\(\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx\) [197]

Optimal result

Integrand size = 26, antiderivative size = 351 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2} \] Output:

-3/64*(d*x+c)^(3/2)*cos(2*b*x+2*a)/b+1/192*(d*x+c)^(3/2)*cos(6*b*x+6*a)/b+ 
1/4608*d^(3/2)*3^(1/2)*Pi^(1/2)*cos(6*a-6*b*c/d)*FresnelS(2*b^(1/2)*3^(1/2 
)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)-9/512*d^(3/2)*Pi^(1/2)*cos(2*a-2 
*b*c/d)*FresnelS(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))/b^(5/2)+1/4608* 
d^(3/2)*3^(1/2)*Pi^(1/2)*FresnelC(2*b^(1/2)*3^(1/2)/Pi^(1/2)*(d*x+c)^(1/2) 
/d^(1/2))*sin(6*a-6*b*c/d)/b^(5/2)-9/512*d^(3/2)*Pi^(1/2)*FresnelC(2*b^(1/ 
2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*sin(2*a-2*b*c/d)/b^(5/2)+9/256*d*(d*x+c 
)^(1/2)*sin(2*b*x+2*a)/b^2-1/768*d*(d*x+c)^(1/2)*sin(6*b*x+6*a)/b^2
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.74 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {i e^{-\frac {6 i (b c+a d)}{d}} (c+d x)^{5/2} \left (-81 e^{4 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {2 i b (c+d x)}{d}\right )+81 e^{4 i a+\frac {8 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {2 i b (c+d x)}{d}\right )+\sqrt {3} \left (e^{12 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {6 i b (c+d x)}{d}\right )-e^{\frac {12 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {6 i b (c+d x)}{d}\right )\right )\right )}{6912 \sqrt {2} d \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \] Input:

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

Output:

((I/6912)*(c + d*x)^(5/2)*(-81*E^((4*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d* 
x))/d]*Gamma[5/2, ((-2*I)*b*(c + d*x))/d] + 81*E^((4*I)*a + ((8*I)*b*c)/d) 
*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[5/2, ((2*I)*b*(c + d*x))/d] + Sqrt[3]*(E 
^((12*I)*a)*Sqrt[(I*b*(c + d*x))/d]*Gamma[5/2, ((-6*I)*b*(c + d*x))/d] - E 
^(((12*I)*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[5/2, ((6*I)*b*(c + d*x) 
)/d])))/(Sqrt[2]*d*E^(((6*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^(3/2) 
)
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4906, 2009}

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.

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

\(\Big \downarrow \) 4906

\(\displaystyle \int \left (\frac {3}{32} (c+d x)^{3/2} \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^{3/2} \sin (6 a+6 b x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\frac {\pi }{3}} d^{3/2} \sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 \sqrt {\pi } d^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 \sqrt {\pi } d^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}-\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}\)

Input:

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

Output:

(-3*(c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(64*b) + ((c + d*x)^(3/2)*Cos[6*a + 
6*b*x])/(192*b) + (d^(3/2)*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelS[(2*Sqr 
t[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1536*b^(5/2)) - (9*d^(3/2)*Sqrt[ 
Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[ 
Pi])])/(512*b^(5/2)) + (d^(3/2)*Sqrt[Pi/3]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]* 
Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(1536*b^(5/2)) - (9*d^(3/2)* 
Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - 
(2*b*c)/d])/(512*b^(5/2)) + (9*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(256*b^2) 
 - (d*Sqrt[c + d*x]*Sin[6*a + 6*b*x])/(768*b^2)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{64 b}}{d}\)	\(383\)
default	\(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{64 b}}{d}\)	\(383\)

Input:

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

Output:

2/d*(-3/128/b*d*(d*x+c)^(3/2)*cos(2*b/d*(d*x+c)+2*(a*d-b*c)/d)+9/128/b*d*( 
1/4/b*d*(d*x+c)^(1/2)*sin(2*b/d*(d*x+c)+2*(a*d-b*c)/d)-1/8/b*d*Pi^(1/2)/(b 
/d)^(1/2)*(cos(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1 
/2)/d)+sin(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/ 
d)))+1/384/b*d*(d*x+c)^(3/2)*cos(6*b/d*(d*x+c)+6*(a*d-b*c)/d)-1/128/b*d*(1 
/12/b*d*(d*x+c)^(1/2)*sin(6*b/d*(d*x+c)+6*(a*d-b*c)/d)-1/144/b*d*2^(1/2)*P 
i^(1/2)*6^(1/2)/(b/d)^(1/2)*(cos(6*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)* 
6^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)+sin(6*(a*d-b*c)/d)*FresnelC(2^(1/2) 
/Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.93 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {\sqrt {3} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {3} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 81 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 81 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 96 \, {\left (8 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{6} - 12 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} + 2 \, b^{2} d x + 2 \, b^{2} c - {\left (2 \, b d \cos \left (b x + a\right )^{5} - 2 \, b d \cos \left (b x + a\right )^{3} - 3 \, b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{4608 \, b^{3}} \] Input:

