Integrand size = 26, antiderivative size = 459 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{10}} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{80 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{10}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{80 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{48 b^{3/2}}-\frac {\sqrt {d} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{8 b^{3/2}}+\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b} \]
1/800*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))*d^(1/2)*10^(1/2)*Pi^(1/2)/b^(3/2)+1/800*FresnelC(b^(1/2)*10^(1/2)/P i^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(5*a-5*b*c/d)*d^(1/2)*10^(1/2)*Pi^(1/2)/ b^(3/2)+1/288*cos(3*a-3*b*c/d)*FresnelS(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^( 1/2)/d^(1/2))*d^(1/2)*6^(1/2)*Pi^(1/2)/b^(3/2)+1/288*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)*d^(1/2)*6^(1/2)*Pi^(1 /2)/b^(3/2)-1/16*cos(a-b*c/d)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1 /2)/d^(1/2))*d^(1/2)*2^(1/2)*Pi^(1/2)/b^(3/2)-1/16*FresnelC(b^(1/2)*2^(1/2 )/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*d^(1/2)*2^(1/2)*Pi^(1/2)/b^ (3/2)+1/8*sin(b*x+a)*(d*x+c)^(1/2)/b-1/48*sin(3*b*x+3*a)*(d*x+c)^(1/2)/b-1 /80*sin(5*b*x+5*a)*(d*x+c)^(1/2)/b
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.81 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {i e^{-\frac {5 i (b c+a d)}{d}} \sqrt {c+d x} \left (-450 e^{6 i a+\frac {4 i b c}{d}} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )+450 e^{4 i a+\frac {6 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )+25 \sqrt {3} e^{2 i \left (4 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {3 i b (c+d x)}{d}\right )-25 \sqrt {3} e^{2 i a+\frac {8 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {3 i b (c+d x)}{d}\right )+9 \sqrt {5} e^{10 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {5 i b (c+d x)}{d}\right )-9 \sqrt {5} e^{\frac {10 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {5 i b (c+d x)}{d}\right )\right )}{7200 b \sqrt {\frac {b^2 (c+d x)^2}{d^2}}} \]
((I/7200)*Sqrt[c + d*x]*(-450*E^((6*I)*a + ((4*I)*b*c)/d)*Sqrt[(I*b*(c + d *x))/d]*Gamma[3/2, ((-I)*b*(c + d*x))/d] + 450*E^((4*I)*a + ((6*I)*b*c)/d) *Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, (I*b*(c + d*x))/d] + 25*Sqrt[3]*E^( (2*I)*(4*a + (b*c)/d))*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, ((-3*I)*b*(c + d *x))/d] - 25*Sqrt[3]*E^((2*I)*a + ((8*I)*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d ]*Gamma[3/2, ((3*I)*b*(c + d*x))/d] + 9*Sqrt[5]*E^((10*I)*a)*Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, ((-5*I)*b*(c + d*x))/d] - 9*Sqrt[5]*E^(((10*I)*b*c)/ d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, ((5*I)*b*(c + d*x))/d]))/(b*E^((( 5*I)*(b*c + a*d))/d)*Sqrt[(b^2*(c + d*x)^2)/d^2])
Time = 1.03 (sec) , antiderivative size = 459, 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.
| \(\displaystyle \int \sqrt {c+d x} \sin ^2(a+b x) \cos ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {1}{8} \sqrt {c+d x} \cos (a+b x)-\frac {1}{16} \sqrt {c+d x} \cos (3 a+3 b x)-\frac {1}{16} \sqrt {c+d x} \cos (5 a+5 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\frac {\pi }{10}} \sqrt {d} \sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{80 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {\sqrt {\frac {\pi }{10}} \sqrt {d} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{80 b^{3/2}}+\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b}\) |
-1/8*(Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqr t[c + d*x])/Sqrt[d]])/b^(3/2) + (Sqrt[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*F resnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (Sqrt[ d]*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(80*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqr t[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(80*b^(3/2)) + (Sqr t[d]*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3 *a - (3*b*c)/d])/(48*b^(3/2)) - (Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt [2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(3/2)) + (Sqrt[c + d *x]*Sin[a + b*x])/(8*b) - (Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(48*b) - (Sqrt[ c + d*x]*Sin[5*a + 5*b*x])/(80*b)
3.2.92.3.1 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 = 0.85 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(\frac {\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{16 b \sqrt {\frac {b}{d}}}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{288 b \sqrt {\frac {b}{d}}}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{800 b \sqrt {\frac {b}{d}}}}{d}\) | \(444\) |
| default | \(\frac {\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{16 b \sqrt {\frac {b}{d}}}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{288 b \sqrt {\frac {b}{d}}}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 a d -5 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{800 b \sqrt {\frac {b}{d}}}}{d}\) | \(444\) |
