\(\int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx\) [134]

Optimal result

Integrand size = 26, antiderivative size = 534 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{800 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{96 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2} \] Output:

-1/8*(d*x+c)^(3/2)*cos(b*x+a)/b-1/48*(d*x+c)^(3/2)*cos(3*b*x+3*a)/b+1/80*( 
d*x+c)^(3/2)*cos(5*b*x+5*a)/b-3/32*d^(3/2)*2^(1/2)*Pi^(1/2)*cos(a-b*c/d)*F 
resnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)-1/576*d^(3 
/2)*6^(1/2)*Pi^(1/2)*cos(3*a-3*b*c/d)*FresnelS(b^(1/2)*6^(1/2)/Pi^(1/2)*(d 
*x+c)^(1/2)/d^(1/2))/b^(5/2)+3/8000*d^(3/2)*10^(1/2)*Pi^(1/2)*cos(5*a-5*b* 
c/d)*FresnelS(b^(1/2)*10^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)+3/8 
000*d^(3/2)*10^(1/2)*Pi^(1/2)*FresnelC(b^(1/2)*10^(1/2)/Pi^(1/2)*(d*x+c)^( 
1/2)/d^(1/2))*sin(5*a-5*b*c/d)/b^(5/2)-1/576*d^(3/2)*6^(1/2)*Pi^(1/2)*Fres 
nelC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)/b^(5 
/2)-3/32*d^(3/2)*2^(1/2)*Pi^(1/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c 
)^(1/2)/d^(1/2))*sin(a-b*c/d)/b^(5/2)+3/16*d*(d*x+c)^(1/2)*sin(b*x+a)/b^2+ 
1/96*d*(d*x+c)^(1/2)*sin(3*b*x+3*a)/b^2-3/800*d*(d*x+c)^(1/2)*sin(5*b*x+5* 
a)/b^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.62 (sec) , antiderivative size = 1126, normalized size of antiderivative = 2.11 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[d]*(-12*Sqrt[b]*Sqrt[d]*E^(((3*I)*b*c)/d)*Sqrt[c + d*x]*(-I + 2*b*x 
+ E^((6*I)*(a + b*x))*(I + 2*b*x)) - (1 - I)*(2*b*c + I*d)*E^(((3*I)*b*(2* 
c + d*x))/d)*Sqrt[6*Pi]*Erf[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt 
[d]] + (1 + I)*((2*I)*b*c + d)*E^((3*I)*(2*a + b*x))*Sqrt[6*Pi]*Erfi[((1 + 
 I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]))/(2304*b^(5/2)*E^(((3*I)*(a 
*d + b*(c + d*x)))/d)) - (Sqrt[d]*(-20*Sqrt[b]*Sqrt[d]*E^(((5*I)*b*c)/d)*S 
qrt[c + d*x]*(-3*I + 10*b*x + E^((10*I)*(a + b*x))*(3*I + 10*b*x)) - (1 - 
I)*(10*b*c + (3*I)*d)*E^(((5*I)*b*(2*c + d*x))/d)*Sqrt[10*Pi]*Erf[((1 + I) 
*Sqrt[5/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + (1 + I)*((10*I)*b*c + 3*d)*E^ 
((5*I)*(2*a + b*x))*Sqrt[10*Pi]*Erfi[((1 + I)*Sqrt[5/2]*Sqrt[b]*Sqrt[c + d 
*x])/Sqrt[d]]))/(32000*b^(5/2)*E^(((5*I)*(a*d + b*(c + d*x)))/d)) + ((I/16 
)*c*d*(-(E^((2*I)*a)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, ((-I)*b*(c + d* 
x))/d]) + E^(((2*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, (I*b*(c + d 
*x))/d]))/(b^2*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x]) + (c*(-1/6*(E^((3*I)*( 
a - (b*c)/d))*Sqrt[c + d*x]*Gamma[3/2, ((-3*I)*b*(c + d*x))/d])/(Sqrt[3]*b 
*Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma[3/2, ((3*I)*b*(c + d*x 
))/d])/(6*Sqrt[3]*b*E^((3*I)*(a - (b*c)/d))*Sqrt[(I*b*(c + d*x))/d])))/16 
- (c*(-1/10*(E^((5*I)*(a - (b*c)/d))*Sqrt[c + d*x]*Gamma[3/2, ((-5*I)*b*(c 
 + d*x))/d])/(Sqrt[5]*b*Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma 
[3/2, ((5*I)*b*(c + d*x))/d])/(10*Sqrt[5]*b*E^((5*I)*(a - (b*c)/d))*Sqr...
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 534, 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.

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

\(\Big \downarrow \) 4906

\(\displaystyle \int \left (\frac {1}{8} (c+d x)^{3/2} \sin (a+b x)+\frac {1}{16} (c+d x)^{3/2} \sin (3 a+3 b x)-\frac {1}{16} (c+d x)^{3/2} \sin (5 a+5 b x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}\)

Input:

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

Output:

-1/8*((c + d*x)^(3/2)*Cos[a + b*x])/b - ((c + d*x)^(3/2)*Cos[3*a + 3*b*x]) 
/(48*b) + ((c + d*x)^(3/2)*Cos[5*a + 5*b*x])/(80*b) - (3*d^(3/2)*Sqrt[Pi/2 
]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/( 
16*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*S 
qrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*C 
os[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]) 
/(800*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt 
[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(800*b^(5/2)) - (d^(3/2)*Sqrt[Pi 
/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c) 
/d])/(96*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqr 
t[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(5/2)) + (3*d*Sqrt[c + d*x]*S 
in[a + b*x])/(16*b^2) + (d*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(96*b^2) - (3*d 
*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(800*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 = 1.94 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{8 b}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{16 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}-\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{10 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{100 b \sqrt {\frac {b}{d}}}\right )}{80 b}}{d}\)	\(580\)
default	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{8 b}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{16 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}-\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{10 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{100 b \sqrt {\frac {b}{d}}}\right )}{80 b}}{d}\)	\(580\)

