Integrand size = 47, antiderivative size = 451 \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 \left (c^2 C-B c d+A d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {\sqrt {2} \left (d^2 (A-3 B+3 C-4 A m)+c d (3 A-B+3 C+4 B m)-2 c^2 (C+2 C m)\right ) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},\frac {3}{2},\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{3 (c-d)^2 d (c+d) f (1+2 m) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {2} \left (B c d (1-2 m)+2 c^2 C (1+m)-d^2 (A+3 C-2 A m)\right ) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},\frac {3}{2},\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}{3 a (c-d)^2 d (c+d) f (3+2 m) \sqrt {1-\sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
2/3*(A*d^2-B*c*d+C*c^2)*cos(f*x+e)*(a+a*sin(f*x+e))^m/d/(c^2-d^2)/f/(c+d*s in(f*x+e))^(3/2)+1/3*(d^2*(-4*A*m+A-3*B+3*C)+c*d*(4*B*m+3*A-B+3*C)-2*c^2*( 2*C*m+C))*AppellF1(1/2+m,3/2,1/2,3/2+m,-d*(1+sin(f*x+e))/(c-d),1/2+1/2*sin (f*x+e))*cos(f*x+e)*(a+a*sin(f*x+e))^m*2^(1/2)*((c+d*sin(f*x+e))/(c-d))^(1 /2)/(c-d)^2/d/(c+d)/f/(1+2*m)/(1-sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)+ 1/3*(B*c*d*(1-2*m)+2*c^2*C*(1+m)-d^2*(-2*A*m+A+3*C))*AppellF1(3/2+m,3/2,1/ 2,5/2+m,-d*(1+sin(f*x+e))/(c-d),1/2+1/2*sin(f*x+e))*cos(f*x+e)*(a+a*sin(f* x+e))^(1+m)*2^(1/2)*((c+d*sin(f*x+e))/(c-d))^(1/2)/a/(c-d)^2/d/(c+d)/f/(3+ 2*m)/(1-sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2)
\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx \]
Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2)) /(c + d*Sin[e + f*x])^(5/2),x]
Integrate[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2)) /(c + d*Sin[e + f*x])^(5/2), x]
Time = 1.41 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.234, Rules used = {3042, 3522, 27, 3042, 3466, 3042, 3267, 157, 27, 156, 155}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^m \left (A+B \sin (e+f x)+C \sin (e+f x)^2\right )}{(c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {2 \int -\frac {(\sin (e+f x) a+a)^m \left (a ((c C-B d) (3 d-2 c m)+A d (3 c-2 d m))+a \left (2 C (m+1) c^2+B d (1-2 m) c-d^2 (-2 m A+A+3 C)\right ) \sin (e+f x)\right )}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 a d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m \left (a ((c C-B d) (3 d-2 c m)+A d (3 c-2 d m))+a \left (2 C (m+1) c^2+B d (1-2 m) c-d^2 (-2 m A+A+3 C)\right ) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{3/2}}dx}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^m \left (a ((c C-B d) (3 d-2 c m)+A d (3 c-2 d m))+a \left (2 C (m+1) c^2+B d (1-2 m) c-d^2 (-2 m A+A+3 C)\right ) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{3/2}}dx}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3466 |
| \(\displaystyle \frac {a \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \int \frac {(\sin (e+f x) a+a)^m}{(c+d \sin (e+f x))^{3/2}}dx+\left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \int \frac {(\sin (e+f x) a+a)^{m+1}}{(c+d \sin (e+f x))^{3/2}}dx}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \int \frac {(\sin (e+f x) a+a)^m}{(c+d \sin (e+f x))^{3/2}}dx+\left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \int \frac {(\sin (e+f x) a+a)^{m+1}}{(c+d \sin (e+f x))^{3/2}}dx}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3267 |
| \(\displaystyle \frac {\frac {a^3 \cos (e+f x) \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}}}{\sqrt {a-a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}+\frac {a^2 \cos (e+f x) \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}}}{\sqrt {a-a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}d\sin (e+f x)}{f \sqrt {a-a \sin (e+f x)} \sqrt {a \sin (e+f x)+a}}}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 157 |
| \(\displaystyle \frac {\frac {a^3 \sqrt {1-\sin (e+f x)} \cos (e+f x) \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \int \frac {\sqrt {2} (\sin (e+f x) a+a)^{m-\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} (c+d \sin (e+f x))^{3/2}}d\sin (e+f x)}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}+\frac {a^2 \sqrt {1-\sin (e+f x)} \cos (e+f x) \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \int \frac {\sqrt {2} (\sin (e+f x) a+a)^{m+\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} (c+d \sin (e+f x))^{3/2}}d\sin (e+f x)}{\sqrt {2} f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a^3 \sqrt {1-\sin (e+f x)} \cos (e+f x) \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} (c+d \sin (e+f x))^{3/2}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}+\frac {a^2 \sqrt {1-\sin (e+f x)} \cos (e+f x) \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} (c+d \sin (e+f x))^{3/2}}d\sin (e+f x)}{f (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a}}}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 156 |
