3.560 \(\int \frac{d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx\)

Optimal. Leaf size=250 \[ \frac{2 (d-a e) \sqrt{a+b \cos (x)+c \sin (x)} E\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{\left (a^2-b^2-c^2\right ) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 e \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} \text{EllipticF}\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{\sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}} \]

[Out]

(2*(d - a*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]])/((a^2 - b^2 - c^2)*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) + (2*e*EllipticF[(x - ArcTa
n[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])])/
Sqrt[a + b*Cos[x] + c*Sin[x]] + (2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/((a^2 - b^2 - c^2)*Sqrt[a + b*Co
s[x] + c*Sin[x]])

________________________________________________________________________________________

Rubi [A]  time = 0.323051, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{2 (d-a e) \sqrt{a+b \cos (x)+c \sin (x)} E\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{\left (a^2-b^2-c^2\right ) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 e \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left (\frac{1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right )}{\sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d - a*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]])/((a^2 - b^2 - c^2)*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) + (2*e*EllipticF[(x - ArcTa
n[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])])/
Sqrt[a + b*Cos[x] + c*Sin[x]] + (2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/((a^2 - b^2 - c^2)*Sqrt[a + b*Co
s[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))^{3/2}} \, dx &=\frac{2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{\left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}}-\frac{2 \int \frac{\frac{1}{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)}{\sqrt{a+b \cos (x)+c \sin (x)}} \, dx}{a^2-b^2-c^2}\\ &=\frac{2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{\left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}}+e \int \frac{1}{\sqrt{a+b \cos (x)+c \sin (x)}} \, dx+\frac{(d-a e) \int \sqrt{a+b \cos (x)+c \sin (x)} \, dx}{a^2-b^2-c^2}\\ &=\frac{2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{\left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}}+\frac{\left ((d-a e) \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}{\left (a^2-b^2-c^2\right ) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{\left (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}{\sqrt{a+b \cos (x)+c \sin (x)}}\\ &=\frac{2 (d-a e) 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)}}{\left (a^2-b^2-c^2\right ) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 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}}}}{\sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{\left (a^2-b^2-c^2\right ) \sqrt{a+b \cos (x)+c \sin (x)}}\\ \end{align*}

Mathematica [C]  time = 6.48662, size = 3176, normalized size = 12.7 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[a + b*Cos[x] + c*Sin[x]]*((2*(b^2 + c^2)*(-d + a*e))/(b*c*(-a^2 + b^2 + c^2)) - (2*(-(a*c*d) + a^2*c*e -
b^2*d*Sin[x] - c^2*d*Sin[x] + a*b^2*e*Sin[x] + a*c^2*e*Sin[x]))/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[x] + c*Sin[x]
))) - (2*a*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*
(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a
/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[
x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(
c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt
[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)) + (2*b^2*e*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x +
ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan
[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c
^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)
/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a
+ c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)) + (2*c*e*AppellF1[1/2, 1/2, 1/2, 3/2, -(
(a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sq
rt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[
b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^
2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)
/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*(-a^2 + b^2 + c^2)) - (b^2*d*(
-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1
- a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b
*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2
+ c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - Arc
Tan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c
^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b
*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(c*(-a^2 + b^2 + c^2)) - (c*d*(-((c*
AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(
b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt
[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^
2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c
/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b
^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt
[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(-a^2 + b^2 + c^2) + (a*b^2*e*(-((c*Appel
lF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqr
t[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 +
c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^
2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]
*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])]
)) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 +
c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(c*(-a^2 + b^2 + c^2)) + (a*c*e*(-((c*AppellF1
[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1
+ c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2
/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*
Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sq
rt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])]))
- ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2
/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(-a^2 + b^2 + c^2)

________________________________________________________________________________________

Maple [B]  time = 9.936, size = 2596, normalized size = 10.4 \begin{align*} \text{result 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))^(3/2),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)*(2*(b^2+c^2)^(1/2)*e*(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-arct
an(-b,c)))*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)/(-(-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)*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))-(b^2+c^2)*cos(x-arctan(-b,c))^
2*(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)-a*(b^2+c^2)^(1
/2)*(a*e-d)/(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)*((1+sin(x-arctan(-b,c)))*(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))-(a*e-d)*(b^2+c^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)*((1+si
n(x-arctan(-b,c)))*(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)*((-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-arcta
n(-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)))+(1/2*a*e-1/2*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+sin(x-arctan(-b,c)))*(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)*(b^2+c^2)^(1/2)/a*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))+(b^2+c^2)^(1/2)*(-b^2-c^2)*cos(x-arctan(-b,c))^2/(a^2-
b^2-c^2)*(a*e-d)/(-(-(b^2+c^2)^(1/2)*sin(x-arctan(-b,c))-a)*cos(x-arctan(-b,c))^2*(b^2+c^2))^(1/2)-a*(b^2+c^2)
*(a*e-d)/(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)*((1+sin(x-arctan(-b,c)))*(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*(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*(-1/2*(b^2+c^2)^(3/2)*(a*e-d)/(a^2-b^2-c^2)+1/2*(b^2+c^2)^(1/2)*(2*b^2+2*c^2)/(a^2
-b^2-c^2)*(a*e-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+sin(x-arctan(-b,c)))*(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*(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)))-1/2*(a*b^2*e+a*c^2*e-b^2*d-c^2*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+sin(x-arctan(-b,c)))*(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*(b^2+c^2))^(1/2)/a*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)

________________________________________________________________________________________

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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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}}{2 \, a b \cos \left (x\right ) +{\left (b^{2} - c^{2}\right )} \cos \left (x\right )^{2} + a^{2} + c^{2} + 2 \,{\left (b c \cos \left (x\right ) + a c\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))^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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))**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

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{3}{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))^(3/2),x, algorithm="giac")

[Out]

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