\(\int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx\) [503]

Optimal result

Integrand size = 27, antiderivative size = 250 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\frac {2 (d-a e) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{\left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}+\frac {2 e \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (c (d-a e) \cos (x)-b (d-a e) \sin (x))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}} \] Output:

2*(-a*e+d)*EllipticE(sin(1/2*x-1/2*arctan(c,b)),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)/((a+b 
*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)+2*e*InverseJacobiAM(1/2*x-1/2 
*arctan(c,b),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*((a+b*co 
s(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a+b*cos(x)+c*sin(x))^(1/2)+2*(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))^(1 
/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 6.44 (sec) , antiderivative size = 3176, normalized size of antiderivative = 12.70 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

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/(Sq 
rt[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*Sq 
rt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*S 
qrt[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*AppellF 
1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/...
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3635, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3635

\(\displaystyle \frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2 \int -\frac {a d-\left (b^2+c^2\right ) e+b (d-a e) \cos (x)+c (d-a e) \sin (x)}{2 \sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a d-\left (b^2+c^2\right ) e+b (d-a e) \cos (x)+c (d-a e) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a d-\left (b^2+c^2\right ) e+b (d-a e) \cos (x)+c (d-a e) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+(d-a e) \int \sqrt {a+b \cos (x)+c \sin (x)}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+(d-a e) \int \sqrt {a+b \cos (x)+c \sin (x)}dx}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+\frac {(d-a e) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+\frac {(d-a e) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {e \left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {\frac {e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {2 e \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 (d-a e) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}}{a^2-b^2-c^2}+\frac {2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left (a^2-b^2-c^2\right ) \sqrt {a+b \cos (x)+c \sin (x)}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3598

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Simp[Sqrt[a + b*Cos[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 + S 
qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc 
Tan[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 3606

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( 
x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq 
rt[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] /; 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 3628

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] :> Simp[B/b   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] 
, x] + Simp[(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 3635

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] + Simp[1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a + b*Co 
s[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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2903\) vs. \(2(235)=470\).

Time = 2.82 (sec) , antiderivative size = 2904, normalized size of antiderivative = 11.62

method	result	size
default	\(\text {Expression too large to display}\)	\(2904\)
parts	\(\text {Expression too large to display}\)	\(113141\)

Input:

int((d+b*e*cos(x)+c*e*sin(x))/(a+b*cos(x)+c*sin(x))^(3/2),x,method=_RETURN 
VERBOSE)
 

Output:

