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

Optimal result

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

[Out]

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+2/315*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)+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)+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-2/9*a*B*cos(f*x+e)*(c+d*sin(f*x+e))^3*(a+a*sin(f*x+e))^(1/2)/d/f

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {3055, 3060, 2840, 2830, 2725} \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x)}{315 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {2 \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac {4 a (5 c-d) \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{315 d f}-\frac {2 a B \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f} \]

[In]

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

[Out]

(2*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[e + f*x])/(315*d^2*f*Sqrt[a
+ a*Sin[e + f*x]]) + (4*a*(5*c - d)*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin
[e + f*x]])/(315*d*f) + (2*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/
2))/(105*f) + (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]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^3)/(9*d*f)

Rule 2725

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] && EqQ[a^2 - b^2, 0]

Rule 2830

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] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[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 2840

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] + Dist[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 3055

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-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] + Dist[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 3060

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] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\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}+\frac {2 \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \left (\frac {1}{2} a (9 A d+B (c+6 d))-\frac {1}{2} a (3 B c-9 A d-10 B d) \sin (e+f x)\right ) \, dx}{9 d} \\ & = \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}-\frac {\left (a \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{21 d^2} \\ & = \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}-\frac {\left (2 \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{105 d^2} \\ & = \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}-\frac {\left (a \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 )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{315 d^2} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.92 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.91 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (4200 A c^2+3276 B c^2+6552 A c d+5928 B c d+2964 A d^2+2689 B d^2-4 \left (9 A d (14 c+13 d)+B \left (63 c^2+234 c d+137 d^2\right )\right ) \cos (2 (e+f x))+35 B d^2 \cos (4 (e+f x))+840 A c^2 \sin (e+f x)+1512 B c^2 \sin (e+f x)+3024 A c d \sin (e+f x)+3036 B c d \sin (e+f x)+1518 A d^2 \sin (e+f x)+1598 B d^2 \sin (e+f x)-180 B c d \sin (3 (e+f x))-90 A d^2 \sin (3 (e+f x))-170 B d^2 \sin (3 (e+f x))\right )}{1260 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

-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*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)]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.70

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

[In]

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

[Out]

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+858*c*d*B+409*d^2*B)/cos(f*x+e)/(a+
a*sin(f*x+e))^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.46 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {2 \, {\left (35 \, B a d^{2} \cos \left (f x + e\right )^{5} - 5 \, {\left (18 \, B a c d + {\left (9 \, A + 10 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{4} + 84 \, {\left (5 \, A + 3 \, B\right )} a c^{2} + 24 \, {\left (21 \, A + 19 \, B\right )} a c d + 4 \, {\left (57 \, A + 47 \, B\right )} a d^{2} - {\left (63 \, B a c^{2} + 18 \, {\left (7 \, A + 13 \, B\right )} a c d + {\left (117 \, A + 172 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (21 \, {\left (5 \, A + 6 \, B\right )} a c^{2} + 6 \, {\left (42 \, A + 43 \, B\right )} a c d + {\left (129 \, A + 134 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (21 \, {\left (25 \, A + 21 \, B\right )} a c^{2} + 6 \, {\left (147 \, A + 143 \, B\right )} a c d + {\left (429 \, A + 409 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right ) - {\left (35 \, B a d^{2} \cos \left (f x + e\right )^{4} + 84 \, {\left (5 \, A + 3 \, B\right )} a c^{2} + 24 \, {\left (21 \, A + 19 \, B\right )} a c d + 4 \, {\left (57 \, A + 47 \, B\right )} a d^{2} + 5 \, {\left (18 \, B a c d + {\left (9 \, A + 17 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (21 \, B a c^{2} + 6 \, {\left (7 \, A + 8 \, B\right )} a c d + {\left (24 \, A + 29 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (21 \, {\left (5 \, A + 9 \, B\right )} a c^{2} + 6 \, {\left (63 \, A + 67 \, B\right )} a c d + {\left (201 \, A + 221 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

[In]

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

[Out]

-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)*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 - (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]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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)

none

Time = 0.38 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.69 \[ \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*s
gn(cos(-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*p
i + 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))*sqr
t(a)/f

Mupad [F(-1)]

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

[In]

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

[Out]

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