\(\int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx\) [777]

Optimal result

Integrand size = 29, antiderivative size = 1237 \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {3+b} (c-d) \sqrt {c+d} \left (9720 b c d^3-3645 d^4+18 b^2 d^2 \left (1877 c^2+846 d^2\right )+24 b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{1920 b^2 (b c-3 d) d^2 f}-\frac {\sqrt {c+d} (b c+3 d) \left (756 b c d^3-243 d^4+84 b^3 c d \left (c^2-20 d^2\right )-18 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{128 b^3 \sqrt {3+b} d^3 f}-\frac {\left (9720 b c d^3-3645 d^4+18 b^2 d^2 \left (1877 c^2+846 d^2\right )+24 b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {3+b \sin (e+f x)}}-\frac {\left (8253 b c d^2+405 d^3+3 b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}+\frac {(3+b)^{3/2} \left (3645 d^4-810 b d^3 (11 c+3 d)+270 b^2 d^2 \left (64 c^2+23 c d+22 d^2\right )+6 b^3 d \left (165 c^3+917 c^2 d+2392 c d^2+516 d^3\right )-b^4 \left (45 c^4-30 c^3 d-1692 c^2 d^2-1544 c d^3-1024 d^4\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt {c+d} f}-\frac {\left (330 b c d+837 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-21 d) \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {3+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f} \]

[Out]

-1/128*(a*d+b*c)*(28*a^3*b*c*d^3-3*a^4*d^4+28*a*b^3*c*d*(c^2-20*d^2)-2*a^2*b^2*d^2*(89*c^2+20*d^2)-b^4*(3*c^4+
40*c^2*d^2+240*d^4))*EllipticPi((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),b*(c+d)/
(a+b)/d,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(
c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^3/d^3/f/(a+b)^(1/2)+1/
1920*(c-d)*(360*a^3*b*c*d^3-45*a^4*d^4+2*a^2*b^2*d^2*(1877*c^2+846*d^2)+8*a*b^3*d*(45*c^3+791*c*d^2)-b^4*(45*c
^4-1692*c^2*d^2-1024*d^4))*EllipticE((a+b)^(1/2)*(c+d*sin(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),((a
-b)*(c+d)/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(a+b)^(1/2)*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/
(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)))^(1/2)/b^2/d^2/(-a*d+b*c)/f-1/
240*(110*a*b*c*d+93*a^2*d^2-b^2*(15*c^2-64*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)*(a+b*sin(f*x+e))^(1/2)/d/f+
3/40*b*(-7*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)*(a+b*sin(f*x+e))^(1/2)/d/f-1/5*b^2*cos(f*x+e)*(c+d*sin(f
*x+e))^(7/2)*(a+b*sin(f*x+e))^(1/2)/d/f+1/1920*(a+b)^(3/2)*(45*a^4*d^4-30*a^3*b*d^3*(11*c+3*d)+30*a^2*b^2*d^2*
(64*c^2+23*c*d+22*d^2)+2*a*b^3*d*(165*c^3+917*c^2*d+2392*c*d^2+516*d^3)-b^4*(45*c^4-30*c^3*d-1692*c^2*d^2-1544
*c*d^3-1024*d^4))*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)
/(a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e))*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-
a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)/b^3/d^2/f/(c+d)^(1/2)-1/1920*(360*a^3*b*c*d^3-45*a^4*d^4
+2*a^2*b^2*d^2*(1877*c^2+846*d^2)+8*a*b^3*d*(45*c^3+791*c*d^2)-b^4*(45*c^4-1692*c^2*d^2-1024*d^4))*cos(f*x+e)*
(c+d*sin(f*x+e))^(1/2)/b/d^2/f/(a+b*sin(f*x+e))^(1/2)-1/960*(917*a^2*b*c*d^2+15*a^3*d^3+a*b^2*d*(345*c^2+772*d
^2)-b^3*(45*c^3-516*c*d^2))*cos(f*x+e)*(a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2)/b/d/f

Rubi [A] (verified)

