\(\int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\) [301]

Optimal result

Integrand size = 37, antiderivative size = 429 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f} \]

[Out]

-2/1155*a*(11*A*d*(c^2-10*c*d+73*d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3))*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/
d/f-2/11*a*B*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3/d/f-2/3465*a^3*(15*c^2+10*c*d+7*d^2)*(11*A*d
*(c^2-10*c*d+73*d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3))*cos(f*x+e)/d^3/f/(a+a*sin(f*x+e))^(1/2)+2/693*a^3*(
11*A*(3*c-19*d)*d-B*(15*c^2-65*c*d+194*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^3/d^3/f/(a+a*sin(f*x+e))^(1/2)-4/3465
*a^2*(5*c-d)*(11*A*d*(c^2-10*c*d+73*d^2)-B*(5*c^3-40*c^2*d+169*c*d^2-710*d^3))*cos(f*x+e)*(a+a*sin(f*x+e))^(1/
2)/d^2/f+2/99*a^2*(-11*A*d+5*B*c-14*B*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3*(a+a*sin(f*x+e))^(1/2)/d^2/f

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {3055, 3060, 2840, 2830, 2725} \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3465 d^2 f}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 d f}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f} \]

[In]

Int[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]

[Out]

(-2*a^3*(15*c^2 + 10*c*d + 7*d^2)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3)
)*Cos[e + f*x])/(3465*d^3*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^2*(5*c - d)*(11*A*d*(c^2 - 10*c*d + 73*d^2) - B*(
5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d^2*f) - (2*a*(11*A*d*(c
^2 - 10*c*d + 73*d^2) - B*(5*c^3 - 40*c^2*d + 169*c*d^2 - 710*d^3))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(
1155*d*f) + (2*a^3*(11*A*(3*c - 19*d)*d - B*(15*c^2 - 65*c*d + 194*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/
(693*d^3*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(5*B*c - 11*A*d - 14*B*d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*
(c + d*Sin[e + f*x])^3)/(99*d^2*f) - (2*a*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3)/(1
1*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) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {2 \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \left (\frac {1}{2} a (11 A d+3 B (c+2 d))-\frac {1}{2} a (5 B c-11 A d-14 B d) \sin (e+f x)\right ) \, dx}{11 d} \\ & = \frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {4 \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \left (\frac {1}{4} a^2 \left (11 A d (c+15 d)-B \left (5 c^2-11 c d-138 d^2\right )\right )-\frac {1}{4} a^2 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{99 d^2} \\ & = \frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {\left (a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{231 d^3} \\ & = -\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {\left (2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\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}{1155 d^3} \\ & = -\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {\left (a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{3465 d^3} \\ & = -\frac {2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.73 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.76 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (164472 A c^2+137280 B c^2+274560 A c d+248732 B c d+124366 A d^2+114640 B d^2-8 \left (11 A \left (63 c^2+360 c d+254 d^2\right )+2 B \left (990 c^2+2794 c d+1625 d^2\right )\right ) \cos (2 (e+f x))+70 d (22 B c+11 A d+32 B d) \cos (4 (e+f x))+51744 A c^2 \sin (e+f x)+66660 B c^2 \sin (e+f x)+133320 A c d \sin (e+f x)+137104 B c d \sin (e+f x)+68552 A d^2 \sin (e+f x)+69890 B d^2 \sin (e+f x)-1980 B c^2 \sin (3 (e+f x))-3960 A c d \sin (3 (e+f x))-11440 B c d \sin (3 (e+f x))-5720 A d^2 \sin (3 (e+f x))-8675 B d^2 \sin (3 (e+f x))+315 B d^2 \sin (5 (e+f x))\right )}{27720 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[(a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x])^2,x]

[Out]

-1/27720*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(164472*A*c^2 + 137280*B*c^2 +
274560*A*c*d + 248732*B*c*d + 124366*A*d^2 + 114640*B*d^2 - 8*(11*A*(63*c^2 + 360*c*d + 254*d^2) + 2*B*(990*c^
2 + 2794*c*d + 1625*d^2))*Cos[2*(e + f*x)] + 70*d*(22*B*c + 11*A*d + 32*B*d)*Cos[4*(e + f*x)] + 51744*A*c^2*Si
n[e + f*x] + 66660*B*c^2*Sin[e + f*x] + 133320*A*c*d*Sin[e + f*x] + 137104*B*c*d*Sin[e + f*x] + 68552*A*d^2*Si
n[e + f*x] + 69890*B*d^2*Sin[e + f*x] - 1980*B*c^2*Sin[3*(e + f*x)] - 3960*A*c*d*Sin[3*(e + f*x)] - 11440*B*c*
d*Sin[3*(e + f*x)] - 5720*A*d^2*Sin[3*(e + f*x)] - 8675*B*d^2*Sin[3*(e + f*x)] + 315*B*d^2*Sin[5*(e + f*x)]))/
(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

