\(\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx\) [194]

Optimal result

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

3/16*d*(d*x+c)^(1/2)*cos(b*x+a)/b^2-1/96*d*(d*x+c)^(1/2)*cos(3*b*x+3*a)/b^ 
2-3/800*d*(d*x+c)^(1/2)*cos(5*b*x+5*a)/b^2-3/32*d^(3/2)*2^(1/2)*Pi^(1/2)*c 
os(a-b*c/d)*FresnelC(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)*FresnelC(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)*FresnelC(b^(1/2)*10^(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)*FresnelS(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)*FresnelS(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)*FresnelS(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)+1/8*(d*x+c)^(3/2)*sin(b 
*x+a)/b-1/48*(d*x+c)^(3/2)*sin(3*b*x+3*a)/b-1/80*(d*x+c)^(3/2)*sin(5*b*x+5 
*a)/b
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.26 (sec) , antiderivative size = 1088, normalized size of antiderivative = 2.04 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2304*(Sqrt[d]*(12*Sqrt[b]*Sqrt[d]*E^(((3*I)*b*c)/d)*Sqrt[c + d*x]*(1 + 
(2*I)*b*x + E^((6*I)*(a + b*x))*(1 - (2*I)*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*b*c - I*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]]))/(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)*Sqrt[c + d*x]*(3 + (10*I)*b*x + E^((10*I)*(a + b*x))*(3 - (10*I)*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*b*c - (3 
*I)*d)*E^((5*I)*(2*a + b*x))*Sqrt[10*Pi]*Erfi[((1 + I)*Sqrt[5/2]*Sqrt[b]*S 
qrt[c + d*x])/Sqrt[d]]))/(32000*b^(5/2)*E^(((5*I)*(a*d + b*(c + d*x)))/d)) 
 + (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]))/(16*b^2*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x]) - (c*(c + d*x)^(3/2) 
*(-((E^((6*I)*a)*Gamma[3/2, ((-3*I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d 
)^(3/2)) - (E^(((6*I)*b*c)/d)*Gamma[3/2, ((3*I)*b*(c + d*x))/d])/((I*b*(c 
+ d*x))/d)^(3/2)))/(96*Sqrt[3]*d*E^(((3*I)*(b*c + a*d))/d)) - (c*(c + d*x) 
^(3/2)*(-((E^((10*I)*a)*Gamma[3/2, ((-5*I)*b*(c + d*x))/d])/(((-I)*b*(c + 
d*x))/d)^(3/2)) - (E^(((10*I)*b*c)/d)*Gamma[3/2, ((5*I)*b*(c + d*x))/d])/( 
(I*b*(c + d*x))/d)^(3/2)))/(160*Sqrt[5]*d*E^(((5*I)*(b*c + a*d))/d)) + ...
 

Rubi [A] (verified)

Time = 1.22 (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.

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(3*d*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^2) - (d*Sqrt[c + d*x]*Cos[3*a + 3*b 
*x])/(96*b^2) - (3*d*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/(800*b^2) - (3*d^(3/2 
)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(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]*FresnelC 
[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sq 
rt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x] 
)/Sqrt[d]])/(800*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/10]*FresnelS[(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]*FresnelS[(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]*FresnelS[(Sqrt[b]*Sqr 
t[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(5/2)) + ((c + d*x 
)^(3/2)*Sin[a + b*x])/(8*b) - ((c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(48*b) - 
((c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(80*b)
 

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 = 1.99 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}\)	\(583\)
default	\(\frac {\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}\)	\(583\)

Input:

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

Output:

2/d*(1/16/b*d*(d*x+c)^(3/2)*sin(b/d*(d*x+c)+(a*d-b*c)/d)-3/16/b*d*(-1/2/b* 
d*(d*x+c)^(1/2)*cos(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)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^ 
(1/2)/d)-sin((a*d-b*c)/d)*FresnelS(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)*sin(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)*cos(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)*FresnelC(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)*FresnelS(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)*sin(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)*co 
s(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)*FresnelC(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)*FresnelS(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 = 446, normalized size of antiderivative = 0.84 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) - 480 \, {\left (9 \, b d \cos \left (b x + a\right )^{5} - 5 \, b d \cos \left (b x + a\right )^{3} - 30 \, b d \cos \left (b x + a\right ) + 10 \, {\left (3 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} - 2 \, b^{2} d x - 2 \, b^{2} c - {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{72000 \, b^{3}} \] Input:

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

Output:

1/72000*(27*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_c 
os(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_cos(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_cos(sqr 
t(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) + 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fr 
esnel_sin(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_sin(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_si 
n(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) - 480*(9*b* 
d*cos(b*x + a)^5 - 5*b*d*cos(b*x + a)^3 - 30*b*d*cos(b*x + a) + 10*(3*(b^2 
*d*x + b^2*c)*cos(b*x + a)^4 - 2*b^2*d*x - 2*b^2*c - (b^2*d*x + b^2*c)*cos 
(b*x + a)^2)*sin(b*x + a))*sqrt(d*x + c))/b^3
 

Sympy [F]

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

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

Output:

Integral((c + d*x)**(3/2)*sin(a + b*x)**2*cos(a + b*x)**3, 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 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/288000*sqrt(2)*(1800*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(5*((d*x + c)*b - b 
*c + a*d)/d)/d^2 + 3000*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(3*((d*x + c)*b - b 
*c + a*d)/d)/d^2 - 18000*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(((d*x + c)*b - b* 
c + a*d)/d)/d^2 + 540*sqrt(2)*sqrt(d*x + c)*b^3*cos(5*((d*x + c)*b - b*c + 
 a*d)/d)/d + 1500*sqrt(2)*sqrt(d*x + c)*b^3*cos(3*((d*x + c)*b - b*c + a*d 
)/d)/d - 27000*sqrt(2)*sqrt(d*x + c)*b^3*cos(((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))* 
erf(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)) - 6 
750*(-(I + 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (I - 1)*s 
qrt(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^...
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.01 (sec) , antiderivative size = 2278, normalized size of antiderivative = 4.27 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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