Integrand size = 16, antiderivative size = 195 \[ \int (c+d x)^{5/2} \sin (a+b x) \, dx=\frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{4 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{4 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{4 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{2 b^2} \]
-(d*x+c)^(5/2)*cos(b*x+a)/b+5/2*d*(d*x+c)^(3/2)*sin(b*x+a)/b^2-15/8*d^(5/2 )*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^(7/2)+15/8*d^(5/2)*FresnelS(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^(7/2)+15/4*d^2*cos(b*x+ a)*(d*x+c)^(1/2)/b^3
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.65 \[ \int (c+d x)^{5/2} \sin (a+b x) \, dx=\frac {i d^3 e^{-\frac {i (b c+a d)}{d}} \left (e^{2 i a} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},-\frac {i b (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {7}{2},\frac {i b (c+d x)}{d}\right )\right )}{2 b^4 \sqrt {c+d x}} \]
((I/2)*d^3*(E^((2*I)*a)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[7/2, ((-I)*b*(c + d*x))/d] - E^(((2*I)*b*c)/d)*Sqrt[(I*b*(c + d*x))/d]*Gamma[7/2, (I*b*(c + d*x))/d]))/(b^4*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x])
Time = 0.96 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 3777, 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.
| \(\displaystyle \int (c+d x)^{5/2} \sin (a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^{5/2} \sin (a+b x)dx\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {5 d \int (c+d x)^{3/2} \cos (a+b x)dx}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 d \int (c+d x)^{3/2} \sin \left (a+b x+\frac {\pi }{2}\right )dx}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {5 d \left (\frac {3 d \int -\sqrt {c+d x} \sin (a+b x)dx}{2 b}+\frac {(c+d x)^{3/2} \sin (a+b x)}{b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sin (a+b x)dx}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \int \sqrt {c+d x} \sin (a+b x)dx}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \int \frac {\cos (a+b x)}{\sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {5 d \left (\frac {(c+d x)^{3/2} \sin (a+b x)}{b}-\frac {3 d \left (\frac {d \left (\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}-\frac {\sqrt {c+d x} \cos (a+b x)}{b}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{5/2} \cos (a+b x)}{b}\) |
-(((c + d*x)^(5/2)*Cos[a + b*x])/b) + (5*d*((-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))/(2*b)
3.1.38.3.1 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 = 0.12 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{b}}{d}\) | \(233\) |
| default | \(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{b}+\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{b}}{d}\) | \(233\) |
2/d*(-1/2/b*d*(d*x+c)^(5/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)+5/2/b*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 = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97 \[ \int (c+d x)^{5/2} \sin (a+b x) \, dx=-\frac {15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 15 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 2 \, \sqrt {d x + c} {\left ({\left (4 \, b^{3} d^{2} x^{2} + 8 \, b^{3} c d x + 4 \, b^{3} c^{2} - 15 \, b d^{2}\right )} \cos \left (b x + a\right ) - 10 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \sin \left (b x + a\right )\right )}}{8 \, b^{4}} \]
-1/8*(15*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_cos(sqr t(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 15*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fres nel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) + 2*sqrt (d*x + c)*((4*b^3*d^2*x^2 + 8*b^3*c*d*x + 4*b^3*c^2 - 15*b*d^2)*cos(b*x + a) - 10*(b^2*d^2*x + b^2*c*d)*sin(b*x + a)))/b^4
\[ \int (c+d x)^{5/2} \sin (a+b x) \, dx=\int \left (c + d x\right )^{\frac {5}{2}} \sin {\left (a + b x \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.35 \[ \int (c+d x)^{5/2} \sin (a+b x) \, dx=\frac {\sqrt {2} {\left (40 \, \sqrt {2} {\left (d x + c\right )}^{\frac {3}{2}} b^{2} d \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 4 \, {\left (4 \, \sqrt {2} {\left (d x + c\right )}^{\frac {5}{2}} b^{3} - 15 \, \sqrt {2} \sqrt {d x + c} b d^{2}\right )} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 15 \, {\left (-\left (i - 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) - \left (i + 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) - 15 \, {\left (\left (i + 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right ) + \left (i - 1\right ) \, \sqrt {\pi } d^{3} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right )\right )}}{32 \, b^{4}} \]
1/32*sqrt(2)*(40*sqrt(2)*(d*x + c)^(3/2)*b^2*d*sin(((d*x + c)*b - b*c + a* d)/d) - 4*(4*sqrt(2)*(d*x + c)^(5/2)*b^3 - 15*sqrt(2)*sqrt(d*x + c)*b*d^2) *cos(((d*x + c)*b - b*c + a*d)/d) - 15*(-(I - 1)*sqrt(pi)*d^3*(b^2/d^2)^(1 /4)*cos(-(b*c - a*d)/d) - (I + 1)*sqrt(pi)*d^3*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) - 15*((I + 1)*sqrt(pi)*d^3*(b^2/d ^2)^(1/4)*cos(-(b*c - a*d)/d) + (I - 1)*sqrt(pi)*d^3*(b^2/d^2)^(1/4)*sin(- (b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)))/b^4
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 1239, normalized size of antiderivative = 6.35 \[ \int (c+d x)^{5/2} \sin (a+b x) \, dx=\text {Too large to display} \]
1/16*(8*(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/s qrt(b^2*d^2) + 1)))*c^3 + 6*c*d^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)/d^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)/ d^2) - d^3*((sqrt(2)*sqrt(pi)*(8*b^3*c^3 + 12*I*b^2*c^2*d - 18*b*c*d^2 - 1 5*I*d^3)*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^3) - 2*I*(4*I*(d*x + c)^(5/2)*b^2*d - 12*I*(d*x + c)^(3/2)*b^2*c*d + 12*I*sqr t(d*x + c)*b^2*c^2*d + 10*(d*x + c)^(3/2)*b*d^2 - 18*sqrt(d*x + c)*b*c*d^2 - 15*I*sqrt(d*x + c)*d^3)*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^3)/d^3 + (sqrt(2)*sqrt(pi)*(8*b^3*c^3 - 12*I*b^2*c^2*d - 18*b*c*d^2 + 15*I*d^...
Timed out. \[ \int (c+d x)^{5/2} \sin (a+b x) \, dx=\int \sin \left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2} \,d x \]