Integrand size = 16, antiderivative size = 169 \[ \int (c+d x)^{3/2} \cos (a+b x) \, dx=\frac {3 d \sqrt {c+d x} \cos (a+b x)}{2 b^2}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^{5/2}}+\frac {(c+d x)^{3/2} \sin (a+b x)}{b} \]
(d*x+c)^(3/2)*sin(b*x+a)/b-3/4*d^(3/2)*cos(a-b*c/d)*FresnelC(b^(1/2)*2^(1/ 2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/b^(5/2)+3/4*d^(3/2)*Fr esnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*2^(1/2 )*Pi^(1/2)/b^(5/2)+3/2*d*cos(b*x+a)*(d*x+c)^(1/2)/b^2
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.72 \[ \int (c+d x)^{3/2} \cos (a+b x) \, dx=\frac {d e^{-\frac {i (b c+a d)}{d}} \sqrt {c+d x} \left (\frac {e^{2 i a} \Gamma \left (\frac {5}{2},-\frac {i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}+\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {5}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{2 b^2} \]
(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]))/(2*b^2*E^((I*(b*c + a*d))/d))
Time = 0.81 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3777, 25, 3042, 3777, 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.
| \(\displaystyle \int (c+d x)^{3/2} \cos (a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^{3/2} \sin \left (a+b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 d \int -\sqrt {c+d x} \sin (a+b x)dx}{2 b}+\frac {(c+d x)^{3/2} \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sin (a+b x)dx}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sin (a+b x)dx}{2 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \int \frac {\cos (a+b x)}{\sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\) |
(-3*d*(-((Sqrt[c + d*x]*Cos[a + b*x])/b) + (d*((Sqrt[2*Pi]*Cos[a - (b*c)/d ]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) - (Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])))/(2*b)))/(2*b) + ((c + d*x)^(3/2)*Sin[a + b* x])/b
3.1.42.3.1 Defintions of rubi rules used
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]
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]
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]
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]
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]
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]
Time = 0.93 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{b}}{d}\) | \(189\) |
| default | \(\frac {\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {a d -b c}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{b}}{d}\) | \(189\) |
2/d*(1/2/b*d*(d*x+c)^(3/2)*sin(b*(d*x+c)/d+(a*d-b*c)/d)-3/2/b*d*(-1/2/b*d* (d*x+c)^(1/2)*cos(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)*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))))
Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int (c+d x)^{3/2} \cos (a+b x) \, dx=-\frac {3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 2 \, {\left (3 \, b d \cos \left (b x + a\right ) + 2 \, {\left (b^{2} d x + b^{2} c\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{4 \, b^{3}} \]
-1/4*(3*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_cos(sqrt (2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 3*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fresne l_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) - 2*(3*b*d *cos(b*x + a) + 2*(b^2*d*x + b^2*c)*sin(b*x + a))*sqrt(d*x + c))/b^3
\[ \int (c+d x)^{3/2} \cos (a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \cos {\left (a + b x \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.43 \[ \int (c+d x)^{3/2} \cos (a+b x) \, dx=\frac {\sqrt {2} {\left (8 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) + 12 \, \sqrt {2} \sqrt {d x + c} b d \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 3 \, {\left (-\left (i - 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (i + 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) - 3 \, {\left (\left (i + 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (i - 1\right ) \, \sqrt {\pi } d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right )\right )}}{16 \, b^{3}} \]
1/16*sqrt(2)*(8*sqrt(2)*(d*x + c)^(3/2)*b^2*sin(((d*x + c)*b - b*c + a*d)/ d) + 12*sqrt(2)*sqrt(d*x + c)*b*d*cos(((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)*sqr t(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
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 779, normalized size of antiderivative = 4.61 \[ \int (c+d x)^{3/2} \cos (a+b x) \, dx=-\frac {4 \, {\left (\frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {i \, \sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{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 {i \, \sqrt {2} \sqrt {\pi } {\left (4 \, b^{2} c^{2} + 4 i \, b c d - 3 \, d^{2}\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{2}} + \frac {2 \, {\left (-2 i \, {\left (d x + c\right )}^{\frac {3}{2}} b d + 4 i \, \sqrt {d x + c} b c d - 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (\frac {-i \, {\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b^{2}}}{d^{2}} + \frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (4 \, b^{2} c^{2} - 4 i \, b c d - 3 \, d^{2}\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b^{2}} + \frac {2 \, {\left (2 i \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 4 i \, \sqrt {d x + c} b c d - 3 \, \sqrt {d x + c} d^{2}\right )} e^{\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b^{2}}}{d^{2}}\right )} + 4 \, {\left (-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c + i \, d\right )} d \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {i \, b c - i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {i \, \sqrt {2} \sqrt {\pi } {\left (2 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{2 \, d}\right ) e^{\left (\frac {-i \, b c + i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {2 i \, \sqrt {d x + c} d e^{\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{d}\right )}}{b} - \frac {2 i \, \sqrt {d x + c} d e^{\left (\frac {-i \, {\left (d x + c\right )} b + i \, b c - i \, a d}{d}\right )}}{b}\right )} c}{8 \, d} \]
-1/8*(4*(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)) - 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)))*c^2 + d^2*((I*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*(-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^2 + (-I*sqrt(2)*sqr t(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*(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^2) + 4*(-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) + I*sqrt(2)*sqrt(pi)*(2*b*c - I*d)*d*er f(-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*I*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 2*I*sqrt(d*x + c)*d*e^((- I*(d*x + c)*b + I*b*c - I*a*d)/d)/b)*c)/d
Timed out. \[ \int (c+d x)^{3/2} \cos (a+b x) \, dx=\int \cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^{3/2} \,d x \]