3.205 \(\int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=275 \[ -\frac{16 c^2 (B (5-2 m)-A (2 m+7)) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) \left (4 m^2+16 m+15\right )}-\frac{64 c^3 (B (5-2 m)-A (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}-\frac{2 c (B (5-2 m)-A (2 m+7)) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}-\frac{2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)} \]

[Out]

(-64*c^3*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*
m^2)*Sqrt[c - c*Sin[e + f*x]]) - (16*c^2*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[
c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(15 + 16*m + 4*m^2)) - (2*c*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*
Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m
*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m))

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Rubi [A]  time = 0.503458, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {2973, 2740, 2738} \[ -\frac{16 c^2 (B (5-2 m)-A (2 m+7)) \cos (e+f x) \sqrt{c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) \left (4 m^2+16 m+15\right )}-\frac{64 c^3 (B (5-2 m)-A (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt{c-c \sin (e+f x)}}-\frac{2 c (B (5-2 m)-A (2 m+7)) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}-\frac{2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-64*c^3*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(f*(5 + 2*m)*(7 + 2*m)*(3 + 8*m + 4*
m^2)*Sqrt[c - c*Sin[e + f*x]]) - (16*c^2*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*Sqrt[
c - c*Sin[e + f*x]])/(f*(7 + 2*m)*(15 + 16*m + 4*m^2)) - (2*c*(B*(5 - 2*m) - A*(7 + 2*m))*Cos[e + f*x]*(a + a*
Sin[e + f*x])^m*(c - c*Sin[e + f*x])^(3/2))/(f*(5 + 2*m)*(7 + 2*m)) - (2*B*Cos[e + f*x]*(a + a*Sin[e + f*x])^m
*(c - c*Sin[e + f*x])^(5/2))/(f*(7 + 2*m))

Rule 2973

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*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n)/(f*(
m + n + 1)), x] - Dist[(B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] &&
!LtQ[m, -2^(-1)] && NeQ[m + n + 1, 0]

Rule 2740

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Sim
p[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(m + n)), x] + Dist[(a*(2*m - 1))/(m
+ n), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E
qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] &&  !LtQ[n, -1] &&  !(IGtQ[n - 1/2, 0] && LtQ[n, m])
&&  !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])

Rule 2738

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

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx &=-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}+\frac{\left (B c \left (-\frac{5}{2}+m\right )+A c \left (\frac{7}{2}+m\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx}{c \left (\frac{7}{2}+m\right )}\\ &=-\frac{2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}-\frac{(8 c (B (5-2 m)-A (7+2 m))) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{(5+2 m) (7+2 m)}\\ &=-\frac{16 c^2 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac{2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}-\frac{\left (32 c^2 (B (5-2 m)-A (7+2 m))\right ) \int (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)} \, dx}{(3+2 m) (5+2 m) (7+2 m)}\\ &=-\frac{64 c^3 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \sqrt{c-c \sin (e+f x)}}-\frac{16 c^2 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt{c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac{2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}\\ \end{align*}