(-(-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)*((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))^2+c^2*sin(x-arctan(-b,c))^2-a^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)*sin(x-arctan(-b,c))+a))^(1/2)*a*b^2+(cos(x-arctan(-b,c))^2*((b^2 
+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(1/2)*a*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)*sin(x-arctan(-b,c))* 
b^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))*c^2)/(b^2*sin(x-arctan(-b,c))^2+c^2*sin(x-ar 
ctan(-b,c))^2+2*sin(x-arctan(-b,c))*a*(b^2+c^2)^(1/2)+a^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)/(cos(x-arctan(-b 
,c))^2*((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*(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-ar 
ctan(-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)*(-(b^2+c^2)^(1/2)*(sin(x-arctan(-b,c) 
)-1)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(x-arctan(-b,c))^2*((b^2+c^2)^(1/2)*si 
n(x-arctan(-b,c))+a)*(b^2+c^2))^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(x-ar 
ctan(-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...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 2050, normalized size of antiderivative = 8.20 \[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/3*((I*a^2*b*d + a^2*c*d + I*(2*a^3*b - 3*a*b^3 - 3*a*b*c^2)*e - (3*a*c^3 
 - (2*a^3 - 3*a*b^2)*c)*e + (I*a*b^2*d + a*b*c*d + I*(2*a^2*b^2 - 3*b^4 - 
3*b^2*c^2)*e - (3*b*c^3 - (2*a^2*b - 3*b^3)*c)*e)*cos(x) + (I*a*b*c*d + a* 
c^2*d - I*(3*b*c^3 - (2*a^2*b - 3*b^3)*c)*e - (3*c^4 - (2*a^2 - 3*b^2)*c^2 
)*e)*sin(x))*sqrt(1/2*b + 1/2*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)) + (-I*a^2*b*d + a^2*c 
*d - I*(2*a^3*b - 3*a*b^3 - 3*a*b*c^2)*e - (3*a*c^3 - (2*a^3 - 3*a*b^2)*c) 
*e + (-I*a*b^2*d + a*b*c*d - I*(2*a^2*b^2 - 3*b^4 - 3*b^2*c^2)*e - (3*b*c^ 
3 - (2*a^2*b - 3*b^3)*c)*e)*cos(x) + (-I*a*b*c*d + a*c^2*d + I*(3*b*c^3 - 
(2*a^2*b - 3*b^3)*c)*e - (3*c^4 - (2*a^2 - 3*b^2)*c^2)*e)*sin(x))*sqrt(1/2 
*b - 1/2*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*(-I*(a*b^2 + a*c^2)*d + I*(a^2*b^2...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(cos(x)*int(sqrt(cos(x)*b + sin(x)*c + a)/(cos(x)**2*b**2 + 2*cos(x)*sin(x 
)*b*c + 2*cos(x)*a*b + sin(x)**2*c**2 + 2*sin(x)*a*c + a**2),x)*b**2*d + c 
os(x)*int((sqrt(cos(x)*b + sin(x)*c + a)*cos(x))/(cos(x)**2*b**2 + 2*cos(x 
)*sin(x)*b*c + 2*cos(x)*a*b + sin(x)**2*c**2 + 2*sin(x)*a*c + a**2),x)*b** 
3*e + cos(x)*int((sqrt(cos(x)*b + sin(x)*c + a)*cos(x))/(cos(x)**2*b**2 + 
2*cos(x)*sin(x)*b*c + 2*cos(x)*a*b + sin(x)**2*c**2 + 2*sin(x)*a*c + a**2) 
,x)*b*c**2*e + 2*sqrt(cos(x)*b + sin(x)*c + a)*c*e + int(sqrt(cos(x)*b + s 
in(x)*c + a)/(cos(x)**2*b**2 + 2*cos(x)*sin(x)*b*c + 2*cos(x)*a*b + sin(x) 
**2*c**2 + 2*sin(x)*a*c + a**2),x)*sin(x)*b*c*d + int(sqrt(cos(x)*b + sin( 
x)*c + a)/(cos(x)**2*b**2 + 2*cos(x)*sin(x)*b*c + 2*cos(x)*a*b + sin(x)**2 
*c**2 + 2*sin(x)*a*c + a**2),x)*a*b*d + int((sqrt(cos(x)*b + sin(x)*c + a) 
*cos(x))/(cos(x)**2*b**2 + 2*cos(x)*sin(x)*b*c + 2*cos(x)*a*b + sin(x)**2* 
c**2 + 2*sin(x)*a*c + a**2),x)*sin(x)*b**2*c*e + int((sqrt(cos(x)*b + sin( 
x)*c + a)*cos(x))/(cos(x)**2*b**2 + 2*cos(x)*sin(x)*b*c + 2*cos(x)*a*b + s 
in(x)**2*c**2 + 2*sin(x)*a*c + a**2),x)*sin(x)*c**3*e + int((sqrt(cos(x)*b 
 + sin(x)*c + a)*cos(x))/(cos(x)**2*b**2 + 2*cos(x)*sin(x)*b*c + 2*cos(x)* 
a*b + sin(x)**2*c**2 + 2*sin(x)*a*c + a**2),x)*a*b**2*e + int((sqrt(cos(x) 
*b + sin(x)*c + a)*cos(x))/(cos(x)**2*b**2 + 2*cos(x)*sin(x)*b*c + 2*cos(x 
)*a*b + sin(x)**2*c**2 + 2*sin(x)*a*c + a**2),x)*a*c**2*e)/(b*(cos(x)*b + 
sin(x)*c + a))