\(\int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx\) [500]

Optimal result

Integrand size = 27, antiderivative size = 294 \[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\frac {2 \left (20 a d+3 a^2 e+9 \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{15 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) (5 d+3 a e) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{15 \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))-\frac {2}{15} \sqrt {a+b \cos (x)+c \sin (x)} (c (5 d+3 a e) \cos (x)-b (5 d+3 a e) \sin (x)) \] Output:

2/15*(20*a*d+3*a^2*e+9*(b^2+c^2)*e)*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+b*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)-2/15*(a^2-b^2-c^2) 
*(3*a*e+5*d)*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*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^ 
(1/2)/(a+b*cos(x)+c*sin(x))^(1/2)-2/5*(a+b*cos(x)+c*sin(x))^(3/2)*(c*e*cos 
(x)-b*e*sin(x))-2/15*(a+b*cos(x)+c*sin(x))^(1/2)*(c*(3*a*e+5*d)*cos(x)-b*( 
3*a*e+5*d)*sin(x))
 

Mathematica [C] (warning: unable to verify)

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

Time = 6.39 (sec) , antiderivative size = 5218, normalized size of antiderivative = 17.75 \[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3625, 27, 3042, 3625, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3625

\(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {a+b \cos (x)+c \sin (x)} \left (a \left (5 a d+3 \left (b^2+c^2\right ) e\right )+a b (5 d+3 a e) \cos (x)+a c (5 d+3 a e) \sin (x)\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {a+b \cos (x)+c \sin (x)} \left (a \left (5 a d+3 \left (b^2+c^2\right ) e\right )+a b (5 d+3 a e) \cos (x)+a c (5 d+3 a e) \sin (x)\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {a+b \cos (x)+c \sin (x)} \left (a \left (5 a d+3 \left (b^2+c^2\right ) e\right )+a b (5 d+3 a e) \cos (x)+a c (5 d+3 a e) \sin (x)\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3625

\(\displaystyle \frac {\frac {2 \int \frac {\left (15 d a^2+12 \left (b^2+c^2\right ) e a+5 \left (b^2+c^2\right ) d\right ) a^2+b \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \sin (x) a^2}{2 \sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (15 d a^2+12 \left (b^2+c^2\right ) e a+5 \left (b^2+c^2\right ) d\right ) a^2+b \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \sin (x) a^2}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (15 d a^2+12 \left (b^2+c^2\right ) e a+5 \left (b^2+c^2\right ) d\right ) a^2+b \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \sin (x) a^2}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {\frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\)

Input:

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

Output:

(-2*(a + b*Cos[x] + c*Sin[x])^(3/2)*(c*e*Cos[x] - b*e*Sin[x]))/5 + ((-2*Sq 
rt[a + b*Cos[x] + c*Sin[x]]*(a*c*(5*d + 3*a*e)*Cos[x] - a*b*(5*d + 3*a*e)* 
Sin[x]))/3 + ((2*a^2*(20*a*d + 3*a^2*e + 9*(b^2 + c^2)*e)*EllipticE[(x - A 
rcTan[b, c])/2, (2*Sqrt[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*(a^2 - b^2 - c^2)*(5*d + 3*a*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sq 
rt[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]])/(3*a))/(5*a)
 

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 3625

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2162\) vs. \(2(269)=538\).

Time = 0.81 (sec) , antiderivative size = 2163, normalized size of antiderivative = 7.36

method	result	size
default	\(\text {Expression too large to display}\)	\(2163\)
parts	\(\text {Expression too large to display}\)	\(13503\)

Input:

int((a+b*cos(x)+c*sin(x))^(3/2)*(d+b*e*cos(x)+c*e*sin(x)),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)*((b^2+c^2)^(1/2) 
*(2*a*b^2*e+2*a*c^2*e+b^2*d+c^2*d)*(-2/3/(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)+2/3*(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)*(-(b 
^2+c^2)^(1/2)*(sin(x-arctan(-b,c))-1)/(a+(b^2+c^2)^(1/2)))^(1/2)/(cos(x-ar 
ctan(-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))-4/3/(b^2+c^2)^(1/2)*a*(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)*(-(b^2+c^2)^(1/2)*(sin(x-arctan(-b,c))-1)/(a+(b^2+c^2)^(1/2)))^(1/2)/(c 
os(x-arctan(-b,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)*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))))+e*(b^4+2*b^2*c^2 
+c^4)*(-2/5/(b^2+c^2)^(1/2)*sin(x-arctan(-b,c))*(cos(x-arctan(-b,c))^2*((b 
^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a))^(1/2)+8/15/(b^2+c^2)*a*(cos(x-arc...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 1688, normalized size of antiderivative = 5.74 \[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Too large to display} \] Input:

