\(\int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} (A+C \sin ^2(e+f x)) \, dx\) [11]

Optimal result

Integrand size = 39, antiderivative size = 385 \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=-\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {\sqrt {2} (c-d) (2 c (C+2 C m)+d (C (5-2 m)+A (7+2 m))) \operatorname {AppellF1}\left (\frac {1}{2}+m,\frac {1}{2},-\frac {3}{2},\frac {3}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c+d \sin (e+f x)}}{d f (1+2 m) (7+2 m) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}+\frac {2 \sqrt {2} C (c-d) (d m-c (1+m)) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},-\frac {3}{2},\frac {5}{2}+m,\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) \cos (e+f x) (a+a \sin (e+f x))^{1+m} \sqrt {c+d \sin (e+f x)}}{a d f (3+2 m) (7+2 m) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}} \]

[Out]

-2*C*cos(f*x+e)*(a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(5/2)/d/f/(7+2*m)+(c-d)*(2*c*(2*C*m+C)+d*(C*(5-2*m)+A*(7+2
*m)))*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))^(1/2)/d/f/(1+2*m)/(7+2*m)/(1-sin(f*x+e))^(1/2)/((c+d*sin(f*x+e))/(c-d))^(1/2)+2*C*(c-
d)*(d*m-c*(1+m))*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))^(1/2)/a/d/f/(3+2*m)/(7+2*m)/(1-sin(f*x+e))^(1/2)/((c+d*sin(f*x+e))/(c-
d))^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3125, 3066, 2867, 145, 144, 143} \[ \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {\sqrt {2} (c-d) \cos (e+f x) (A d (2 m+7)+2 c (2 C m+C)+C d (5-2 m)) (a \sin (e+f x)+a)^m \sqrt {c+d \sin (e+f x)} \operatorname {AppellF1}\left (m+\frac {1}{2},\frac {1}{2},-\frac {3}{2},m+\frac {3}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{d f (2 m+1) (2 m+7) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}+\frac {2 \sqrt {2} C (c-d) (d m-c (m+1)) \cos (e+f x) (a \sin (e+f x)+a)^{m+1} \sqrt {c+d \sin (e+f x)} \operatorname {AppellF1}\left (m+\frac {3}{2},\frac {1}{2},-\frac {3}{2},m+\frac {5}{2},\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a d f (2 m+3) (2 m+7) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}-\frac {2 C \cos (e+f x) (a \sin (e+f x)+a)^m (c+d \sin (e+f x))^{5/2}}{d f (2 m+7)} \]

[In]

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

[Out]

(-2*C*Cos[e + f*x]*(a + a*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2))/(d*f*(7 + 2*m)) + (Sqrt[2]*(c - d)*(C*d*
(5 - 2*m) + A*d*(7 + 2*m) + 2*c*(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]*(a + a*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]])/(d*f*(1 + 2*m)*(7
+ 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c + d*Sin[e + f*x])/(c - d)]) + (2*Sqrt[2]*C*(c - d)*(d*m - c*(1 + m))*App
ellF1[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]*(a +
a*Sin[e + f*x])^(1 + m)*Sqrt[c + d*Sin[e + f*x]])/(a*d*f*(3 + 2*m)*(7 + 2*m)*Sqrt[1 - Sin[e + f*x]]*Sqrt[(c +
d*Sin[e + f*x])/(c - d)])

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(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] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((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*
(b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 145

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^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[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rule 2867

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dis
t[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]

Rule 3066

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A*b - a*B)/b, Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n, x
], x] + Dist[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]

Rule 3125

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Si
mp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^
(-1)] && NeQ[m + n + 2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {2 \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \left (\frac {1}{2} a \left (2 A d \left (\frac {7}{2}+m\right )+2 C \left (\frac {5 d}{2}+c m\right )\right )+a C (d m-c (1+m)) \sin (e+f x)\right ) \, dx}{a d (7+2 m)} \\ & = -\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {(2 C (d m-c (1+m))) \int (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^{3/2} \, dx}{a d (7+2 m)}+\frac {(C d (5-2 m)+A d (7+2 m)+2 c (C+2 C m)) \int (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{3/2} \, dx}{d (7+2 m)} \\ & = -\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {(2 a C (d m-c (1+m)) \cos (e+f x)) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^{3/2}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (7+2 m) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (C d (5-2 m)+A d (7+2 m)+2 c (C+2 C m)) \cos (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^{3/2}}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (7+2 m) \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {\left (\sqrt {2} a C (d m-c (1+m)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} (c+d x)^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (C d (5-2 m)+A d (7+2 m)+2 c (C+2 C m)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} (c+d x)^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}} \\ & = -\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {\left (\sqrt {2} C (a c-a d) (d m-c (1+m)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} \sqrt {c+d \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)} \sqrt {\frac {a (c+d \sin (e+f x))}{a c-a d}}}+\frac {\left (a (a c-a d) (C d (5-2 m)+A d (7+2 m)+2 c (C+2 C m)) \cos (e+f x) \sqrt {\frac {a-a \sin (e+f x)}{a}} \sqrt {c+d \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+a x)^{-\frac {1}{2}+m} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^{3/2}}{\sqrt {\frac {1}{2}-\frac {x}{2}}} \, dx,x,\sin (e+f x)\right )}{\sqrt {2} d f (7+2 m) (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)} \sqrt {\frac {a (c+d \sin (e+f x))}{a c-a d}}} \\ & = -\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^{5/2}}{d f (7+2 m)}+\frac {\sqrt {2} (c-d) (C d (5-2 m)+A d (7+2 m)+2 c (C+2 C m)) \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 {c+d \sin (e+f x)}}{d f (1+2 m) (7+2 m) \sqrt {1-\sin (e+f x)} \sqrt {\frac {c+d \sin (e+f x)}{c-d}}}+\frac {2 \sqrt {2} C (c-d) (d m-c (1+m)) \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) \sqrt {1-\sin (e+f x)} (a+a \sin (e+f x))^{1+m} \sqrt {c+d \sin (e+f x)}}{d f (3+2 m) (7+2 m) (a-a \sin (e+f x)) \sqrt {\frac {c+d \sin (e+f x)}{c-d}}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(-(C*c*cos(f*x + e)^2 - (A + C)*c + (C*d*cos(f*x + e)^2 - (A + C)*d)*sin(f*x + e))*sqrt(d*sin(f*x + e)
+ c)*(a*sin(f*x + e) + a)^m, x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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