\(\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\) [294]

Optimal result

Integrand size = 37, antiderivative size = 294 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x)}{315 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {4 a (5 c-d) \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 d f}+\frac {2 \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (3 B c-9 A d-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a B \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{9 d f} \] Output:

2/315*a^2*(15*c^2+10*c*d+7*d^2)*(3*A*(c-13*d)*d-B*(c^2-7*c*d+34*d^2))*cos( 
f*x+e)/d^2/f/(a+a*sin(f*x+e))^(1/2)+4/315*a*(5*c-d)*(3*A*(c-13*d)*d-B*(c^2 
-7*c*d+34*d^2))*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/d/f+2/105*(3*A*(c-13*d)* 
d-B*(c^2-7*c*d+34*d^2))*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f+2/63*a^2*(-9*A 
*d+3*B*c-10*B*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/d^2/f/(a+a*sin(f*x+e))^(1/2 
)-2/9*a*B*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^3/d/f
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.67 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.91 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (4200 A c^2+3276 B c^2+6552 A c d+5928 B c d+2964 A d^2+2689 B d^2-4 \left (9 A d (14 c+13 d)+B \left (63 c^2+234 c d+137 d^2\right )\right ) \cos (2 (e+f x))+35 B d^2 \cos (4 (e+f x))+840 A c^2 \sin (e+f x)+1512 B c^2 \sin (e+f x)+3024 A c d \sin (e+f x)+3036 B c d \sin (e+f x)+1518 A d^2 \sin (e+f x)+1598 B d^2 \sin (e+f x)-180 B c d \sin (3 (e+f x))-90 A d^2 \sin (3 (e+f x))-170 B d^2 \sin (3 (e+f x))\right )}{1260 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:

Integrate[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f 
*x])^2,x]
 

Output:

-1/1260*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x]) 
]*(4200*A*c^2 + 3276*B*c^2 + 6552*A*c*d + 5928*B*c*d + 2964*A*d^2 + 2689*B 
*d^2 - 4*(9*A*d*(14*c + 13*d) + B*(63*c^2 + 234*c*d + 137*d^2))*Cos[2*(e + 
 f*x)] + 35*B*d^2*Cos[4*(e + f*x)] + 840*A*c^2*Sin[e + f*x] + 1512*B*c^2*S 
in[e + f*x] + 3024*A*c*d*Sin[e + f*x] + 3036*B*c*d*Sin[e + f*x] + 1518*A*d 
^2*Sin[e + f*x] + 1598*B*d^2*Sin[e + f*x] - 180*B*c*d*Sin[3*(e + f*x)] - 9 
0*A*d^2*Sin[3*(e + f*x)] - 170*B*d^2*Sin[3*(e + f*x)]))/(f*(Cos[(e + f*x)/ 
2] + Sin[(e + f*x)/2]))
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.91, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {3042, 3455, 27, 3042, 3460, 3042, 3240, 27, 3042, 3230, 3042, 3125}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2dx\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2 (a (9 A d+B (c+6 d))-a (3 B c-9 A d-10 B d) \sin (e+f x))dx}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2 (a (9 A d+B (c+6 d))-a (3 B c-9 A d-10 B d) \sin (e+f x))dx}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2 (a (9 A d+B (c+6 d))-a (3 B c-9 A d-10 B d) \sin (e+f x))dx}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3460

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \int \sqrt {\sin (e+f x) a+a} (c+d \sin (e+f x))^2dx}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3240

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \left (\frac {2 \int \frac {1}{2} \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \left (\frac {\int \sqrt {\sin (e+f x) a+a} \left (a \left (5 c^2+3 d^2\right )+2 a (5 c-d) d \sin (e+f x)\right )dx}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3230

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \left (\frac {\frac {1}{3} a \left (15 c^2+10 c d+7 d^2\right ) \int \sqrt {\sin (e+f x) a+a}dx-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

\(\Big \downarrow \) 3125

\(\displaystyle \frac {\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {a \sin (e+f x)+a}}-\frac {3 a \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \left (\frac {-\frac {2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a d (5 c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{5 a}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f}\right )}{7 d}}{9 d}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f}\)

Input:

Int[(a + a*Sin[e + f*x])^(3/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2 
,x]
 

Output:

(-2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3)/(9*d 
*f) + ((2*a^2*(3*B*c - 9*A*d - 10*B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3 
)/(7*d*f*Sqrt[a + a*Sin[e + f*x]]) - (3*a*(3*A*(c - 13*d)*d - B*(c^2 - 7*c 
*d + 34*d^2))*((-2*d^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*a*f) + 
((-2*a^2*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f 
*x]]) - (4*a*(5*c - d)*d*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f))/(5* 
a)))/(7*d))/(9*d)
 

