Integrand size = 26, antiderivative size = 534 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{800 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{96 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2} \]
-1/8*(d*x+c)^(3/2)*cos(b*x+a)/b-1/48*(d*x+c)^(3/2)*cos(3*b*x+3*a)/b+1/80*( d*x+c)^(3/2)*cos(5*b*x+5*a)/b+3/8000*d^(3/2)*cos(5*a-5*b*c/d)*FresnelS(b^( 1/2)*10^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*10^(1/2)*Pi^(1/2)/b^(5/2)+3/ 8000*d^(3/2)*FresnelC(b^(1/2)*10^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin (5*a-5*b*c/d)*10^(1/2)*Pi^(1/2)/b^(5/2)-1/576*d^(3/2)*cos(3*a-3*b*c/d)*Fre snelS(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*6^(1/2)*Pi^(1/2)/b^( 5/2)-1/576*d^(3/2)*FresnelC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2) )*sin(3*a-3*b*c/d)*6^(1/2)*Pi^(1/2)/b^(5/2)-3/32*d^(3/2)*cos(a-b*c/d)*Fres nelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/b^(5 /2)-3/32*d^(3/2)*FresnelC(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)+3/16*d*sin(b*x+a)*(d*x+c)^(1/2)/b^2+ 1/96*d*sin(3*b*x+3*a)*(d*x+c)^(1/2)/b^2-3/800*d*sin(5*b*x+5*a)*(d*x+c)^(1/ 2)/b^2
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 1126, normalized size of antiderivative = 2.11 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {\sqrt {d} e^{-\frac {3 i (a d+b (c+d x))}{d}} \left (-12 \sqrt {b} \sqrt {d} e^{\frac {3 i b c}{d}} \sqrt {c+d x} \left (-i+2 b x+e^{6 i (a+b x)} (i+2 b x)\right )-(1-i) (2 b c+i d) e^{\frac {3 i b (2 c+d x)}{d}} \sqrt {6 \pi } \text {erf}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+(1+i) (2 i b c+d) e^{3 i (2 a+b x)} \sqrt {6 \pi } \text {erfi}\left (\frac {(1+i) \sqrt {\frac {3}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )}{2304 b^{5/2}}-\frac {\sqrt {d} e^{-\frac {5 i (a d+b (c+d x))}{d}} \left (-20 \sqrt {b} \sqrt {d} e^{\frac {5 i b c}{d}} \sqrt {c+d x} \left (-3 i+10 b x+e^{10 i (a+b x)} (3 i+10 b x)\right )-(1-i) (10 b c+3 i d) e^{\frac {5 i b (2 c+d x)}{d}} \sqrt {10 \pi } \text {erf}\left (\frac {(1+i) \sqrt {\frac {5}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )+(1+i) (10 i b c+3 d) e^{5 i (2 a+b x)} \sqrt {10 \pi } \text {erfi}\left (\frac {(1+i) \sqrt {\frac {5}{2}} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )}{32000 b^{5/2}}+\frac {i c d e^{-\frac {i (b c+a d)}{d}} \left (-e^{2 i a} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{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 {3}{2},\frac {i b (c+d x)}{d}\right )\right )}{16 b^2 \sqrt {c+d x}}+\frac {1}{16} c \left (-\frac {e^{3 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},-\frac {3 i b (c+d x)}{d}\right )}{6 \sqrt {3} b \sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{-3 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},\frac {3 i b (c+d x)}{d}\right )}{6 \sqrt {3} b \sqrt {\frac {i b (c+d x)}{d}}}\right )-\frac {1}{16} c \left (-\frac {e^{5 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},-\frac {5 i b (c+d x)}{d}\right )}{10 \sqrt {5} b \sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{-5 i \left (a-\frac {b c}{d}\right )} \sqrt {c+d x} \Gamma \left (\frac {3}{2},\frac {5 i b (c+d x)}{d}\right )}{10 \sqrt {5} b \sqrt {\frac {i b (c+d x)}{d}}}\right )+\frac {\sqrt {d} \left (e^{i \left (a-\frac {b c}{d}\right )} \left (-2 \sqrt {b} \sqrt {d} e^{\frac {i b (c+d x)}{d}} (3 i+2 b x) \sqrt {c+d x}+\sqrt [4]{-1} (2 i b c+3 d) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )+i \left (2 \sqrt {b} \sqrt {d} (3+2 i b x) \sqrt {c+d x}+(1+i) (2 b c+3 i d) \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {(1+i) \sqrt {b} \sqrt {c+d x}}{\sqrt {2} \sqrt {d}}\right ) \left (\cos \left (b \left (\frac {c}{d}+x\right )\right )+i \sin \left (b \left (\frac {c}{d}+x\right )\right )\right )\right ) (\cos (a+b x)-i \sin (a+b x))\right )}{64 b^{5/2}} \]
(Sqrt[d]*(-12*Sqrt[b]*Sqrt[d]*E^(((3*I)*b*c)/d)*Sqrt[c + d*x]*(-I + 2*b*x + E^((6*I)*(a + b*x))*(I + 2*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*I)*b*c + 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]]))/(2304*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)*S qrt[c + d*x]*(-3*I + 10*b*x + E^((10*I)*(a + b*x))*(3*I + 10*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*I)*b*c + 3*d)*E^ ((5*I)*(2*a + b*x))*Sqrt[10*Pi]*Erfi[((1 + I)*Sqrt[5/2]*Sqrt[b]*Sqrt[c + d *x])/Sqrt[d]]))/(32000*b^(5/2)*E^(((5*I)*(a*d + b*(c + d*x)))/d)) + ((I/16 )*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]))/(b^2*E^((I*(b*c + a*d))/d)*Sqrt[c + d*x]) + (c*(-1/6*(E^((3*I)*( a - (b*c)/d))*Sqrt[c + d*x]*Gamma[3/2, ((-3*I)*b*(c + d*x))/d])/(Sqrt[3]*b *Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma[3/2, ((3*I)*b*(c + d*x ))/d])/(6*Sqrt[3]*b*E^((3*I)*(a - (b*c)/d))*Sqrt[(I*b*(c + d*x))/d])))/16 - (c*(-1/10*(E^((5*I)*(a - (b*c)/d))*Sqrt[c + d*x]*Gamma[3/2, ((-5*I)*b*(c + d*x))/d])/(Sqrt[5]*b*Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma [3/2, ((5*I)*b*(c + d*x))/d])/(10*Sqrt[5]*b*E^((5*I)*(a - (b*c)/d))*Sqr...
