\(\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx\) [200]

Optimal result

Integrand size = 26, antiderivative size = 351 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2} \]

[Out]

-3/64*(d*x+c)^(3/2)*cos(2*b*x+2*a)/b+1/192*(d*x+c)^(3/2)*cos(6*b*x+6*a)/b+1/4608*d^(3/2)*cos(6*a-6*b*c/d)*Fres
nelS(2*b^(1/2)*3^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*3^(1/2)*Pi^(1/2)/b^(5/2)+1/4608*d^(3/2)*FresnelC(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)*3^(1/2)*Pi^(1/2)/b^(5/2)-9/512*d^(3/2)*cos(2*a-2*
b*c/d)*FresnelS(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(5/2)-9/512*d^(3/2)*FresnelC(2*b^(1/2)*(d
*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*sin(2*a-2*b*c/d)*Pi^(1/2)/b^(5/2)+9/256*d*sin(2*b*x+2*a)*(d*x+c)^(1/2)/b^2-1/768
*d*sin(6*b*x+6*a)*(d*x+c)^(1/2)/b^2

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {\sqrt {\frac {\pi }{3}} d^{3/2} \sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 \sqrt {\pi } d^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {\sqrt {\frac {\pi }{3}} d^{3/2} \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1536 b^{5/2}}-\frac {9 \sqrt {\pi } d^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}-\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b} \]

[In]

Int[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(-3*(c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(64*b) + ((c + d*x)^(3/2)*Cos[6*a + 6*b*x])/(192*b) + (d^(3/2)*Sqrt[Pi/3
]*Cos[6*a - (6*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1536*b^(5/2)) - (9*d^(3/2)*Sqr
t[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(512*b^(5/2)) + (d^(3/2)*Sq
rt[Pi/3]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(1536*b^(5/2)) - (9*d^(3
/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(512*b^(5/2)) + (9*d
*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(256*b^2) - (d*Sqrt[c + d*x]*Sin[6*a + 6*b*x])/(768*b^2)

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[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]

Rule 3432

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]

Rule 3433

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]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{32} (c+d x)^{3/2} \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^{3/2} \sin (6 a+6 b x)\right ) \, dx \\ & = -\left (\frac {1}{32} \int (c+d x)^{3/2} \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^{3/2} \sin (2 a+2 b x) \, dx \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}-\frac {d \int \sqrt {c+d x} \cos (6 a+6 b x) \, dx}{128 b}+\frac {(9 d) \int \sqrt {c+d x} \cos (2 a+2 b x) \, dx}{128 b} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac {d^2 \int \frac {\sin (6 a+6 b x)}{\sqrt {c+d x}} \, dx}{1536 b^2}-\frac {\left (9 d^2\right ) \int \frac {\sin (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{512 b^2} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac {\left (d^2 \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{1536 b^2}-\frac {\left (9 d^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{512 b^2}+\frac {\left (d^2 \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{1536 b^2}-\frac {\left (9 d^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{512 b^2} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac {\left (d \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{768 b^2}-\frac {\left (9 d \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2}+\frac {\left (d \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{768 b^2}-\frac {\left (9 d \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2} \\ & = -\frac {3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \cos \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 )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \cos \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 )}{512 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{3}} \operatorname {FresnelC}\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 )}{1536 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{512 b^{5/2}}+\frac {9 d \sqrt {c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac {d \sqrt {c+d x} \sin (6 a+6 b x)}{768 b^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.74 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {i e^{-\frac {6 i (b c+a d)}{d}} (c+d x)^{5/2} \left (-81 e^{4 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {2 i b (c+d x)}{d}\right )+81 e^{4 i a+\frac {8 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {2 i b (c+d x)}{d}\right )+\sqrt {3} \left (e^{12 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {6 i b (c+d x)}{d}\right )-e^{\frac {12 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {6 i b (c+d x)}{d}\right )\right )\right )}{6912 \sqrt {2} d \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \]

[In]

Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

((I/6912)*(c + d*x)^(5/2)*(-81*E^((4*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d*x))/d]*Gamma[5/2, ((-2*I)*b*(c + d*x
))/d] + 81*E^((4*I)*a + ((8*I)*b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[5/2, ((2*I)*b*(c + d*x))/d] + Sqrt[3]*
(E^((12*I)*a)*Sqrt[(I*b*(c + d*x))/d]*Gamma[5/2, ((-6*I)*b*(c + d*x))/d] - E^(((12*I)*b*c)/d)*Sqrt[((-I)*b*(c
+ d*x))/d]*Gamma[5/2, ((6*I)*b*(c + d*x))/d])))/(Sqrt[2]*d*E^(((6*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^(
3/2))