Mathematica [C]  time = 6.82886, size = 667, normalized size = 2.43 \[ \frac{(c-c \sin (e+f x))^{5/2} (a (\sin (e+f x)+1))^m \left (\frac{\left (32 A m^3+304 A m^2+1272 A m+2100 A-8 B m^3-68 B m^2-110 B m-1575 B\right ) \left (\left (\frac{1}{8}-\frac{i}{8}\right ) \sin \left (\frac{1}{2} (e+f x)\right )+\left (\frac{1}{8}+\frac{i}{8}\right ) \cos \left (\frac{1}{2} (e+f x)\right )\right )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7)}+\frac{\left (32 A m^3+304 A m^2+1272 A m+2100 A-8 B m^3-68 B m^2-110 B m-1575 B\right ) \left (\left (\frac{1}{8}+\frac{i}{8}\right ) \sin \left (\frac{1}{2} (e+f x)\right )+\left (\frac{1}{8}-\frac{i}{8}\right ) \cos \left (\frac{1}{2} (e+f x)\right )\right )}{(2 m+1) (2 m+3) (2 m+5) (2 m+7)}+\frac{\left (24 A m^2+184 A m+350 A-12 B m^2-104 B m-385 B\right ) \left (\left (\frac{1}{8}-\frac{i}{8}\right ) \cos \left (\frac{3}{2} (e+f x)\right )-\left (\frac{1}{8}+\frac{i}{8}\right ) \sin \left (\frac{3}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7)}+\frac{\left (24 A m^2+184 A m+350 A-12 B m^2-104 B m-385 B\right ) \left (\left (\frac{1}{8}+\frac{i}{8}\right ) \cos \left (\frac{3}{2} (e+f x)\right )-\left (\frac{1}{8}-\frac{i}{8}\right ) \sin \left (\frac{3}{2} (e+f x)\right )\right )}{(2 m+3) (2 m+5) (2 m+7)}+\frac{(4 A m+14 A-6 B m-35 B) \left (\left (-\frac{1}{8}+\frac{i}{8}\right ) \cos \left (\frac{5}{2} (e+f x)\right )-\left (\frac{1}{8}+\frac{i}{8}\right ) \sin \left (\frac{5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7)}+\frac{(4 A m+14 A-6 B m-35 B) \left (\left (-\frac{1}{8}-\frac{i}{8}\right ) \cos \left (\frac{5}{2} (e+f x)\right )-\left (\frac{1}{8}-\frac{i}{8}\right ) \sin \left (\frac{5}{2} (e+f x)\right )\right )}{(2 m+5) (2 m+7)}+\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) B \cos \left (\frac{7}{2} (e+f x)\right )-\left (\frac{1}{8}+\frac{i}{8}\right ) B \sin \left (\frac{7}{2} (e+f x)\right )}{2 m+7}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) B \cos \left (\frac{7}{2} (e+f x)\right )-\left (\frac{1}{8}-\frac{i}{8}\right ) B \sin \left (\frac{7}{2} (e+f x)\right )}{2 m+7}\right )}{f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(1 + Sin[e + f*x]))^m*(c - c*Sin[e + f*x])^(5/2)*(((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*
B*m^2 + 32*A*m^3 - 8*B*m^3)*((1/8 + I/8)*Cos[(e + f*x)/2] + (1/8 - I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m
)*(5 + 2*m)*(7 + 2*m)) + ((2100*A - 1575*B + 1272*A*m - 110*B*m + 304*A*m^2 - 68*B*m^2 + 32*A*m^3 - 8*B*m^3)*(
(1/8 - I/8)*Cos[(e + f*x)/2] + (1/8 + I/8)*Sin[(e + f*x)/2]))/((1 + 2*m)*(3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((35
0*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*((1/8 - I/8)*Cos[(3*(e + f*x))/2] - (1/8 + I/8)*Sin[(3*
(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((350*A - 385*B + 184*A*m - 104*B*m + 24*A*m^2 - 12*B*m^2)*(
(1/8 + I/8)*Cos[(3*(e + f*x))/2] - (1/8 - I/8)*Sin[(3*(e + f*x))/2]))/((3 + 2*m)*(5 + 2*m)*(7 + 2*m)) + ((14*A
- 35*B + 4*A*m - 6*B*m)*((-1/8 + I/8)*Cos[(5*(e + f*x))/2] - (1/8 + I/8)*Sin[(5*(e + f*x))/2]))/((5 + 2*m)*(7
+ 2*m)) + ((14*A - 35*B + 4*A*m - 6*B*m)*((-1/8 - I/8)*Cos[(5*(e + f*x))/2] - (1/8 - I/8)*Sin[(5*(e + f*x))/2
]))/((5 + 2*m)*(7 + 2*m)) + ((1/8 - I/8)*B*Cos[(7*(e + f*x))/2] - (1/8 + I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2*m
) + ((1/8 + I/8)*B*Cos[(7*(e + f*x))/2] - (1/8 - I/8)*B*Sin[(7*(e + f*x))/2])/(7 + 2*m)))/(f*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^5)

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Maple [F]  time = 0.327, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c-c\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.73941, size = 979, normalized size = 3.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(((4*m^2 + 24*m + 43)*a^m*c^(5/2) - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 2*(4
*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2*(4*m^2 + 8*m + 35)*a^m*c^(5/2)*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 - (12*m^2 + 40*m - 15)*a^m*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + (4*m^2 + 24
*m + 43)*a^m*c^(5/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)*A*e^(2*m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) -
m*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((8*m^3 + 36*m^2 + 46*m + 15)*(sin(f*x + e)^2/(cos(f*x + e) +
1)^2 + 1)^(5/2)) - 2*((4*m^2 + 40*m + 115)*a^m*c^(5/2) - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)/(
cos(f*x + e) + 1) + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - (16*m^3
+ 76*m^2 + 260*m - 175)*a^m*c^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - (16*m^3 + 76*m^2 + 260*m - 175)*a^m
*c^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 2*(12*m^3 + 76*m^2 + 97*m + 175)*a^m*c^(5/2)*sin(f*x + e)^5/(co
s(f*x + e) + 1)^5 - 2*(4*m^3 + 40*m^2 + 115*m)*a^m*c^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + (4*m^2 + 40*m
+ 115)*a^m*c^(5/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)*B*e^(2*m*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1) - m
*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1))/((16*m^4 + 128*m^3 + 344*m^2 + 352*m + (16*m^4 + 128*m^3 + 344*
m^2 + 352*m + 105)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 105)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2))
)/f