Time = 1.17 (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.
| \(\displaystyle \int (c+d x)^{3/2} \sin ^3(a+b x) \cos ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {1}{8} (c+d x)^{3/2} \sin (a+b x)+\frac {1}{16} (c+d x)^{3/2} \sin (3 a+3 b x)-\frac {1}{16} (c+d x)^{3/2} \sin (5 a+5 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b}\) |
-1/8*((c + d*x)^(3/2)*Cos[a + b*x])/b - ((c + d*x)^(3/2)*Cos[3*a + 3*b*x]) /(48*b) + ((c + d*x)^(3/2)*Cos[5*a + 5*b*x])/(80*b) - (3*d^(3/2)*Sqrt[Pi/2 ]*Cos[a - (b*c)/d]*FresnelS[(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]*FresnelS[(Sqrt[b]*S qrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*C os[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]) /(800*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*FresnelC[(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]*FresnelC[(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]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqr t[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(5/2)) + (3*d*Sqrt[c + d*x]*S in[a + b*x])/(16*b^2) + (d*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(96*b^2) - (3*d *Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(800*b^2)
3.2.34.3.1 Defintions of rubi rules used
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]
Time = 0.53 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.09
| 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 -c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \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 {FresnelS}\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 {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 )}{8 b}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \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}\, \sin \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 {FresnelS}\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 {FresnelC}\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}} \cos \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}\, \sin \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 {FresnelS}\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 {FresnelC}\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}\) | \(580\) |
| 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 -c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \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 {FresnelS}\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 {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 )}{8 b}-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \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}\, \sin \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 {FresnelS}\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 {FresnelC}\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}} \cos \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}\, \sin \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 {FresnelS}\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 {FresnelC}\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}\) | \(580\) |
2/d*(-1/16/b*d*(d*x+c)^(3/2)*cos(b/d*(d*x+c)+(a*d-b*c)/d)+3/16/b*d*(1/2/b* d*(d*x+c)^(1/2)*sin(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)*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)))-1/96/b*d*(d*x+c)^(3/2)*cos(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)*sin(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)*FresnelS(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)*FresnelC(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)*cos(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)*sin( 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)*FresnelS(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)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1 /2)*b*(d*x+c)^(1/2)/d))))
Time = 0.30 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.80 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 480 \, {\left (30 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{5} - 50 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{3} - {\left (9 \, b d \cos \left (b x + a\right )^{4} - 13 \, b d \cos \left (b x + a\right )^{2} - 26 \, b d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{72000 \, b^{3}} \]
1/72000*(27*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_s in(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_sin(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_sin(sqr t(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fr esnel_cos(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_cos(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_co s(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) + 480*(30*( b^2*d*x + b^2*c)*cos(b*x + a)^5 - 50*(b^2*d*x + b^2*c)*cos(b*x + a)^3 - (9 *b*d*cos(b*x + a)^4 - 13*b*d*cos(b*x + a)^2 - 26*b*d)*sin(b*x + a))*sqrt(d *x + c))/b^3
\[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.42 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
1/288000*sqrt(2)*(1800*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(5*((d*x + c)*b - b* c + a*d)/d)/d^2 - 3000*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(3*((d*x + c)*b - b* c + a*d)/d)/d^2 - 18000*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(((d*x + c)*b - b*c + a*d)/d)/d^2 - 540*sqrt(2)*sqrt(d*x + c)*b^3*sin(5*((d*x + c)*b - b*c + a*d)/d)/d + 1500*sqrt(2)*sqrt(d*x + c)*b^3*sin(3*((d*x + c)*b - b*c + a*d) /d)/d + 27000*sqrt(2)*sqrt(d*x + c)*b^3*sin(((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))*er f(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)) + 67 50*((I - 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (I + 1)*sqr t(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^2...
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 2314, normalized size of antiderivative = 4.33 \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
1/144000*(300*(30*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*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*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*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*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*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)))*c^2 + d^2*(2250*(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 + 125*(sqrt(6)*sqrt(pi)*(12*b^2 *c^2 - 4*I*b*c*d - d^2)*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...
Timed out. \[ \int (c+d x)^{3/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]