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 3125

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos 
[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[a^2 - b^2, 0]
 

rule 3230

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( 
f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1))   Int[(a + b*Sin[e 
 + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] 
&& EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]
 

rule 3240

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^2, x_Symbol] :> Simp[(-d^2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^ 
(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2))   Int[(a + b*Sin[e + f*x])^ 
m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x] 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ 
[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

rule 3455

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n 
 + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 
) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + 
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 
 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1 
] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3460

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( 
f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp 
[-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + 
b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b 
*d*(2*n + 3))   Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - 
 b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]
 

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (35 B \cos \left (f x +e \right )^{4} d^{2}+\left (-45 A \,d^{2}-90 B c d -85 B \,d^{2}\right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )+\left (-126 A c d -117 A \,d^{2}-63 B \,c^{2}-234 B c d -172 B \,d^{2}\right ) \cos \left (f x +e \right )^{2}+\left (105 A \,c^{2}+378 A c d +201 A \,d^{2}+189 B \,c^{2}+402 B c d +221 B \,d^{2}\right ) \sin \left (f x +e \right )+525 A \,c^{2}+882 A c d +429 A \,d^{2}+441 B \,c^{2}+858 B c d +409 B \,d^{2}\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(207\)
parts	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) d \left (A d +2 B c \right ) \left (15 \sin \left (f x +e \right )^{3}+39 \sin \left (f x +e \right )^{2}+52 \sin \left (f x +e \right )+104\right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) c \left (2 A d +B c \right ) \left (\sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 A \,c^{2} \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 B \,d^{2} \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (35 \sin \left (f x +e \right )^{4}+85 \sin \left (f x +e \right )^{3}+102 \sin \left (f x +e \right )^{2}+136 \sin \left (f x +e \right )+272\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(300\)

Input:

int((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x,method=_R 
ETURNVERBOSE)
 

Output:

2/315*(1+sin(f*x+e))*a^2*(sin(f*x+e)-1)*(35*B*cos(f*x+e)^4*d^2+(-45*A*d^2- 
90*B*c*d-85*B*d^2)*cos(f*x+e)^2*sin(f*x+e)+(-126*A*c*d-117*A*d^2-63*B*c^2- 
234*B*c*d-172*B*d^2)*cos(f*x+e)^2+(105*A*c^2+378*A*c*d+201*A*d^2+189*B*c^2 
+402*B*c*d+221*B*d^2)*sin(f*x+e)+525*A*c^2+882*A*c*d+429*A*d^2+441*B*c^2+8 
58*B*c*d+409*B*d^2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.46 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {2 \, {\left (35 \, B a d^{2} \cos \left (f x + e\right )^{5} - 5 \, {\left (18 \, B a c d + {\left (9 \, A + 10 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{4} + 84 \, {\left (5 \, A + 3 \, B\right )} a c^{2} + 24 \, {\left (21 \, A + 19 \, B\right )} a c d + 4 \, {\left (57 \, A + 47 \, B\right )} a d^{2} - {\left (63 \, B a c^{2} + 18 \, {\left (7 \, A + 13 \, B\right )} a c d + {\left (117 \, A + 172 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (21 \, {\left (5 \, A + 6 \, B\right )} a c^{2} + 6 \, {\left (42 \, A + 43 \, B\right )} a c d + {\left (129 \, A + 134 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (21 \, {\left (25 \, A + 21 \, B\right )} a c^{2} + 6 \, {\left (147 \, A + 143 \, B\right )} a c d + {\left (429 \, A + 409 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right ) - {\left (35 \, B a d^{2} \cos \left (f x + e\right )^{4} + 84 \, {\left (5 \, A + 3 \, B\right )} a c^{2} + 24 \, {\left (21 \, A + 19 \, B\right )} a c d + 4 \, {\left (57 \, A + 47 \, B\right )} a d^{2} + 5 \, {\left (18 \, B a c d + {\left (9 \, A + 17 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (21 \, B a c^{2} + 6 \, {\left (7 \, A + 8 \, B\right )} a c d + {\left (24 \, A + 29 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (21 \, {\left (5 \, A + 9 \, B\right )} a c^{2} + 6 \, {\left (63 \, A + 67 \, B\right )} a c d + {\left (201 \, A + 221 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \] Input:

integrate((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, al 
gorithm="fricas")
 

Output:

