Integrand size = 24, antiderivative size = 406 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{144 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \]
5/8*d*(d*x+c)^(3/2)*cos(b*x+a)/b^2-5/72*d*(d*x+c)^(3/2)*cos(3*b*x+3*a)/b^2 +1/4*(d*x+c)^(5/2)*sin(b*x+a)/b-1/12*(d*x+c)^(5/2)*sin(3*b*x+3*a)/b-5/864* d^(5/2)*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))*6^(1/2)*Pi^(1/2)/b^(7/2)-5/864*d^(5/2)*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^(7/2)+15 /32*d^(5/2)*cos(a-b*c/d)*FresnelS(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)+15/32*d^(5/2)*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^(7/2)-15/16* d^2*sin(b*x+a)*(d*x+c)^(1/2)/b^3+5/144*d^2*sin(3*b*x+3*a)*(d*x+c)^(1/2)/b^ 3
Result contains complex when optimal does not.
Time = 12.50 (sec) , antiderivative size = 1418, normalized size of antiderivative = 3.49 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx =\text {Too large to display} \]
-1/288*(c*Sqrt[d]*(12*Sqrt[b]*Sqrt[d]*E^(((3*I)*b*c)/d)*Sqrt[c + d*x]*(1 + (2*I)*b*x + E^((6*I)*(a + b*x))*(1 - (2*I)*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*b*c - I*d)*E^((3*I)*(2*a + b*x))*Sqrt[6*P i]*Erfi[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]))/(b^(5/2)*E^(( (3*I)*(a*d + b*(c + d*x)))/d)) + (c^2*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]))/(8*b^2*E^((I*(b*c + a*d))/d)*Sqrt [c + d*x]) - (c^2*(c + d*x)^(3/2)*(-((E^((6*I)*a)*Gamma[3/2, ((-3*I)*b*(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^(3/2)) - (E^(((6*I)*b*c)/d)*Gamma[3/2, ((3*I)*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^(3/2)))/(24*Sqrt[3]*d*E^(((3*I )*(b*c + a*d))/d)) + (c*Sqrt[d]*(E^(I*(a - (b*c)/d))*(2*Sqrt[b]*Sqrt[d]*E^ ((I*b*(c + d*x))/d)*(3 - (2*I)*b*x)*Sqrt[c + d*x] + (-1)^(1/4)*(-2*b*c + ( 3*I)*d)*Sqrt[Pi]*Erfi[((-1)^(1/4)*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]) + (2*Sq rt[b]*Sqrt[d]*(3 + (2*I)*b*x)*Sqrt[c + d*x] + (1 + I)*(2*b*c + (3*I)*d)*Sq rt[Pi/2]*Erf[((1 + I)*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[2]*Sqrt[d])]*(Cos[b*(c/ d + x)] + I*Sin[b*(c/d + x)]))*(Cos[a + b*x] - I*Sin[a + b*x])))/(16*b^(5/ 2)) + (Sqrt[d]*((Cos[a - (b*c)/d] + I*Sin[a - (b*c)/d])*((1 + I)*(4*b^2*c^ 2 - (12*I)*b*c*d - 15*d^2)*Sqrt[Pi/2]*Erfi[((1 + I)*Sqrt[b]*Sqrt[c + d*x]) /(Sqrt[2]*Sqrt[d])] + 2*Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((15*I)*d - (4*I)...