Time = 5.12 (sec) , antiderivative size = 1295, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2872, 3128, 3140, 3132, 2890, 3077, 2897, 3075} \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {\left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {(a+b)^{3/2} \left (-\left (\left (45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right ) b^4\right )+2 a d \left (165 c^3+917 d c^2+2392 d^2 c+516 d^3\right ) b^3+30 a^2 d^2 \left (64 c^2+23 d c+22 d^2\right ) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt {c+d} f}-\frac {\left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a c d \left (45 c^2+791 d^2\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{1920 b^2 d^2 (b c-a d) f}-\frac {\sqrt {c+d} (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt {a+b} d^3 f} \]

[In]

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

[Out]

(Sqrt[a + b]*(c - d)*Sqrt[c + d]*(360*a^3*b*c*d^3 - 45*a^4*d^4 + 8*a*b^3*c*d*(45*c^2 + 791*d^2) + 2*a^2*b^2*d^
2*(1877*c^2 + 846*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[
e + f*x]])/(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-((
(b*c - a*d)*(1 - Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d
)*(a + b*Sin[e + f*x]))]*(a + b*Sin[e + f*x]))/(1920*b^2*d^2*(b*c - a*d)*f) - (Sqrt[c + d]*(b*c + a*d)*(28*a^3
*b*c*d^3 - 3*a^4*d^4 + 28*a*b^3*c*d*(c^2 - 20*d^2) - 2*a^2*b^2*d^2*(89*c^2 + 20*d^2) - b^4*(3*c^4 + 40*c^2*d^2
+ 240*d^4))*EllipticPi[(b*(c + d))/((a + b)*d), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])/(Sqrt[c + d]*Sq
rt[a + b*Sin[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sec[e + f*x]*Sqrt[-(((b*c - a*d)*(1 - Sin[e + f
*x]))/((c + d)*(a + b*Sin[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sin[e + f*x]))/((c - d)*(a + b*Sin[e + f*x]))]*(
a + b*Sin[e + f*x]))/(128*b^3*Sqrt[a + b]*d^3*f) - ((360*a^3*b*c*d^3 - 45*a^4*d^4 + 8*a*b^3*c*d*(45*c^2 + 791*
d^2) + 2*a^2*b^2*d^2*(1877*c^2 + 846*d^2) - b^4*(45*c^4 - 1692*c^2*d^2 - 1024*d^4))*Cos[e + f*x]*Sqrt[c + d*Si
n[e + f*x]])/(1920*b*d^2*f*Sqrt[a + b*Sin[e + f*x]]) - ((917*a^2*b*c*d^2 + 15*a^3*d^3 + a*b^2*d*(345*c^2 + 772
*d^2) - b^3*(45*c^3 - 516*c*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])/(960*b*d*f)
+ ((a + b)^(3/2)*(45*a^4*d^4 - 30*a^3*b*d^3*(11*c + 3*d) + 30*a^2*b^2*d^2*(64*c^2 + 23*c*d + 22*d^2) + 2*a*b^3
*d*(165*c^3 + 917*c^2*d + 2392*c*d^2 + 516*d^3) - b^4*(45*c^4 - 30*c^3*d - 1692*c^2*d^2 - 1544*c*d^3 - 1024*d^
4))*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*
(c - d))/((a - b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + d*Sin[e + f*x]))]
*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/(1920*b^3*d^2*
Sqrt[c + d]*f) - ((110*a*b*c*d + 93*a^2*d^2 - b^2*(15*c^2 - 64*d^2))*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c
+ d*Sin[e + f*x])^(3/2))/(240*d*f) + (3*b*(b*c - 7*a*d)*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f
*x])^(5/2))/(40*d*f) - (b^2*Cos[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(7/2))/(5*d*f)

Rule 2872

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/
(d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a
*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n
- 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m]
|| (EqQ[a, 0] && NeQ[c, 0])))

