\(\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx\) [350]

Optimal result

Integrand size = 23, antiderivative size = 75 \[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\frac {d \sqrt {d \cos (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m) \sqrt [4]{\cos ^2(a+b x)}} \]

[Out]

d*hypergeom([-1/4, 1/2+1/2*m],[3/2+1/2*m],sin(b*x+a)^2)*(c*sin(b*x+a))^(1+m)*(d*cos(b*x+a))^(1/2)/b/c/(1+m)/(c
os(b*x+a)^2)^(1/4)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2657} \[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c (m+1) \sqrt [4]{\cos ^2(a+b x)}} \]

[In]

Int[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^m,x]

[Out]

(d*Sqrt[d*Cos[a + b*x]]*Hypergeometric2F1[-1/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1 + m)
)/(b*c*(1 + m)*(Cos[a + b*x]^2)^(1/4))

Rule 2657

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b^(2*IntPart[
(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^
2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {d \sqrt {d \cos (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m) \sqrt [4]{\cos ^2(a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04 \[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\frac {d^2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^m \tan (a+b x)}{b (1+m) \sqrt {d \cos (a+b x)}} \]

[In]

Integrate[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^m,x]

[Out]

(d^2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*T
an[a + b*x])/(b*(1 + m)*Sqrt[d*Cos[a + b*x]])

Maple [F]

\[\int \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \left (c \sin \left (b x +a \right )\right )^{m}d x\]

[In]

int((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x)

[Out]

int((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x)

Fricas [F]

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

[In]

integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*(c*sin(b*x + a))^m*d*cos(b*x + a), x)

Sympy [F]

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

[In]

integrate((d*cos(b*x+a))**(3/2)*(c*sin(b*x+a))**m,x)

[Out]

Integral((c*sin(a + b*x))**m*(d*cos(a + b*x))**(3/2), x)

Maxima [F]

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

[In]

integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(3/2)*(c*sin(b*x + a))^m, x)

Giac [F]

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

[In]

integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^(3/2)*(c*sin(b*x + a))^m, x)

Mupad [F(-1)]

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

[In]

int((d*cos(a + b*x))^(3/2)*(c*sin(a + b*x))^m,x)

[Out]

int((d*cos(a + b*x))^(3/2)*(c*sin(a + b*x))^m, x)