| \(\displaystyle \frac {\frac {a^3 \sqrt {1-\sin (e+f x)} \cos (e+f x) \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c-d}} \int \frac {(\sin (e+f x) a+a)^{m-\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} \left (\frac {c}{c-d}+\frac {d \sin (e+f x)}{c-d}\right )^{3/2}}d\sin (e+f x)}{f (c-d) (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}+\frac {a^2 \sqrt {1-\sin (e+f x)} \cos (e+f x) \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c-d}} \int \frac {(\sin (e+f x) a+a)^{m+\frac {1}{2}}}{\sqrt {1-\sin (e+f x)} \left (\frac {c}{c-d}+\frac {d \sin (e+f x)}{c-d}\right )^{3/2}}d\sin (e+f x)}{f (c-d) (a-a \sin (e+f x)) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 155 |
| \(\displaystyle \frac {\frac {\sqrt {2} a^2 \sqrt {1-\sin (e+f x)} \cos (e+f x) (a \sin (e+f x)+a)^m \left (c d (3 A+4 B m-B+3 C)+d^2 (-4 A m+A-3 B+3 C)-2 c^2 (2 C m+C)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c-d}} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},\frac {3}{2},m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{f (2 m+1) (c-d) (a-a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\sqrt {2} a \sqrt {1-\sin (e+f x)} \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \left (-d^2 (-2 A m+A+3 C)+B c d (1-2 m)+2 c^2 C (m+1)\right ) \sqrt {\frac {c+d \sin (e+f x)}{c-d}} \operatorname {AppellF1}\left (m+\frac {3}{2},\frac {1}{2},\frac {3}{2},m+\frac {5}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{f (2 m+3) (c-d) (a-a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}}{3 a d \left (c^2-d^2\right )}+\frac {2 \cos (e+f x) \left (A d^2-B c d+c^2 C\right ) (a \sin (e+f x)+a)^m}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
Int[((a + a*Sin[e + f*x])^m*(A + B*Sin[e + f*x] + C*Sin[e + f*x]^2))/(c + d*Sin[e + f*x])^(5/2),x]
(2*(c^2*C - B*c*d + A*d^2)*Cos[e + f*x]*(a + a*Sin[e + f*x])^m)/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((Sqrt[2]*a^2*(d^2*(A - 3*B + 3*C - 4*A*m) + c*d*(3*A - B + 3*C + 4*B*m) - 2*c^2*(C + 2*C*m))*AppellF1[1/2 + m, 1/2, 3/2, 3/2 + m, (1 + Sin[e + f*x])/2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*Sqrt[1 - Sin[e + f*x]]*(a + a*Sin[e + f*x])^m*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/((c - d)*f*(1 + 2*m)*(a - a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]) + (Sqrt[2]*a*(B*c*d*(1 - 2*m) + 2*c^2*C*(1 + m) - d^2* (A + 3*C - 2*A*m))*AppellF1[3/2 + m, 1/2, 3/2, 5/2 + m, (1 + Sin[e + f*x]) /2, -((d*(1 + Sin[e + f*x]))/(c - d))]*Cos[e + f*x]*Sqrt[1 - Sin[e + f*x]] *(a + a*Sin[e + f*x])^(1 + m)*Sqrt[(c + d*Sin[e + f*x])/(c - d)])/((c - d) *f*(3 + 2*m)*(a - a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]))/(3*a*d*(c^2 - d^2))
3.1.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n ]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & !GtQ[Simplify[b/(b*c - a*d)], 0] && !SimplerQ[c + d*x, a + b*x] && !Si mplerQ[e + f*x, a + b*x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^2*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])) Subst[Int[(a + b*x)^(m - 1/2)*((c + d* x)^n/Sqrt[a - b*x]), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m , n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !IntegerQ[m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(A*b - a*B)/b Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x], x ] + Simp[B/b 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, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ [a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A*b + a*B, 0]
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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
\[\int \frac {\left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )}{\left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c+d*sin(f*x+ e))^(5/2),x, algorithm="fricas")
integral((C*cos(f*x + e)^2 - B*sin(f*x + e) - A - C)*sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3* cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c+d*sin(f*x+ e))^(5/2),x, algorithm="maxima")
integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m/( d*sin(f*x + e) + c)^(5/2), x)
\[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
integrate((a+a*sin(f*x+e))^m*(A+B*sin(f*x+e)+C*sin(f*x+e)^2)/(c+d*sin(f*x+ e))^(5/2),x, algorithm="giac")
integrate((C*sin(f*x + e)^2 + B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^m/( d*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int \frac {(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (C\,{\sin \left (e+f\,x\right )}^2+B\,\sin \left (e+f\,x\right )+A\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
int(((a + a*sin(e + f*x))^m*(A + B*sin(e + f*x) + C*sin(e + f*x)^2))/(c + d*sin(e + f*x))^(5/2),x)