Rule 2890

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

Rule 2897

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

Rule 3075

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

Rule 3077

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

Rule 3128

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

Rule 3132

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

Rule 3140

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

Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {(c+d \sin (e+f x))^{5/2} \left (\frac {1}{2} \left (b^3 c+10 a^3 d+7 a b^2 d\right )-b \left (a b c-15 a^2 d-4 b^2 d\right ) \sin (e+f x)-\frac {3}{2} b^2 (b c-7 a d) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{5 d} \\ & = \frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (\frac {1}{4} b \left (5 b^3 c^2+80 a^3 c d+62 a b^2 c d+105 a^2 b d^2\right )+\frac {1}{2} b \left (99 a^2 b c d+27 b^3 c d+40 a^3 d^2-a b^2 \left (5 c^2-91 d^2\right )\right ) \sin (e+f x)+\frac {1}{4} b^2 \left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{20 b d} \\ & = -\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{8} b^2 \left (1053 a^2 b c d^2+3 a^3 d \left (160 c^2+93 d^2\right )+a b^2 d \left (437 c^2+192 d^2\right )+b^3 \left (15 c^3+64 c d^2\right )\right )+\frac {1}{4} b^2 \left (387 a^3 c d^2-3 a b^2 c \left (5 c^2-296 d^2\right )+b^3 d \left (147 c^2+128 d^2\right )+a^2 b d \left (484 c^2+501 d^2\right )\right ) \sin (e+f x)+\frac {1}{8} b^2 \left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{60 b^2 d} \\ & = -\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {\frac {1}{16} b^2 \left (15 a^4 d^4+128 a^3 b c d \left (15 c^2+16 d^2\right )+8 a b^3 c d \left (256 c^2+257 d^2\right )+2 a^2 b^2 d^2 \left (2737 c^2+386 d^2\right )+b^4 \left (15 c^4+772 c^2 d^2\right )\right )-\frac {1}{8} b^2 (b c+a d) \left (15 a b^2 c^3-1606 a^2 b c^2 d-573 b^3 c^2 d+15 a^3 c d^2-3682 a b^2 c d^2-573 a^2 b d^3-1156 b^3 d^3\right ) \sin (e+f x)+\frac {1}{16} b^2 \left (360 a^3 b c d^3-45 a^4 d^4+8 a b^3 c d \left (45 c^2+791 d^2\right )+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{120 b^3 d} \\ & = -\frac {\left (360 a^3 b c d^3-45 a^4 d^4+8 a b^3 c d \left (45 c^2+791 d^2\right )+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {-\frac {1}{16} b^2 \left (15 a^5 d^5-54 a^2 b^3 c d^2 \left (13 c^2+162 d^2\right )-2 a^3 b^2 d^3 \left (7171 c^2+1618 d^2\right )-a^4 b c d^2 \left (3840 c^2+4501 d^2\right )+a b^4 d \left (375 c^4+3092 c^2 d^2-1024 d^4\right )-b^5 \left (45 c^5-1692 c^3 d^2-1024 c d^4\right )\right )+\frac {1}{8} b^2 \left (15 a^5 c d^4+b^5 c^2 d \left (15 c^2+772 d^2\right )+a^4 b d^3 \left (2822 c^2+1161 d^2\right )+2 a^3 b^2 c d^2 \left (674 c^2+4433 d^2\right )-6 a^2 b^3 d \left (65 c^4-1276 c^2 d^2-514 d^4\right )+a b^4 c \left (45 c^4+1502 c^2 d^2+3344 d^4\right )\right ) \sin (e+f x)+\frac {15}{16} b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{240 b^3 d^2} \\ & = -\frac {\left (360 a^3 b c d^3-45 a^4 d^4+8 a b^3 c d \left (45 c^2+791 d^2\right )+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {-\frac {15}{16} a^2 b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )-\frac {1}{16} b^4 \left (15 a^5 d^5-54 a^2 b^3 c d^2 \left (13 c^2+162 d^2\right )-2 a^3 b^2 d^3 \left (7171 c^2+1618 d^2\right )-a^4 b c d^2 \left (3840 c^2+4501 d^2\right )+a b^4 d \left (375 