Input:

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

Output:

2/d*(-1/16/b*d*(d*x+c)^(3/2)*cos(b/d*(d*x+c)+(a*d-b*c)/d)+3/16/b*d*(1/2/b* 
d*(d*x+c)^(1/2)*sin(b/d*(d*x+c)+(a*d-b*c)/d)-1/4/b*d*2^(1/2)*Pi^(1/2)/(b/d 
)^(1/2)*(cos((a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^ 
(1/2)/d)+sin((a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^ 
(1/2)/d)))-1/96/b*d*(d*x+c)^(3/2)*cos(3*b/d*(d*x+c)+3*(a*d-b*c)/d)+1/32/b* 
d*(1/6/b*d*(d*x+c)^(1/2)*sin(3*b/d*(d*x+c)+3*(a*d-b*c)/d)-1/36/b*d*2^(1/2) 
*Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(cos(3*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2 
)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)+sin(3*(a*d-b*c)/d)*FresnelC(2^(1/ 
2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))+1/160/b*d*(d*x+c)^(3/ 
2)*cos(5*b/d*(d*x+c)+5*(a*d-b*c)/d)-3/160/b*d*(1/10/b*d*(d*x+c)^(1/2)*sin( 
5*b/d*(d*x+c)+5*(a*d-b*c)/d)-1/100/b*d*2^(1/2)*Pi^(1/2)*5^(1/2)/(b/d)^(1/2 
)*(cos(5*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*b*(d*x 
+c)^(1/2)/d)+sin(5*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1 
/2)*b*(d*x+c)^(1/2)/d))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.80 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 480 \, {\left (30 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{5} - 50 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{3} - {\left (9 \, b d \cos \left (b x + a\right )^{4} - 13 \, b d \cos \left (b x + a\right )^{2} - 26 \, b d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{72000 \, b^{3}} \] Input:

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

Output:

1/72000*(27*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_s 
in(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 125*sqrt(6)*pi*d^2*sqrt(b/(pi* 
d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)) 
) - 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqr 
t(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fr 
esnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) - 125* 
sqrt(6)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi 
*d)))*sin(-3*(b*c - a*d)/d) + 27*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*fresnel_co 
s(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) + 480*(30*( 
b^2*d*x + b^2*c)*cos(b*x + a)^5 - 50*(b^2*d*x + b^2*c)*cos(b*x + a)^3 - (9 
*b*d*cos(b*x + a)^4 - 13*b*d*cos(b*x + a)^2 - 26*b*d)*sin(b*x + a))*sqrt(d 
*x + c))/b^3
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.42 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/288000*sqrt(2)*(1800*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(5*((d*x + c)*b - b* 
c + a*d)/d)/d^2 - 3000*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(3*((d*x + c)*b - b* 
c + a*d)/d)/d^2 - 18000*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(((d*x + c)*b - b*c 
 + a*d)/d)/d^2 - 540*sqrt(2)*sqrt(d*x + c)*b^3*sin(5*((d*x + c)*b - b*c + 
a*d)/d)/d + 1500*sqrt(2)*sqrt(d*x + c)*b^3*sin(3*((d*x + c)*b - b*c + a*d) 
/d)/d + 27000*sqrt(2)*sqrt(d*x + c)*b^3*sin(((d*x + c)*b - b*c + a*d)/d)/d 
 + 27*((I + 1)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) 
 - (I - 1)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d))*er 
f(sqrt(d*x + c)*sqrt(5*I*b/d)) + 125*(-(I + 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d 
^2)^(1/4)*cos(-3*(b*c - a*d)/d) + (I - 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^( 
1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) + 6750*(-(I + 
 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (I - 1)*sqrt(pi)*b^ 
2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) + 67 
50*((I - 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (I + 1)*sqr 
t(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b 
/d)) + 125*((I - 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d 
)/d) - (I + 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d)) 
*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)) + 27*(-(I - 1)*25^(1/4)*sqrt(pi)*b^2*(b 
^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) + (I + 1)*25^(1/4)*sqrt(pi)*b^2*(b^2/d 
^2)^(1/4)*sin(-5*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-5*I*b/d)))*d^2...
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.04 (sec) , antiderivative size = 2279, normalized size of antiderivative = 4.27 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

1/144000*(300*(30*sqrt(2)*sqrt(pi)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x 
+ c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(-I*b* 
d/sqrt(b^2*d^2) + 1)) + 5*sqrt(6)*sqrt(pi)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d)* 
sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(I*b*c - I*a*d)/d)/(sqrt( 
b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) - 3*sqrt(10)*sqrt(pi)*d*erf(-1/2*I*sqrt(10 
)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-5*(I*b*c - I*a* 
d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + 30*sqrt(2)*sqrt(pi)*d*erf(-1 
/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-I*b 
*c + I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + 5*sqrt(6)*sqrt(pi)* 
d*erf(1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)* 
e^(-3*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) - 3*sqrt( 
10)*sqrt(pi)*d*erf(1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2 
*d^2) + 1)/d)*e^(-5*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 
 1)))*c^2 - 20*(450*sqrt(2)*sqrt(pi)*(2*b*c + I*d)*d*erf(1/2*I*sqrt(2)*sqr 
t(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d)/d)/( 
sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 25*sqrt(6)*sqrt(pi)*(6*b*c - I*d 
)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d 
)*e^(-3*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) - 9*sqr 
t(10)*sqrt(pi)*(10*b*c - I*d)*d*erf(-1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c 
)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-5*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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