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Fricas [B]  time = 2.31827, size = 1354, normalized size = 4.92 \begin{align*} \frac{2 \,{\left ({\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{4} + 64 \,{\left (A + B\right )} c^{2} m -{\left (8 \,{\left (A - 2 \, B\right )} c^{2} m^{3} + 4 \,{\left (11 \, A - 28 \, B\right )} c^{2} m^{2} + 2 \,{\left (31 \, A - 86 \, B\right )} c^{2} m + 3 \,{\left (7 \, A - 20 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \,{\left (7 \, A - 5 \, B\right )} c^{2} +{\left (8 \,{\left (A - B\right )} c^{2} m^{3} + 4 \,{\left (19 \, A - 11 \, B\right )} c^{2} m^{2} + 190 \,{\left (A - B\right )} c^{2} m +{\left (77 \, A - 85 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (8 \,{\left (A - B\right )} c^{2} m^{3} + 60 \,{\left (A - B\right )} c^{2} m^{2} + 2 \,{\left (79 \, A - 63 \, B\right )} c^{2} m +{\left (161 \, A - 145 \, B\right )} c^{2}\right )} \cos \left (f x + e\right ) +{\left (64 \,{\left (A + B\right )} c^{2} m -{\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \,{\left (7 \, A - 5 \, B\right )} c^{2} -{\left (8 \,{\left (A - B\right )} c^{2} m^{3} + 4 \,{\left (11 \, A - 19 \, B\right )} c^{2} m^{2} + 2 \,{\left (31 \, A - 63 \, B\right )} c^{2} m + 3 \,{\left (7 \, A - 15 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \,{\left (8 \,{\left (A - B\right )} c^{2} m^{3} + 60 \,{\left (A - B\right )} c^{2} m^{2} + 2 \,{\left (63 \, A - 79 \, B\right )} c^{2} m +{\left (49 \, A - 65 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m +{\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \cos \left (f x + e\right ) -{\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \sin \left (f x + e\right ) + 105 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((8*B*c^2*m^3 + 36*B*c^2*m^2 + 46*B*c^2*m + 15*B*c^2)*cos(f*x + e)^4 + 64*(A + B)*c^2*m - (8*(A - 2*B)*c^2*m
^3 + 4*(11*A - 28*B)*c^2*m^2 + 2*(31*A - 86*B)*c^2*m + 3*(7*A - 20*B)*c^2)*cos(f*x + e)^3 + 32*(7*A - 5*B)*c^2
+ (8*(A - B)*c^2*m^3 + 4*(19*A - 11*B)*c^2*m^2 + 190*(A - B)*c^2*m + (77*A - 85*B)*c^2)*cos(f*x + e)^2 + 2*(8
*(A - B)*c^2*m^3 + 60*(A - B)*c^2*m^2 + 2*(79*A - 63*B)*c^2*m + (161*A - 145*B)*c^2)*cos(f*x + e) + (64*(A + B
)*c^2*m - (8*B*c^2*m^3 + 36*B*c^2*m^2 + 46*B*c^2*m + 15*B*c^2)*cos(f*x + e)^3 + 32*(7*A - 5*B)*c^2 - (8*(A - B
)*c^2*m^3 + 4*(11*A - 19*B)*c^2*m^2 + 2*(31*A - 63*B)*c^2*m + 3*(7*A - 15*B)*c^2)*cos(f*x + e)^2 - 2*(8*(A - B
)*c^2*m^3 + 60*(A - B)*c^2*m^2 + 2*(63*A - 79*B)*c^2*m + (49*A - 65*B)*c^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-
c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + (16*f*m^4 + 128*f*m^3
+ 344*f*m^2 + 352*f*m + 105*f)*cos(f*x + e) - (16*f*m^4 + 128*f*m^3 + 344*f*m^2 + 352*f*m + 105*f)*sin(f*x +
e) + 105*f)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2