Maple [A] (verified)

Time = 36.06 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (315 B \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right ) d^{2}+\left (385 A \,d^{2}+770 c d B +1120 d^{2} B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-990 A c d -1430 A \,d^{2}-495 B \,c^{2}-2860 c d B -2405 d^{2} B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-693 A \,c^{2}-3960 A c d -3179 A \,d^{2}-1980 B \,c^{2}-6358 c d B -4370 d^{2} B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (3234 A \,c^{2}+8580 A c d +4642 A \,d^{2}+4290 B \,c^{2}+9284 c d B +4930 d^{2} B \right ) \sin \left (f x +e \right )+10626 A \,c^{2}+19140 A c d +9218 A \,d^{2}+9570 B \,c^{2}+18436 c d B +8930 d^{2} B \right )}{3465 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(257\)
parts	\(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) d \left (d A +2 B c \right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )+130 \left (\sin ^{3}\left (f x +e \right )\right )+219 \left (\sin ^{2}\left (f x +e \right )\right )+292 \sin \left (f x +e \right )+584\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) c \left (2 d A +B c \right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )+12 \left (\sin ^{2}\left (f x +e \right )\right )+23 \sin \left (f x +e \right )+46\right )}{21 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 A \,c^{2} \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d^{2} B \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (63 \left (\sin ^{5}\left (f x +e \right )\right )+224 \left (\sin ^{4}\left (f x +e \right )\right )+355 \left (\sin ^{3}\left (f x +e \right )\right )+426 \left (\sin ^{2}\left (f x +e \right )\right )+568 \sin \left (f x +e \right )+1136\right )}{693 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(344\)

[In]

int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/3465*(1+sin(f*x+e))*a^3*(sin(f*x+e)-1)*(315*B*cos(f*x+e)^4*sin(f*x+e)*d^2+(385*A*d^2+770*B*c*d+1120*B*d^2)*c
os(f*x+e)^4+(-990*A*c*d-1430*A*d^2-495*B*c^2-2860*B*c*d-2405*B*d^2)*cos(f*x+e)^2*sin(f*x+e)+(-693*A*c^2-3960*A
*c*d-3179*A*d^2-1980*B*c^2-6358*B*c*d-4370*B*d^2)*cos(f*x+e)^2+(3234*A*c^2+8580*A*c*d+4642*A*d^2+4290*B*c^2+92
84*B*c*d+4930*B*d^2)*sin(f*x+e)+10626*A*c^2+19140*A*c*d+9218*A*d^2+9570*B*c^2+18436*c*d*B+8930*d^2*B)/cos(f*x+
e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.38 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {2 \, {\left (315 \, B a^{2} d^{2} \cos \left (f x + e\right )^{6} + 35 \, {\left (22 \, B a^{2} c d + {\left (11 \, A + 32 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{5} + 1056 \, {\left (7 \, A + 5 \, B\right )} a^{2} c^{2} + 704 \, {\left (15 \, A + 13 \, B\right )} a^{2} c d + 32 \, {\left (143 \, A + 125 \, B\right )} a^{2} d^{2} - 5 \, {\left (99 \, B a^{2} c^{2} + 22 \, {\left (9 \, A + 19 \, B\right )} a^{2} c d + {\left (209 \, A + 320 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (99 \, {\left (7 \, A + 20 \, B\right )} a^{2} c^{2} + 22 \, {\left (180 \, A + 289 \, B\right )} a^{2} c d + {\left (3179 \, A + 4370 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (33 \, {\left (77 \, A + 85 \, B\right )} a^{2} c^{2} + 22 \, {\left (255 \, A + 263 \, B\right )} a^{2} c d + {\left (2893 \, A + 2965 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (33 \, {\left (161 \, A + 145 \, B\right )} a^{2} c^{2} + 22 \, {\left (435 \, A + 419 \, B\right )} a^{2} c d + {\left (4609 \, A + 4465 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right ) + {\left (315 \, B a^{2} d^{2} \cos \left (f x + e\right )^{5} - 1056 \, {\left (7 \, A + 5 \, B\right )} a^{2} c^{2} - 704 \, {\left (15 \, A + 13 \, B\right )} a^{2} c d - 32 \, {\left (143 \, A + 125 \, B\right )} a^{2} d^{2} - 35 \, {\left (22 \, B a^{2} c d + {\left (11 \, A + 23 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} - 5 \, {\left (99 \, B a^{2} c^{2} + 22 \, {\left (9 \, A + 26 \, B\right )} a^{2} c d + 13 \, {\left (22 \, A + 37 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (33 \, {\left (7 \, A + 15 \, B\right )} a^{2} c^{2} + 22 \, {\left (45 \, A + 53 \, B\right )} a^{2} c d + {\left (583 \, A + 655 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (33 \, {\left (49 \, A + 65 \, B\right )} a^{2} c^{2} + 22 \, {\left (195 \, A + 211 \, B\right )} a^{2} c d + {\left (2321 \, A + 2465 \, B\right )} a^{2} 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}}{3465 \, {\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))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