c^4+3092 c^2 d^2-1024 d^4\right )-b^5 \left (45 c^5-1692 c^3 d^2-1024 c d^4\right )\right )+b \left (-\frac {15}{8} a b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )+\frac {1}{8} b^3 \left (15 a^5 c d^4+b^5 c^2 d \left (15 c^2+772 d^2\right )+a^4 b d^3 \left (2822 c^2+1161 d^2\right )+2 a^3 b^2 c d^2 \left (674 c^2+4433 d^2\right )-6 a^2 b^3 d \left (65 c^4-1276 c^2 d^2-514 d^4\right )+a b^4 c \left (45 c^4+1502 c^2 d^2+3344 d^4\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{240 b^5 d^2}-\frac {\left ((b c+a d) \left (28 a^3 b c d^3-3 a^4 d^4+28 a b^3 c d \left (c^2-20 d^2\right )-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{256 b^3 d^2} \\ & = -\frac {\sqrt {c+d} (b c+a d) \left (28 a^3 b c d^3-3 a^4 d^4+28 a b^3 c d \left (c^2-20 d^2\right )-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt {a+b} d^3 f}-\frac {\left (360 a^3 b c d^3-45 a^4 d^4+8 a b^3 c d \left (45 c^2+791 d^2\right )+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}-\frac {\left ((a+b) \left (45 a^5 d^5-15 a^4 b d^4 (25 c+6 d)+30 a^3 b^2 d^3 \left (75 c^2+26 c d+22 d^2\right )-2 a^2 b^3 d^2 \left (795 c^3-572 c^2 d-2062 c d^2-516 d^3\right )+b^5 c \left (45 c^4-30 c^3 d-1692 c^2 d^2-1544 c d^3-1024 d^4\right )-a b^4 d \left (375 c^4+1804 c^3 d+3092 c^2 d^2-512 c d^3-1024 d^4\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{3840 b^3 d^2}+-\frac {\left (-a b \left (-\frac {15}{8} a b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )+\frac {1}{8} b^3 \left (15 a^5 c d^4+b^5 c^2 d \left (15 c^2+772 d^2\right )+a^4 b d^3 \left (2822 c^2+1161 d^2\right )+2 a^3 b^2 c d^2 \left (674 c^2+4433 d^2\right )-6 a^2 b^3 d \left (65 c^4-1276 c^2 d^2-514 d^4\right )+a b^4 c \left (45 c^4+1502 c^2 d^2+3344 d^4\right )\right )\right )+b \left (-\frac {15}{16} a^2 b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )-\frac {1}{16} b^4 \left (15 a^5 d^5-54 a^2 b^3 c d^2 \left (13 c^2+162 d^2\right )-2 a^3 b^2 d^3 \left (7171 c^2+1618 d^2\right )-a^4 b c d^2 \left (3840 c^2+4501 d^2\right )+a b^4 d \left (375 c^4+3092 c^2 d^2-1024 d^4\right )-b^5 \left (45 c^5-1692 c^3 d^2-1024 c d^4\right )\right )\right )\right ) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{240 (a-b) b^5 d^2} \\ & = \frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (360 a^3 b c d^3-45 a^4 d^4+8 a b^3 c d \left (45 c^2+791 d^2\right )+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{1920 b^2 d^2 (b c-a d) f}-\frac {\sqrt {c+d} (b c+a d) \left (28 a^3 b c d^3-3 a^4 d^4+28 a b^3 c d \left (c^2-20 d^2\right )-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right ) \operatorname {EllipticPi}\left (\frac {b (c+d)}{(a+b) d},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt {a+b} d^3 f}-\frac {\left (360 a^3 b c d^3-45 a^4 d^4+8 a b^3 c d \left (45 c^2+791 d^2\right )+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}+\frac {(a+b)^{3/2} \left (45 a^4 d^4-30 a^3 b d^3 (11 c+3 d)+30 a^2 b^2 d^2 \left (64 c^2+23 c d+22 d^2\right )+2 a b^3 d \left (165 c^3+917 c^2 d+2392 c d^2+516 d^3\right )-b^4 \left (45 c^4-30 c^3 d-1692 c^2 d^2-1544 c d^3-1024 d^4\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt {c+d} f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 12.50 (sec) , antiderivative size = 2195, normalized size of antiderivative = 1.77 \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Result too large to show} \]

