(b - I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3
+ 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c
^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3
*b^2*c^4 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(-I*b^2 - I*c^2)*sin(x))/(b^2 + c^2)) - 3*(sq
rt(2)*(4*I*(a*b^4 - a*c^4)*d - I*(a^2*b^4 + 3*b^6 + 3*b^4*c^2 - 3*c^6 - (a^2 + 3*b^2)*c^4)*e)*cos(x)^2 + 2*sqr
t(2)*(4*I*(a^2*b^3 + a^2*b*c^2)*d - I*(a^3*b^3 + 3*a*b^5 + 3*a*b*c^4 + (a^3*b + 6*a*b^3)*c^2)*e)*cos(x) + 2*(s
qrt(2)*(4*I*(a*b^3*c + a*b*c^3)*d - I*(3*b*c^5 + (a^2*b + 6*b^3)*c^3 + (a^2*b^3 + 3*b^5)*c)*e)*cos(x) + sqrt(2
)*(4*I*(a^2*b^2*c + a^2*c^3)*d - I*(3*a*c^5 + (a^3 + 6*a*b^2)*c^3 + (a^3*b^2 + 3*a*b^4)*c)*e))*sin(x) + sqrt(2
)*(4*I*(a^3*b^2 + a*c^4 + (a^3 + a*b^2)*c^2)*d - I*(a^4*b^2 + 3*a^2*b^4 + 3*c^6 + 2*(2*a^2 + 3*b^2)*c^4 + (a^4
+ 7*a^2*b^2 + 3*b^4)*c^2)*e))*sqrt(b + I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 +
3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5
+ 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^
2*c^4 + c^6), weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^
3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3
- 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b -
2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(I*b^2 + I*c^2)*sin(x))/(b^2 + c^2))) - 3*(sqrt(2)*(-4*I*(a*b^4 - a*c^4)*d
+ I*(a^2*b^4 + 3*b^6 + 3*b^4*c^2 - 3*c^6 - (a^2 + 3*b^2)*c^4)*e)*cos(x)^2 + 2*sqrt(2)*(-4*I*(a^2*b^3 + a^2*b*c
^2)*d + I*(a^3*b^3 + 3*a*b^5 + 3*a*b*c^4 + (a^3*b + 6*a*b^3)*c^2)*e)*cos(x) + 2*(sqrt(2)*(-4*I*(a*b^3*c + a*b*
c^3)*d + I*(3*b*c^5 + (a^2*b + 6*b^3)*c^3 + (a^2*b^3 + 3*b^5)*c)*e)*cos(x) + sqrt(2)*(-4*I*(a^2*b^2*c + a^2*c^
3)*d + I*(3*a*c^5 + (a^3 + 6*a*b^2)*c^3 + (a^3*b^2 + 3*a*b^4)*c)*e))*sin(x) + sqrt(2)*(-4*I*(a^3*b^2 + a*c^4 +
(a^3 + a*b^2)*c^2)*d + I*(a^4*b^2 + 3*a^2*b^4 + 3*c^6 + 2*(2*a^2 + 3*b^2)*c^4 + (a^4 + 7*a^2*b^2 + 3*b^4)*c^2
)*e))*sqrt(b - I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*
b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c
^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), weierstrass
PInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 +
c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*
c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*c
os(x) - 3*(-I*b^2 - I*c^2)*sin(x))/(b^2 + c^2))) + 6*(2*(4*(a*b^3*c + a*b*c^3)*d - (3*b*c^5 + (a^2*b + 6*b^3)*
c^3 + (a^2*b^3 + 3*b^5)*c)*e)*cos(x)^2 - 4*(a*b^3*c + a*b*c^3)*d + (3*b*c^5 + (a^2*b + 6*b^3)*c^3 + (a^2*b^3 +
3*b^5)*c)*e - ((c^5 - (5*a^2 - 2*b^2)*c^3 - (5*a^2*b^2 - b^4)*c)*d + 2*(a*c^5 + (a^3 + 2*a*b^2)*c^3 + (a^3*b^
2 + a*b^4)*c)*e)*cos(x) - ((5*a^2*b^3 - b^5 - b*c^4 + (5*a^2*b - 2*b^3)*c^2)*d - 2*(a^3*b^3 + a*b^5 + a*b*c^4
+ (a^3*b + 2*a*b^3)*c^2)*e + (4*(a*b^4 - a*c^4)*d - (a^2*b^4 + 3*b^6 + 3*b^4*c^2 - 3*c^6 - (a^2 + 3*b^2)*c^4)*
e)*cos(x))*sin(x))*sqrt(b*cos(x) + c*sin(x) + a))/(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + c^8 - (a^2 - 3*b^2)*c^6 - (
a^4 + a^2*b^2 - 3*b^4)*c^4 + (a^6 - 3*a^4*b^2 + a^2*b^4 + b^6)*c^2 + (a^4*b^4 - 2*a^2*b^6 + b^8 - c^8 + 2*(a^2
- b^2)*c^6 - (a^4 - 2*a^2*b^2)*c^4 - 2*(a^2*b^4 - b^6)*c^2)*cos(x)^2 + 2*(a^5*b^3 - 2*a^3*b^5 + a*b^7 + a*b*c
^6 - (2*a^3*b - 3*a*b^3)*c^4 + (a^5*b - 4*a^3*b^3 + 3*a*b^5)*c^2)*cos(x) + 2*(a*c^7 - (2*a^3 - 3*a*b^2)*c^5 +
(a^5 - 4*a^3*b^2 + 3*a*b^4)*c^3 + (a^5*b^2 - 2*a^3*b^4 + a*b^6)*c + (b*c^7 - (2*a^2*b - 3*b^3)*c^5 + (a^4*b -
4*a^2*b^3 + 3*b^5)*c^3 + (a^4*b^3 - 2*a^2*b^5 + b^7)*c)*cos(x))*sin(x))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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