Integrand size = 23, antiderivative size = 75 \[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\frac {d \sqrt {d \cos (a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m) \sqrt [4]{\cos ^2(a+b x)}} \] Output:
d*(d*cos(b*x+a))^(1/2)*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)/b/c/(1+m)/(cos(b*x+a)^2)^(1/4)
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04 \[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\frac {d^2 \cos ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^m \tan (a+b x)}{b (1+m) \sqrt {d \cos (a+b x)}} \] Input:
Integrate[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^m,x]
Output:
(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*Tan[a + b*x])/(b*(1 + m)*Sqrt[d*Cos[a + b*x]])
Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3042, 3057}
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.
\(\displaystyle \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^mdx\) |
\(\Big \downarrow \) 3057 |
\(\displaystyle \frac {d \sqrt {d \cos (a+b x)} (c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c (m+1) \sqrt [4]{\cos ^2(a+b x)}}\) |
Input:
Int[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^m,x]
Output:
(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))
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*Frac Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr acPart[(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]
\[\int \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}} \left (c \sin \left (b x +a \right )\right )^{m}d x\]
Input:
int((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x)
Output:
int((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x)
\[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sin \left (b x + a\right )\right )^{m} \,d x } \] Input:
integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x, algorithm="fricas")
Output:
integral(sqrt(d*cos(b*x + a))*(c*sin(b*x + a))^m*d*cos(b*x + a), x)
\[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{m} \left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((d*cos(b*x+a))**(3/2)*(c*sin(b*x+a))**m,x)
Output:
Integral((c*sin(a + b*x))**m*(d*cos(a + b*x))**(3/2), x)
\[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sin \left (b x + a\right )\right )^{m} \,d x } \] Input:
integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x, algorithm="maxima")
Output:
integrate((d*cos(b*x + a))^(3/2)*(c*sin(b*x + a))^m, x)
\[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sin \left (b x + a\right )\right )^{m} \,d x } \] Input:
integrate((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x, algorithm="giac")
Output:
integrate((d*cos(b*x + a))^(3/2)*(c*sin(b*x + a))^m, x)
Timed out. \[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\int {\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^m \,d x \] Input:
int((d*cos(a + b*x))^(3/2)*(c*sin(a + b*x))^m,x)
Output:
int((d*cos(a + b*x))^(3/2)*(c*sin(a + b*x))^m, x)
\[ \int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx=\sqrt {d}\, c^{m} \left (\int \sin \left (b x +a \right )^{m} \sqrt {\cos \left (b x +a \right )}\, \cos \left (b x +a \right )d x \right ) d \] Input:
int((d*cos(b*x+a))^(3/2)*(c*sin(b*x+a))^m,x)
Output:
sqrt(d)*c**m*int(sin(a + b*x)**m*sqrt(cos(a + b*x))*cos(a + b*x),x)*d