Integrand size = 26, antiderivative size = 299 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\sqrt {d} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}+\frac {3 \sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{3}} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{384 b^{3/2}}-\frac {3 \sqrt {d} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{128 b^{3/2}} \]
-1/1152*cos(6*a-6*b*c/d)*FresnelC(2*b^(1/2)*3^(1/2)/Pi^(1/2)*(d*x+c)^(1/2) /d^(1/2))*d^(1/2)*3^(1/2)*Pi^(1/2)/b^(3/2)+1/1152*FresnelS(2*b^(1/2)*3^(1/ 2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(6*a-6*b*c/d)*d^(1/2)*3^(1/2)*Pi^(1/ 2)/b^(3/2)+3/128*cos(2*a-2*b*c/d)*FresnelC(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2) /Pi^(1/2))*d^(1/2)*Pi^(1/2)/b^(3/2)-3/128*FresnelS(2*b^(1/2)*(d*x+c)^(1/2) /d^(1/2)/Pi^(1/2))*sin(2*a-2*b*c/d)*d^(1/2)*Pi^(1/2)/b^(3/2)-3/64*cos(2*b* x+2*a)*(d*x+c)^(1/2)/b+1/192*cos(6*b*x+6*a)*(d*x+c)^(1/2)/b
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.86 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {e^{-\frac {6 i (b c+a d)}{d}} \sqrt {c+d x} \left (-27 e^{4 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {2 i b (c+d x)}{d}\right )-27 e^{4 i a+\frac {8 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {2 i b (c+d x)}{d}\right )+\sqrt {3} \left (e^{12 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},-\frac {6 i b (c+d x)}{d}\right )+e^{\frac {12 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {3}{2},\frac {6 i b (c+d x)}{d}\right )\right )\right )}{1152 \sqrt {2} b \sqrt {\frac {b^2 (c+d x)^2}{d^2}}} \]
(Sqrt[c + d*x]*(-27*E^((4*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d*x))/d]*Gamm a[3/2, ((-2*I)*b*(c + d*x))/d] - 27*E^((4*I)*a + ((8*I)*b*c)/d)*Sqrt[((-I) *b*(c + d*x))/d]*Gamma[3/2, ((2*I)*b*(c + d*x))/d] + Sqrt[3]*(E^((12*I)*a) *Sqrt[(I*b*(c + d*x))/d]*Gamma[3/2, ((-6*I)*b*(c + d*x))/d] + E^(((12*I)*b *c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[3/2, ((6*I)*b*(c + d*x))/d])))/(11 52*Sqrt[2]*b*E^(((6*I)*(b*c + a*d))/d)*Sqrt[(b^2*(c + d*x)^2)/d^2])
Time = 0.66 (sec) , antiderivative size = 299, 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.
\(\displaystyle \int \sqrt {c+d x} \sin ^3(a+b x) \cos ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \int \left (\frac {3}{32} \sqrt {c+d x} \sin (2 a+2 b x)-\frac {1}{32} \sqrt {c+d x} \sin (6 a+6 b x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}+\frac {3 \sqrt {\pi } \sqrt {d} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}-\frac {3 \sqrt {\pi } \sqrt {d} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}\) |
(-3*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b) + (Sqrt[c + d*x]*Cos[6*a + 6*b* x])/(192*b) - (Sqrt[d]*Sqrt[Pi/3]*Cos[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b] *Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(384*b^(3/2)) + (3*Sqrt[d]*Sqrt[Pi]*C os[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])] )/(128*b^(3/2)) + (Sqrt[d]*Sqrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[ c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(384*b^(3/2)) - (3*Sqrt[d]*Sqrt[P i]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c )/d])/(128*b^(3/2))
3.2.98.3.1 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 = 0.83 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {3 d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {3 d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{128 b \sqrt {\frac {b}{d}}}+\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{2304 b \sqrt {\frac {b}{d}}}}{d}\) | \(293\) |
default | \(\frac {-\frac {3 d \sqrt {d x +c}\, \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {3 d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{128 b \sqrt {\frac {b}{d}}}+\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{2304 b \sqrt {\frac {b}{d}}}}{d}\) | \(293\) |
2/d*(-3/128/b*d*(d*x+c)^(1/2)*cos(2*b/d*(d*x+c)+2*(a*d-b*c)/d)+3/256/b*d*P i^(1/2)/(b/d)^(1/2)*(cos(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*b* (d*x+c)^(1/2)/d)-sin(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x +c)^(1/2)/d))+1/384/b*d*(d*x+c)^(1/2)*cos(6*b/d*(d*x+c)+6*(a*d-b*c)/d)-1/4 608/b*d*2^(1/2)*Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(cos(6*(a*d-b*c)/d)*FresnelC( 2^(1/2)/Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin(6*(a*d-b*c)/d) *FresnelS(2^(1/2)/Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))
Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.81 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {\sqrt {3} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \sqrt {3} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 27 \, \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 27 \, \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 48 \, {\left (4 \, b \cos \left (b x + a\right )^{6} - 6 \, b \cos \left (b x + a\right )^{4} + b\right )} \sqrt {d x + c}}{1152 \, b^{2}} \]
-1/1152*(sqrt(3)*pi*d*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_cos(2*s qrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d))) - sqrt(3)*pi*d*sqrt(b/(pi*d))*fresnel _sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/d) - 27*pi *d*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_cos(2*sqrt(d*x + c)*sqrt(b /(pi*d))) + 27*pi*d*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi* d)))*sin(-2*(b*c - a*d)/d) - 48*(4*b*cos(b*x + a)^6 - 6*b*cos(b*x + a)^4 + b)*sqrt(d*x + c))/b^2
\[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int \sqrt {c + d x} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.47 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (\frac {48 \, \sqrt {d x + c} b^{2} \cos \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {432 \, \sqrt {d x + c} b^{2} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - {\left (-\left (i - 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {6 i \, b}{d}}\right ) - 27 \, {\left (\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \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 {2} \sqrt {\pi } b \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 ) - 27 \, {\left (-\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \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 {2} \sqrt {\pi } b \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 ) - {\left (\left (i + 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {6 i \, b}{d}}\right )\right )} d}{9216 \, b^{3}} \]
1/9216*(48*sqrt(d*x + c)*b^2*cos(6*((d*x + c)*b - b*c + a*d)/d)/d - 432*sq rt(d*x + c)*b^2*cos(2*((d*x + c)*b - b*c + a*d)/d)/d - (-(I - 1)*36^(1/4)* sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d) - (I + 1)*36^(1/4 )*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d)) - 27*((I - 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4 )*cos(-2*(b*c - a*d)/d) + (I + 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/ 4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) - 27*(-(I + 1)* 4^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) - (I - 1) *4^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqr t(d*x + c)*sqrt(-2*I*b/d)) - ((I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d^2 )^(1/4)*cos(-6*(b*c - a*d)/d) + (I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*(b^2/d ^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-6*I*b/d)))*d/b^3
Result contains complex when optimal does not.
Time = 1.16 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.76 \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]
1/2304*(sqrt(3)*sqrt(pi)*(12*b*c - I*d)*d*erf(-I*sqrt(3)*sqrt(b*d)*sqrt(d* x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I *b*d/sqrt(b^2*d^2) + 1)*b) + sqrt(3)*sqrt(pi)*(12*b*c + I*d)*d*erf(I*sqrt( 3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6*(-I*b*c + I *a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 12*(sqrt(3)*sqrt(pi)*d *erf(-I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-6 *(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + sqrt(3)*sqrt(p i)*d*erf(I*sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e ^(-6*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) - 9*sqrt(p i)*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)) - 9*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 - 27*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) - 27*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) - 54*sqrt (d*x + c)*d*e^(-2*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 6*sqrt(d*x + c)*d *e^(-6*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^(-2*(-I *(d*x + c)*b + I*b*c - I*a*d)/d)/b + 6*sqrt(d*x + c)*d*e^(-6*(-I*(d*x +...
Timed out. \[ \int \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,\sqrt {c+d\,x} \,d x \]