-2/315*(35*B*a*d^2*cos(f*x + e)^5 - 5*(18*B*a*c*d + (9*A + 10*B)*a*d^2)*co 
s(f*x + e)^4 + 84*(5*A + 3*B)*a*c^2 + 24*(21*A + 19*B)*a*c*d + 4*(57*A + 4 
7*B)*a*d^2 - (63*B*a*c^2 + 18*(7*A + 13*B)*a*c*d + (117*A + 172*B)*a*d^2)* 
cos(f*x + e)^3 + (21*(5*A + 6*B)*a*c^2 + 6*(42*A + 43*B)*a*c*d + (129*A + 
134*B)*a*d^2)*cos(f*x + e)^2 + (21*(25*A + 21*B)*a*c^2 + 6*(147*A + 143*B) 
*a*c*d + (429*A + 409*B)*a*d^2)*cos(f*x + e) - (35*B*a*d^2*cos(f*x + e)^4 
+ 84*(5*A + 3*B)*a*c^2 + 24*(21*A + 19*B)*a*c*d + 4*(57*A + 47*B)*a*d^2 + 
5*(18*B*a*c*d + (9*A + 17*B)*a*d^2)*cos(f*x + e)^3 - 3*(21*B*a*c^2 + 6*(7* 
A + 8*B)*a*c*d + (24*A + 29*B)*a*d^2)*cos(f*x + e)^2 - (21*(5*A + 9*B)*a*c 
^2 + 6*(63*A + 67*B)*a*c*d + (201*A + 221*B)*a*d^2)*cos(f*x + e))*sin(f*x 
+ e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)
 

Sympy [F]

\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )^{2}\, dx \] Input:

integrate((a+a*sin(f*x+e))**(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2,x)
 

Output:

Integral((a*(sin(e + f*x) + 1))**(3/2)*(A + B*sin(e + f*x))*(c + d*sin(e + 
 f*x))**2, x)
 

Maxima [F]

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

integrate((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, al 
gorithm="maxima")
 

Output:

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) 
+ c)^2, x)
 

Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.69 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, al 
gorithm="giac")
 

Output:

1/2520*sqrt(2)*(35*B*a*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-9/4*pi 
 + 9/2*f*x + 9/2*e) + 630*(12*A*a*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) 
+ 8*B*a*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 16*A*a*c*d*sgn(cos(-1/4* 
pi + 1/2*f*x + 1/2*e)) + 14*B*a*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 
7*A*a*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*B*a*d^2*sgn(cos(-1/4*pi 
+ 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 210*(4*A*a*c^2*sgn(c 
os(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*B*a*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2 
*e)) + 12*A*a*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*B*a*c*d*sgn(cos 
(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*A*a*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e 
)) + 5*B*a*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x 
+ 3/2*e) + 126*(2*B*a*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 4*A*a*c*d* 
sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 6*B*a*c*d*sgn(cos(-1/4*pi + 1/2*f*x 
+ 1/2*e)) + 3*A*a*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*B*a*d^2*sgn( 
cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-5/4*pi + 5/2*f*x + 5/2*e) + 45*(4*B* 
a*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*A*a*d^2*sgn(cos(-1/4*pi + 1/ 
2*f*x + 1/2*e)) + 3*B*a*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-7/4* 
pi + 7/2*f*x + 7/2*e))*sqrt(a)/f
 

Mupad [F(-1)]

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

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^2 
,x)
 

Output:

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^2 
, x)
 

Reduce [F]

\[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\sqrt {a}\, a \left (\left (\int \sqrt {\sin \left (f x +e \right )+1}d x \right ) a \,c^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{4}d x \right ) b \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) a \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{3}d x \right ) b \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) a \,d^{2}+\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b \,c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )^{2}d x \right ) b c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a \,c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) a c d +\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )d x \right ) b \,c^{2}\right ) \] Input:

int((a+a*sin(f*x+e))^(3/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x)
 

Output:

sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1),x)*a*c**2 + int(sqrt(sin(e + f*x) + 
1)*sin(e + f*x)**4,x)*b*d**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3, 
x)*a*d**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**3,x)*b*c*d + int(sq 
rt(sin(e + f*x) + 1)*sin(e + f*x)**3,x)*b*d**2 + 2*int(sqrt(sin(e + f*x) + 
 1)*sin(e + f*x)**2,x)*a*c*d + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2, 
x)*a*d**2 + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*b*c**2 + 2*int(s 
qrt(sin(e + f*x) + 1)*sin(e + f*x)**2,x)*b*c*d + int(sqrt(sin(e + f*x) + 1 
)*sin(e + f*x),x)*a*c**2 + 2*int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*a* 
c*d + int(sqrt(sin(e + f*x) + 1)*sin(e + f*x),x)*b*c**2)