[In]

Integrate[(3 + b*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

((-4*(-(b*c) + 3*d)*(-15*b^4*c^4 + 103680*b*c^3*d + 13368*b^3*c^3*d + 132318*b^2*c^2*d^2 + 3236*b^4*c^2*d^2 +
120312*b*c*d^3 + 31320*b^3*c*d^3 - 1215*d^4 + 29124*b^2*d^4 + 1024*b^4*d^4)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x
)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3
*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Cs
c[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c +
d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - 4*(-(b
*c) + 3*d)*(-180*b^3*c^4 + 57276*b^2*c^3*d + 2292*b^4*c^3*d + 171828*b*c^2*d^2 + 51060*b^3*c^2*d^2 - 4860*c*d^
3 + 153180*b^2*c*d^3 + 4624*b^4*c*d^3 + 61884*b*d^4 + 13872*b^3*d^4)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^
2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/
Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e
+ Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin
[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt[((c +
d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(-e
+ Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec
[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) +
3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*d*Sqrt[3 + b*S
in[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) + 2*(45*b^4*c^4 - 1080*b^3*c^3*d - 33786*b^2*c^2*d^2 - 1692*b^4*c^2*d^
2 - 9720*b*c*d^3 - 18984*b^3*c*d^3 + 3645*d^4 - 15228*b^2*d^4 - 1024*b^4*d^4)*((Cos[e + f*x]*Sqrt[c + d*Sin[e
+ f*x]])/(d*Sqrt[3 + b*Sin[e + f*x]]) + (Sqrt[(3 - b)/(3 + b)]*(3 + b)*Cos[(-e + Pi/2 - f*x)/2]*EllipticE[ArcS
in[(Sqrt[(3 - b)/(3 + b)]*Sin[(-e + Pi/2 - f*x)/2])/Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]], (2*(-(b*c) + 3*d))/((
3 - b)*(c + d))]*Sqrt[c + d*Sin[e + f*x]])/(b*d*Sqrt[((3 + b)*Cos[(-e + Pi/2 - f*x)/2]^2)/(3 + b*Sin[e + f*x])
]*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[(3 + b*Sin[e + f*x])/(3 + b)]*Sqrt[((3 + b)*(c + d*Sin[e + f*x]))/((c + d)*(3
+ b*Sin[e + f*x]))]) - (2*(-(b*c) + 3*d)*((((3 + b)*c + 3*d)*Sqrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d
)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]],
(2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 -
f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x])
)/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - ((b*c + 3*d)*Sqrt[((c
+ d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticPi[(-(b*c) + 3*d)/((3 + b)*d), ArcSin[Sqrt[((-3 - b)*Csc[(
-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*S
ec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c)
+ 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*d*Sqrt[3 + b
*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])))/(b*d)))/(3840*b*d*f) + (Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e +
f*x]]*(-1/960*((15*b^3*c^3 + 3867*b^2*c^2*d + 11601*b*c*d^2 + 898*b^3*c*d^2 + 405*d^3 + 2694*b^2*d^3)*Cos[e +
f*x])/(b*d) + (21*b*d*(b*c + 3*d)*Cos[3*(e + f*x)])/160 - ((93*b^2*c^2 + 1086*b*c*d + 837*d^2 + 88*b^2*d^2)*S
in[2*(e + f*x)])/480 + (b^2*d^2*Sin[4*(e + f*x)])/40))/f

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 40.48 (sec) , antiderivative size = 680680, normalized size of antiderivative = 550.27

method	result	size
default	\(\text {Expression too large to display}\)	\(680680\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (3+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

[In]

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

[Out]

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