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

Output:

1/4608*(sqrt(3)*pi*d^2*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_sin(2* 
sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d))) + sqrt(3)*pi*d^2*sqrt(b/(pi*d))*fres 
nel_cos(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/d) - 81 
*pi*d^2*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_sin(2*sqrt(d*x + c)*s 
qrt(b/(pi*d))) - 81*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(2*sqrt(d*x + c)*sqrt 
(b/(pi*d)))*sin(-2*(b*c - a*d)/d) + 96*(8*(b^2*d*x + b^2*c)*cos(b*x + a)^6 
 - 12*(b^2*d*x + b^2*c)*cos(b*x + a)^4 + 2*b^2*d*x + 2*b^2*c - (2*b*d*cos( 
b*x + a)^5 - 2*b*d*cos(b*x + a)^3 - 3*b*d*cos(b*x + a))*sin(b*x + a))*sqrt 
(d*x + c))/b^3
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.46 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

1/36864*(192*(d*x + c)^(3/2)*b^3*cos(6*((d*x + c)*b - b*c + a*d)/d)/d - 17 
28*(d*x + c)^(3/2)*b^3*cos(2*((d*x + c)*b - b*c + a*d)/d)/d - 48*sqrt(d*x 
+ c)*b^2*sin(6*((d*x + c)*b - b*c + a*d)/d) + 1296*sqrt(d*x + c)*b^2*sin(2 
*((d*x + c)*b - b*c + a*d)/d) + ((I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^ 
2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d) - (I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d 
*(b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d)) + 
 81*(-(I + 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*cos(-2*(b*c - a 
*d)/d) + (I - 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*sin(-2*(b*c 
- a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) + 81*((I - 1)*4^(1/4)*sqrt(2)* 
sqrt(pi)*b*d*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) - (I + 1)*4^(1/4)*sqrt( 
2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*s 
qrt(-2*I*b/d)) + (-(I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*c 
os(-6*(b*c - a*d)/d) + (I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/ 
4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-6*I*b/d)))*d/b^4
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.60 (sec) , antiderivative size = 1489, normalized size of antiderivative = 4.24 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/9216*(48*(sqrt(3)*sqrt(pi)*d*erf(-I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I* 
b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt( 
b^2*d^2) + 1)) + sqrt(3)*sqrt(pi)*d*erf(I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)* 
(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d 
/sqrt(b^2*d^2) + 1)) - 9*sqrt(pi)*d*erf(-I*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/ 
sqrt(b^2*d^2) + 1)/d)*e^(-2*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2* 
d^2) + 1)) - 9*sqrt(pi)*d*erf(I*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d 
^2) + 1)/d)*e^(-2*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1 
)))*c^2 - 8*(sqrt(3)*sqrt(pi)*(12*b*c - I*d)*d*erf(-I*sqrt(3)*sqrt(b*d)*sq 
rt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(I*b*c - I*a*d)/d)/(sqrt(b* 
d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) + sqrt(3)*sqrt(pi)*(12*b*c + I*d)*d*erf(I* 
sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(-I*b* 
c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 27*sqrt(pi)*(4*b* 
c - I*d)*d*erf(-I*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^( 
-2*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) - 27*sqrt(pi 
)*(4*b*c + I*d)*d*erf(I*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1) 
/d)*e^(-2*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 5 
4*sqrt(d*x + c)*d*e^(-2*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 6*sqrt(d*x 
+ c)*d*e^(-6*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^( 
-2*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b + 6*sqrt(d*x + c)*d*e^(-6*(-I*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(c + d*x)*cos(a + b*x)**3*sin(a + b*x)**3*x,x)*d + int(sqrt(c + d* 
x)*cos(a + b*x)**3*sin(a + b*x)**3,x)*c