-2/3465*(315*B*a^2*d^2*cos(f*x + e)^6 + 35*(22*B*a^2*c*d + (11*A + 32*B)*a^2*d^2)*cos(f*x + e)^5 + 1056*(7*A +
5*B)*a^2*c^2 + 704*(15*A + 13*B)*a^2*c*d + 32*(143*A + 125*B)*a^2*d^2 - 5*(99*B*a^2*c^2 + 22*(9*A + 19*B)*a^2
*c*d + (209*A + 320*B)*a^2*d^2)*cos(f*x + e)^4 - (99*(7*A + 20*B)*a^2*c^2 + 22*(180*A + 289*B)*a^2*c*d + (3179
*A + 4370*B)*a^2*d^2)*cos(f*x + e)^3 + (33*(77*A + 85*B)*a^2*c^2 + 22*(255*A + 263*B)*a^2*c*d + (2893*A + 2965
*B)*a^2*d^2)*cos(f*x + e)^2 + 2*(33*(161*A + 145*B)*a^2*c^2 + 22*(435*A + 419*B)*a^2*c*d + (4609*A + 4465*B)*a
^2*d^2)*cos(f*x + e) + (315*B*a^2*d^2*cos(f*x + e)^5 - 1056*(7*A + 5*B)*a^2*c^2 - 704*(15*A + 13*B)*a^2*c*d -
32*(143*A + 125*B)*a^2*d^2 - 35*(22*B*a^2*c*d + (11*A + 23*B)*a^2*d^2)*cos(f*x + e)^4 - 5*(99*B*a^2*c^2 + 22*(
9*A + 26*B)*a^2*c*d + 13*(22*A + 37*B)*a^2*d^2)*cos(f*x + e)^3 + 3*(33*(7*A + 15*B)*a^2*c^2 + 22*(45*A + 53*B)
*a^2*c*d + (583*A + 655*B)*a^2*d^2)*cos(f*x + e)^2 + 2*(33*(49*A + 65*B)*a^2*c^2 + 22*(195*A + 211*B)*a^2*c*d
+ (2321*A + 2465*B)*a^2*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(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2,x)

[Out]

Timed out

Maxima [F]

\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{2} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(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)^(5/2)*(d*sin(f*x + e) + c)^2, x)

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.59 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm="giac")

[Out]

1/55440*sqrt(2)*(315*B*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-11/4*pi + 11/2*f*x + 11/2*e) + 6930*(4
0*A*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 30*B*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 60*A*a^2*
c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 52*B*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 26*A*a^2*d^2*sgn(
cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 23*B*a^2*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) + 2310*(20*A*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 22*B*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e
)) + 44*A*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 40*B*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 20*
A*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 19*B*a^2*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) + 693*(8*A*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 20*B*a^2*c^2*sgn(cos(-1/4*pi + 1/2
*f*x + 1/2*e)) + 40*A*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 48*B*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1
/2*e)) + 24*A*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 25*B*a^2*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) + 495*(4*B*a^2*c^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 8*A*a^2*c*d*sgn(cos(-1
/4*pi + 1/2*f*x + 1/2*e)) + 20*B*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 10*A*a^2*d^2*sgn(cos(-1/4*pi +
1/2*f*x + 1/2*e)) + 13*B*a^2*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) + 385*(4*
B*a^2*c*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 2*A*a^2*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 5*B*a^2*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))*sqrt(a)/f

Mupad [F(-1)]

Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2,x)

[Out]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^2, x)