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

Optimal. Leaf size=378 \[ -\frac{2 (d-a e) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} \text{EllipticF}\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{3 \left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt{a+b \cos (x)+c \sin (x)} E\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac{2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt{a+b \cos (x)+c \sin (x)}} \]

[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]])

________________________________________________________________________________________

Rubi [A]  time = 0.561126, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3156, 3149, 3119, 2653, 3127, 2661} \[ -\frac{2 (d-a e) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{3 \left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right ) \sqrt{a+b \cos (x)+c \sin (x)} E\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac{2 \left (c \cos (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )-b \sin (x) \left (a^2 (-e)+4 a d-3 e \left (b^2+c^2\right )\right )\right )}{3 \left (a^2-b^2-c^2\right )^2 \sqrt{a+b \cos (x)+c \sin (x)}} \]

Antiderivative was successfully verified.

[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 3156

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]

Rule 3149

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 3119

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 2653

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

Rule 3127

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 2661

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

Rubi steps

\begin{align*} \int \frac{d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx &=\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) F\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]  time = 6.84663, size = 5554, normalized size = 14.69 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[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]  time = 43.325, size = 3164, normalized size = 8.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*(1/4/a/(a^2-b^2-c^2)*(a*e-d)*(b^2+c^2)^(3/2)*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin
(x-arctan(-b,c))+a))^(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^2-b^2-c^
2)*(a*e-d)*(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(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)/(cos(x-arctan
(-b,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(1/2)+2*(1/24*(a*e-d)*(b^2+c^2)^(1/2)/(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)*((1+sin(x-arctan(-b,c)))*(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))^(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-arct
an(-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)*
((1+sin(x-arctan(-b,c)))*(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)*s
in(x-arctan(-b,c))+a))^(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-a
rctan(-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-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)*((1+sin(x-arctan(-b,c)))*(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))^(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*(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)*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(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^2-b^2-c^2)*(a*e-d)/(b^2+c^2)^(1/2)*(cos(x-arctan(-b,c))^2*(b^2+c^2)*(
(b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(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
)*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(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)*((1
+sin(x-arctan(-b,c)))*(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)*((b^2+c^2)^
(1/2)*sin(x-arctan(-b,c))+a))^(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)*(b^2+c^2)^(1/2)/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)*((1+s
in(x-arctan(-b,c)))*(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)*((b^2+c^2)^(1
/2)*sin(x-arctan(-b,c))+a))^(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)*si
n(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-ar
ctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)*((1+sin(x-arctan(-b,c)))*(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)*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b e \cos \left (x\right ) + c e \sin \left (x\right ) + d\right )} \sqrt{b \cos \left (x\right ) + c \sin \left (x\right ) + a}}{{\left (b^{3} - 3 \, b c^{2}\right )} \cos \left (x\right )^{3} + a^{3} + 3 \, a c^{2} + 3 \,{\left (a b^{2} - a c^{2}\right )} \cos \left (x\right )^{2} + 3 \,{\left (a^{2} b + b c^{2}\right )} \cos \left (x\right ) +{\left (6 \, a b c \cos \left (x\right ) + 3 \, a^{2} c + c^{3} +{\left (3 \, b^{2} c - c^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*e*cos(x) + c*e*sin(x) + d)*sqrt(b*cos(x) + c*sin(x) + a)/((b^3 - 3*b*c^2)*cos(x)^3 + a^3 + 3*a*c^2
+ 3*(a*b^2 - a*c^2)*cos(x)^2 + 3*(a^2*b + b*c^2)*cos(x) + (6*a*b*c*cos(x) + 3*a^2*c + c^3 + (3*b^2*c - c^3)*c
os(x)^2)*sin(x)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)