\(\int (c+d x)^{3/2} \sin (a+b x) \, dx\) [39]

Optimal result

Integrand size = 16, antiderivative size = 170 \[ \int (c+d x)^{3/2} \sin (a+b x) \, dx=-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{2 b^2} \] Output:

-(d*x+c)^(3/2)*cos(b*x+a)/b-3/4*d^(3/2)*2^(1/2)*Pi^(1/2)*cos(a-b*c/d)*Fres 
nelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)-3/4*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))*s 
in(a-b*c/d)/b^(5/2)+3/2*d*(d*x+c)^(1/2)*sin(b*x+a)/b^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.74 \[ \int (c+d x)^{3/2} \sin (a+b x) \, dx=-\frac {i d e^{-\frac {i (b c+a d)}{d}} \sqrt {c+d x} \left (\frac {e^{2 i a} \Gamma \left (\frac {5}{2},-\frac {i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {5}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{2 b^2} \] Input:

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

Output:

((-1/2*I)*d*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[5/2, ((-I)*b*(c + d*x))/d])/ 
Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2*I)*b*c)/d)*Gamma[5/2, (I*b*(c + d*x)) 
/d])/Sqrt[(I*b*(c + d*x))/d]))/(b^2*E^((I*(b*c + a*d))/d))
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3777, 3042, 3777, 25, 3042, 3787, 3042, 3785, 3786, 3832, 3833}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

\(\displaystyle \frac {3 d \int \sqrt {c+d x} \cos (a+b x)dx}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d \int \sqrt {c+d x} \sin \left (a+b x+\frac {\pi }{2}\right )dx}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\)

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3787

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\)

\(\Big \downarrow \) 3785

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

\(\Big \downarrow \) 3786

\(\displaystyle \frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}+\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}+\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}+\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\)

Input:

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

Output:

-(((c + d*x)^(3/2)*Cos[a + b*x])/b) + (3*d*(-1/2*(d*((Sqrt[2*Pi]*Cos[a - ( 
b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqr 
t[d]) + (Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*S 
in[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])))/b + (Sqrt[c + d*x]*Sin[a + b*x])/b))/ 
(2*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, 
d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d 
   Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f 
}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

rule 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( 
d*e - c*f)/d]   Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d 
, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
 

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{b}}{d}\)	\(188\)
default	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{b}}{d}\)	\(188\)

Input:

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

Output:

2/d*(-1/2/b*d*(d*x+c)^(3/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)+3/2/b*d*(1/2/b*d* 
(d*x+c)^(1/2)*sin(b*(d*x+c)/d+(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))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int (c+d x)^{3/2} \sin (a+b x) \, dx=-\frac {3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 2 \, {\left (3 \, b d \sin \left (b x + a\right ) - 2 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )\right )} \sqrt {d x + c}}{4 \, b^{3}} \] Input:

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

Output:

-1/4*(3*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt 
(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) + 3*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fresne 
l_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) - 2*(3*b*d 
*sin(b*x + a) - 2*(b^2*d*x + b^2*c)*cos(b*x + a))*sqrt(d*x + c))/b^3
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.06 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.42 \[ \int (c+d x)^{3/2} \sin (a+b x) \, dx=-\frac {\sqrt {2} {\left (8 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 12 \, \sqrt {2} \sqrt {d x + c} b d \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + 3 \, {\left (\left (i + 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (i - 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + 3 \, {\left (-\left (i - 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (i + 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right )\right )}}{16 \, b^{3}} \] Input:

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

Output:

-1/16*sqrt(2)*(8*sqrt(2)*(d*x + c)^(3/2)*b^2*cos(((d*x + c)*b - b*c + a*d) 
/d) - 12*sqrt(2)*sqrt(d*x + c)*b*d*sin(((d*x + c)*b - b*c + a*d)/d) + 3*(( 
I + 1)*sqrt(pi)*d^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (I - 1)*sqrt(pi) 
*d^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) + 
 3*(-(I - 1)*sqrt(pi)*d^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (I + 1)*sq 
rt(pi)*d^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I* 
b/d)))/b^3
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 758, normalized size of antiderivative = 4.46 \[ \int (c+d x)^{3/2} \sin (a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(4*(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)) + 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/sq 
rt(b^2*d^2) + 1)))*c^2 - 4*(sqrt(2)*sqrt(pi)*(2*b*c + I*d)*d*erf(1/2*I*sqr 
t(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) + 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) + 2* 
sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 2*sqrt(d*x + c)* 
d*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b)*c + sqrt(2)*sqrt(pi)*(4*b^2*c^ 
2 + 4*I*b*c*d - 3*d^2)*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^2) + sqrt(2)*sqrt(pi)*(4*b^2*c^2 - 4*I*b*c*d - 3*d^2)*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^2) + 2*I*(2*I*(d*x + 
c)^(3/2)*b*d - 4*I*sqrt(d*x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^((I*(d*x + 
 c)*b - I*b*c + I*a*d)/d)/b^2 + 2*I*(2*I*(d*x + c)^(3/2)*b*d - 4*I*sqrt(d* 
x + c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d) 
/b^2)/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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