2/d*(1/16/b*d*(d*x+c)^(1/2)*sin(b/d*(d*x+c)+(a*d-b*c)/d)-1/32/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)^(1/2)*sin(3*b/d*(d*x+c)+3*(a*d-b*c )/d)+1/576/b*d*2^(1/2)*Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(cos(3*(a*d-b*c)/d)*Fr esnelS(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)^(1/2)*sin(5*b/d*(d*x+c)+5*(a*d-b*c)/d)+1/1600/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 = 0.28 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.80 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {9 \, \sqrt {10} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 25 \, \sqrt {6} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 450 \, \sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 450 \, \sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 25 \, \sqrt {6} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 9 \, \sqrt {10} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) - 480 \, {\left (3 \, b \cos \left (b x + a\right )^{4} - b \cos \left (b x + a\right )^{2} - 2 \, b\right )} \sqrt {d x + c} \sin \left (b x + a\right )}{7200 \, b^{2}} \]
1/7200*(9*sqrt(10)*pi*d*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_sin(s qrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d))) + 25*sqrt(6)*pi*d*sqrt(b/(pi*d))*cos (-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 450 *sqrt(2)*pi*d*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt( d*x + c)*sqrt(b/(pi*d))) - 450*sqrt(2)*pi*d*sqrt(b/(pi*d))*fresnel_cos(sqr t(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) + 25*sqrt(6)*pi*d*s qrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b* c - a*d)/d) + 9*sqrt(10)*pi*d*sqrt(b/(pi*d))*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) - 480*(3*b*cos(b*x + a)^4 - b* cos(b*x + a)^2 - 2*b)*sqrt(d*x + c)*sin(b*x + a))/b^2
\[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\int \sqrt {c + d x} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.48 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {\sqrt {2} {\left (\frac {180 \, \sqrt {2} \sqrt {d x + c} b^{3} \sin \left (\frac {5 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d^{2}} + \frac {300 \, \sqrt {2} \sqrt {d x + c} b^{3} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d^{2}} - \frac {1800 \, \sqrt {2} \sqrt {d x + c} b^{3} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}{d^{2}} - 9 \, {\left (\frac {\left (i + 1\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d} - \frac {\left (i - 1\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {5 i \, b}{d}}\right ) - 25 \, {\left (\frac {\left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d} - \frac {\left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {3 i \, b}{d}}\right ) - 450 \, {\left (-\frac {\left (i + 1\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right )}{d} + \frac {\left (i - 1\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) - 450 \, {\left (\frac {\left (i - 1\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right )}{d} - \frac {\left (i + 1\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) - 25 \, {\left (-\frac {\left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d} + \frac {\left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {3 i \, b}{d}}\right ) - 9 \, {\left (-\frac {\left (i - 1\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d} + \frac {\left (i + 1\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {5 i \, b}{d}}\right )\right )} d^{2}}{28800 \, b^{4}} \]
-1/28800*sqrt(2)*(180*sqrt(2)*sqrt(d*x + c)*b^3*sin(5*((d*x + c)*b - b*c + a*d)/d)/d^2 + 300*sqrt(2)*sqrt(d*x + c)*b^3*sin(3*((d*x + c)*b - b*c + a* d)/d)/d^2 - 1800*sqrt(2)*sqrt(d*x + c)*b^3*sin(((d*x + c)*b - b*c + a*d)/d )/d^2 - 9*((I + 1)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d )/d)/d - (I - 1)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/ d)/d)*erf(sqrt(d*x + c)*sqrt(5*I*b/d)) - 25*((I + 1)*9^(1/4)*sqrt(pi)*b^2* (b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d)/d - (I - 1)*9^(1/4)*sqrt(pi)*b^2*(b^ 2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) - 4 50*(-(I + 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d)/d + (I - 1)* sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt (I*b/d)) - 450*((I - 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d)/d - (I + 1)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d)/d)*erf(sqrt(d* x + c)*sqrt(-I*b/d)) - 25*(-(I - 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*c os(-3*(b*c - a*d)/d)/d + (I + 1)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin( -3*(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)) - 9*(-(I - 1)*25^(1 /4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d)/d + (I + 1)*25^(1/4 )*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)* sqrt(-5*I*b/d)))*d^2/b^4
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 1270, normalized size of antiderivative = 2.77 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]
-1/14400*(-450*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)/(sqr t(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 25*I*sqrt(6)*sqrt(pi)*(6*b*c - I*d) *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)*b) - 9*I*sq rt(10)*sqrt(pi)*(10*b*c - I*d)*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)*b) + 450*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) + 25*I*sqrt(6)*sqrt(pi) *(6*b*c + I*d)*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)*b) + 9*I*sqrt(10)*sqrt(pi)*(10*b*c + I*d)*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)*b) + 30*(30*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)) + 5*I*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*I* sqrt(10)*sqrt(pi)*d*erf(-1/2*I*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/...
Timed out. \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^2\,\sqrt {c+d\,x} \,d x \]