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

Output:

-2/45*((-5*I*(a^2*b + 3*b^3 + 3*b*c^2)*d - 5*(3*c^3 + (a^2 + 3*b^2)*c)*d + 
 6*I*(a^3*b - 3*a*b^3 - 3*a*b*c^2)*e - 6*(3*a*c^3 - (a^3 - 3*a*b^2)*c)*e)* 
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)) + (5*I*(a^2*b + 3*b^3 + 3*b*c^2)*d 
 - 5*(3*c^3 + (a^2 + 3*b^2)*c)*d - 6*I*(a^3*b - 3*a*b^3 - 3*a*b*c^2)*e - 6 
*(3*a*c^3 - (a^3 - 3*a*b^2)*c)*e)*sqrt(1/2*b - 1/2*I*c)*weierstrassPInvers 
e(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*(20*I*(a*b^2 + a*c^2)*d + 3*I*(a^2*b^2 + 3*b^4 + 3*c^4 + (a^2 + 6*b^2 
)*c^2)*e)*sqrt(1/2*b + 1/2*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)...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

(48*sqrt(cos(x)*b + sin(x)*c + a)*cos(x)**2*b**7*e + 60*sqrt(cos(x)*b + si 
n(x)*c + a)*cos(x)**2*b**5*c**2*e + 18*sqrt(cos(x)*b + sin(x)*c + a)*cos(x 
)**2*b**3*c**4*e + 36*sqrt(cos(x)*b + sin(x)*c + a)*cos(x)*sin(x)*b**6*c*e 
 + 30*sqrt(cos(x)*b + sin(x)*c + a)*cos(x)*sin(x)*b**4*c**3*e + 6*sqrt(cos 
(x)*b + sin(x)*c + a)*cos(x)*sin(x)*b**2*c**5*e + 16*sqrt(cos(x)*b + sin(x 
)*c + a)*cos(x)*a*b**6*e + 20*sqrt(cos(x)*b + sin(x)*c + a)*cos(x)*a*b**4* 
c**2*e + 16*sqrt(cos(x)*b + sin(x)*c + a)*cos(x)*a*b**2*c**4*e - 20*sqrt(c 
os(x)*b + sin(x)*c + a)*cos(x)*b**4*c**2*d - 10*sqrt(cos(x)*b + sin(x)*c + 
 a)*cos(x)*b**2*c**4*d + 60*sqrt(cos(x)*b + sin(x)*c + a)*sin(x)**2*b**7*e 
 + 138*sqrt(cos(x)*b + sin(x)*c + a)*sin(x)**2*b**5*c**2*e + 90*sqrt(cos(x 
)*b + sin(x)*c + a)*sin(x)**2*b**3*c**4*e + 18*sqrt(cos(x)*b + sin(x)*c + 
a)*sin(x)**2*b*c**6*e + 76*sqrt(cos(x)*b + sin(x)*c + a)*sin(x)*a*b**5*c*e 
 + 80*sqrt(cos(x)*b + sin(x)*c + a)*sin(x)*a*b**3*c**3*e + 16*sqrt(cos(x)* 
b + sin(x)*c + a)*sin(x)*a*b*c**5*e + 60*sqrt(cos(x)*b + sin(x)*c + a)*sin 
(x)*b**5*c*d + 70*sqrt(cos(x)*b + sin(x)*c + a)*sin(x)*b**3*c**3*d + 20*sq 
rt(cos(x)*b + sin(x)*c + a)*sin(x)*b*c**5*d - 32*sqrt(cos(x)*b + sin(x)*c 
+ a)*a**2*b**5*e - 40*sqrt(cos(x)*b + sin(x)*c + a)*a**2*b**3*c**2*e - 2*s 
qrt(cos(x)*b + sin(x)*c + a)*a**2*b*c**4*e + 120*sqrt(cos(x)*b + sin(x)*c 
+ a)*a*b**5*d + 100*sqrt(cos(x)*b + sin(x)*c + a)*a*b**3*c**2*d + 20*sqrt( 
cos(x)*b + sin(x)*c + a)*a*b*c**4*d + 240*int(sqrt(cos(x)*b + sin(x)*c ...