\(\int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx\) [56]

Optimal result

Integrand size = 22, antiderivative size = 168 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx=-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}-\frac {3 d^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{32 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{32 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{16 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4491, 12, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx=-\frac {3 \sqrt {\pi } d^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{32 b^{5/2}}-\frac {3 \sqrt {\pi } d^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{32 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{16 b^2}-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b} \]

[In]

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

[Out]

-1/4*((c + d*x)^(3/2)*Cos[2*a + 2*b*x])/b - (3*d^(3/2)*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[
c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(32*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*
Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(32*b^(5/2)) + (3*d*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(16*b^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3377

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

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[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 3432

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 3433

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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{2} (c+d x)^{3/2} \sin (2 a+2 b x) \, dx \\ & = \frac {1}{2} \int (c+d x)^{3/2} \sin (2 a+2 b x) \, dx \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}+\frac {(3 d) \int \sqrt {c+d x} \cos (2 a+2 b x) \, dx}{8 b} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{16 b^2}-\frac {\left (3 d^2\right ) \int \frac {\sin (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{32 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{16 b^2}-\frac {\left (3 d^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{32 b^2}-\frac {\left (3 d^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{32 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{16 b^2}-\frac {\left (3 d \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^2}-\frac {\left (3 d \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^2} \\ & = -\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{4 b}-\frac {3 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 )}{32 b^{5/2}}-\frac {3 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 )}{32 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{16 b^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.02 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.76 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx=\frac {e^{-\frac {2 i (b c+a d)}{d}} (c+d x)^{3/2} \left (-\frac {e^{4 i a} \Gamma \left (\frac {5}{2},-\frac {2 i b (c+d x)}{d}\right )}{\left (-\frac {i b (c+d x)}{d}\right )^{3/2}}-\frac {e^{\frac {4 i b c}{d}} \Gamma \left (\frac {5}{2},\frac {2 i b (c+d x)}{d}\right )}{\left (\frac {i b (c+d x)}{d}\right )^{3/2}}\right )}{16 \sqrt {2} b} \]

[In]

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

[Out]

((c + d*x)^(3/2)*(-((E^((4*I)*a)*Gamma[5/2, ((-2*I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^(3/2)) - (E^(((4*I
)*b*c)/d)*Gamma[5/2, ((2*I)*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^(3/2)))/(16*Sqrt[2]*b*E^(((2*I)*(b*c + a*d))/
d))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}}{d}\)	\(187\)
default	\(\frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}}{d}\)	\(187\)

[In]

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

[Out]

2/d*(-1/8/b*d*(d*x+c)^(3/2)*cos(2*b/d*(d*x+c)+2*(a*d-b*c)/d)+3/8/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))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx=-\frac {3 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 4 \, {\left (2 \, b^{2} d x + 3 \, b d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 2 \, b^{2} c - 4 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2}\right )} \sqrt {d x + c}}{32 \, b^{3}} \]

[In]

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

[Out]

-1/32*(3*pi*d^2*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d))) + 3*pi*d^2*sq
rt(b/(pi*d))*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) - 4*(2*b^2*d*x + 3*b*d*cos(b*x
+ a)*sin(b*x + a) + 2*b^2*c - 4*(b^2*d*x + b^2*c)*cos(b*x + a)^2)*sqrt(d*x + c))/b^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.52 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx=-\frac {\sqrt {2} {\left (32 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 24 \, \sqrt {2} \sqrt {d x + c} b d \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 3 \, {\left (\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) + 3 \, {\left (-\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right )\right )}}{256 \, b^{3}} \]

[In]

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

[Out]

-1/256*sqrt(2)*(32*sqrt(2)*(d*x + c)^(3/2)*b^2*cos(2*((d*x + c)*b - b*c + a*d)/d) - 24*sqrt(2)*sqrt(d*x + c)*b
*d*sin(2*((d*x + c)*b - b*c + a*d)/d) + 3*((I + 1)*4^(1/4)*sqrt(pi)*d^2*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d)
- (I - 1)*4^(1/4)*sqrt(pi)*d^2*(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) + 3*(-(
I - 1)*4^(1/4)*sqrt(pi)*d^2*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) + (I + 1)*4^(1/4)*sqrt(pi)*d^2*(b^2/d^2)^(1/
4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-2*I*b/d)))/b^3

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 747, normalized size of antiderivative = 4.45 \[ \int (c+d x)^{3/2} \cos (a+b x) \sin (a+b x) \, dx=\frac {16 \, {\left (\frac {\sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}\right )} c^{2} + d^{2} {\left (\frac {\frac {\sqrt {\pi } {\left (16 \, b^{2} c^{2} - 8 i \, b c d - 3 \, d^{2}\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{2}} + \frac {2 i \, {\left (4 i \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 8 i \, \sqrt {d x + c} b c d - 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (-\frac {2 \, {\left (-i \, {\left (d x + c\right )} b + i \, b c - i \, a d\right )}}{d}\right )}}{b^{2}}}{d^{2}} + \frac {\frac {\sqrt {\pi } {\left (16 \, b^{2} c^{2} + 8 i \, b c d - 3 \, d^{2}\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{2}} + \frac {2 i \, {\left (4 i \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 8 i \, \sqrt {d x + c} b c d + 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (-\frac {2 \, {\left (i \, {\left (d x + c\right )} b - i \, b c + i \, a d\right )}}{d}\right )}}{b^{2}}}{d^{2}}\right )} - 8 \, {\left (\frac {\sqrt {\pi } {\left (4 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, b c - i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {\sqrt {\pi } {\left (4 \, b c + i \, d\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, b c + i \, a d\right )}}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {2 \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (i \, {\left (d x + c\right )} b - i \, b c + i \, a d\right )}}{d}\right )}}{b} + \frac {2 \, \sqrt {d x + c} d e^{\left (-\frac {2 \, {\left (-i \, {\left (d x + c\right )} b + i \, b c - i \, a d\right )}}{d}\right )}}{b}\right )} c}{64 \, d} \]

[In]

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

[Out]

1/64*(16*(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)/(sqr
t(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + 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 + d^2*((sqrt(pi)*(16*b^2*c^2 - 8*I*b*c*d -
3*d^2)*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^2) + 2*I*(4*I*(d*x + c)^(3/2)*b*d - 8*I*sqrt(d*x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^(-
2*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + (sqrt(pi)*(16*b^2*c^2 + 8*I*b*c*d - 3*d^2)*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^
2) + 2*I*(4*I*(d*x + c)^(3/2)*b*d - 8*I*sqrt(d*x + c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^(-2*(I*(d*x + c)*b - I*b*
c + I*a*d)/d)/b^2)/d^2) - 8*(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) + 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) + 2*sqrt(d*x + c)*d*e^(-2*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 2*sqrt(d*x + c)*d*e^(-2*(-I*(d*x + c)*b +
I*b*c - I*a*d)/d)/b)*c)/d

Mupad [F(-1)]

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

[In]

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

[Out]

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