Time = 1.36 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
| \(\displaystyle \int (c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x) \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {1}{4} (c+d x)^{5/2} \cos (a+b x)-\frac {1}{4} (c+d x)^{5/2} \cos (3 a+3 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{144 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{16 b^3}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}\) |
(5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(8*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(72*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqr t[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(16*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi /6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[ d]])/(144*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sq rt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2)*Sq rt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c )/d])/(16*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(16*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(1 44*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(12*b)
3.1.58.3.1 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 = 1.44 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.17
| method | result | size |
| derivativedivides | \(\frac {\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{4 b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{12 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\) | \(474\) |
| default | \(\frac {\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{4 b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{4 b}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{12 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{2 b}\right )}{12 b}}{d}\) | \(474\) |
2/d*(1/8/b*d*(d*x+c)^(5/2)*sin(b/d*(d*x+c)+(a*d-b*c)/d)-5/8/b*d*(-1/2/b*d* (d*x+c)^(3/2)*cos(b/d*(d*x+c)+(a*d-b*c)/d)+3/2/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))))-1/24 /b*d*(d*x+c)^(5/2)*sin(3*b/d*(d*x+c)+3*(a*d-b*c)/d)+5/24/b*d*(-1/6/b*d*(d* x+c)^(3/2)*cos(3*b/d*(d*x+c)+3*(a*d-b*c)/d)+1/2/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)))))
Time = 0.27 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.91 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 405 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 405 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 5 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left (10 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 30 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) - {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 35 \, b d^{2} - {\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{864 \, b^{4}} \]
-1/864*(5*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin( sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 405*sqrt(2)*pi*d^3*sqrt(b/(pi*d))* cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 40 5*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/( pi*d)))*sin(-(b*c - a*d)/d) + 5*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos( sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 24*(10*(b^2* d^2*x + b^2*c*d)*cos(b*x + a)^3 - 30*(b^2*d^2*x + b^2*c*d)*cos(b*x + a) - (12*b^3*d^2*x^2 + 24*b^3*c*d*x + 12*b^3*c^2 - 35*b*d^2 - (12*b^3*d^2*x^2 + 24*b^3*c*d*x + 12*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^2)*sin(b*x + a))*sqrt(d *x + c))/b^4
Timed out. \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=\text {Timed out} \]
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.35 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (240 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 2160 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right ) - 5 \, {\left (-\left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + \left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {3 i \, b}{d}}\right ) - 405 \, {\left (\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } b 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 {2} \sqrt {\pi } b 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 ) - 405 \, {\left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } b 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 {2} \sqrt {\pi } b 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 ) - 5 \, {\left (\left (i - 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \left (i + 1\right ) \cdot 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {3 i \, b}{d}}\right ) + 24 \, {\left (\frac {12 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 5 \, \sqrt {d x + c} b^{2} d\right )} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 216 \, {\left (\frac {4 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )\right )} d}{3456 \, b^{5}} \]
-1/3456*(240*(d*x + c)^(3/2)*b^3*cos(3*((d*x + c)*b - b*c + a*d)/d) - 2160 *(d*x + c)^(3/2)*b^3*cos(((d*x + c)*b - b*c + a*d)/d) - 5*(-(I + 1)*9^(1/4 )*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) + (I - 1)*9 ^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*erf(s qrt(d*x + c)*sqrt(3*I*b/d)) - 405*((I + 1)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2 )^(1/4)*cos(-(b*c - a*d)/d) - (I - 1)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/ 4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) - 405*(-(I - 1)*sqr t(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (I + 1)*sqrt(2)* sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt (-I*b/d)) - 5*((I - 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos( -3*(b*c - a*d)/d) - (I + 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4) *sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)) + 24*(12*(d*x + c)^(5/2)*b^4/d - 5*sqrt(d*x + c)*b^2*d)*sin(3*((d*x + c)*b - b*c + a*d)/d) - 216*(4*(d*x + c)^(5/2)*b^4/d - 15*sqrt(d*x + c)*b^2*d)*sin(((d*x + c)*b - b*c + a*d)/d))*d/b^5
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 2478, normalized size of antiderivative = 6.10 \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]
-1/1728*(72*(3*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(6)*sqrt(pi)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d)*s qrt(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(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(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)))*c^3 + 18*c*d^2*(9*(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)/(s qrt(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(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(b^2*d^2) + 1)*b^2) + 6*(-2* I*(d*x + c)^(3/2)*b*d + 4*I*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*d^2)*e^(-3 *(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + 9*(-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/s...
Timed out. \[ \int (c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx=\int \cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{5/2} \,d x \]