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{64 b}}{d}\)	\(383\)
default	\(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{64 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 a d -2 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{192 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {6 b \left (d x +c \right )}{d}+\frac {6 a d -6 c b}{d}\right )}{12 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {6 a d -6 c b}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {6}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{64 b}}{d}\)	\(383\)

[In]

int((d*x+c)^(3/2)*cos(b*x+a)^3*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2/d*(-3/128/b*d*(d*x+c)^(3/2)*cos(2*b/d*(d*x+c)+2*(a*d-b*c)/d)+9/128/b*d*(1/4/b*d*(d*x+c)^(1/2)*sin(2*b/d*(d*x
+c)+2*(a*d-b*c)/d)-1/8/b*d*Pi^(1/2)/(b/d)^(1/2)*(cos(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^
(1/2)/d)+sin(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))+1/384/b*d*(d*x+c)^(3/2)*cos(6
*b/d*(d*x+c)+6*(a*d-b*c)/d)-1/128/b*d*(1/12/b*d*(d*x+c)^(1/2)*sin(6*b/d*(d*x+c)+6*(a*d-b*c)/d)-1/144/b*d*2^(1/
2)*Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(cos(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)+sin(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))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.93 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {\sqrt {3} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {3} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 81 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 81 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 96 \, {\left (8 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{6} - 12 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} + 2 \, b^{2} d x + 2 \, b^{2} c - {\left (2 \, b d \cos \left (b x + a\right )^{5} - 2 \, b d \cos \left (b x + a\right )^{3} - 3 \, b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{4608 \, b^{3}} \]

[In]

integrate((d*x+c)^(3/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4608*(sqrt(3)*pi*d^2*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d))
) + sqrt(3)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/d) -
81*pi*d^2*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d))) - 81*pi*d^2*sqrt(b/
(pi*d))*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) + 96*(8*(b^2*d*x + b^2*c)*cos(b*x +
a)^6 - 12*(b^2*d*x + b^2*c)*cos(b*x + a)^4 + 2*b^2*d*x + 2*b^2*c - (2*b*d*cos(b*x + a)^5 - 2*b*d*cos(b*x + a)^
3 - 3*b*d*cos(b*x + a))*sin(b*x + a))*sqrt(d*x + c))/b^3

Sympy [F]

\[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**(3/2)*cos(b*x+a)**3*sin(b*x+a)**3,x)

[Out]

Integral((c + d*x)**(3/2)*sin(a + b*x)**3*cos(a + b*x)**3, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.46 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (\frac {192 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {1728 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - 48 \, \sqrt {d x + c} b^{2} \sin \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 1296 \, \sqrt {d x + c} b^{2} \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + {\left (\left (i + 1\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \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 d \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 ) + 81 \, {\left (-\left (i + 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \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 d \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 ) + 81 \, {\left (\left (i - 1\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \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 d \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 d \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 d \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}{36864 \, b^{4}} \]

[In]

integrate((d*x+c)^(3/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/36864*(192*(d*x + c)^(3/2)*b^3*cos(6*((d*x + c)*b - b*c + a*d)/d)/d - 1728*(d*x + c)^(3/2)*b^3*cos(2*((d*x +
c)*b - b*c + a*d)/d)/d - 48*sqrt(d*x + c)*b^2*sin(6*((d*x + c)*b - b*c + a*d)/d) + 1296*sqrt(d*x + c)*b^2*sin
(2*((d*x + c)*b - b*c + a*d)/d) + ((I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d)
- (I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d
)) + 81*(-(I + 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) + (I - 1)*4^(1/4)*sqrt(2)
*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) + 81*((I - 1)*4^(1/4)*sq
rt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) - (I + 1)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4
)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-2*I*b/d)) + (-(I - 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)
^(1/4)*cos(-6*(b*c - a*d)/d) + (I + 1)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*er
f(sqrt(d*x + c)*sqrt(-6*I*b/d)))*d/b^4

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.79 (sec) , antiderivative size = 1514, normalized size of antiderivative = 4.31 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(3/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="giac")

[Out]

-1/9216*(48*(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(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)) - 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)) - 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^2 + d^2*((sqrt(3)*sqrt(pi)*(48*b^2*c^2 - 8*I*b*c*d - d^2)*d*er
f(-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/s
qrt(b^2*d^2) + 1)*b^2) - 6*I*(-4*I*(d*x + c)^(3/2)*b*d + 8*I*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*d^2)*e^(-6*(-
I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + (sqrt(3)*sqrt(pi)*(48*b^2*c^2 + 8*I*b*c*d - d^2)*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^2) - 6*I*(-4*I*(d*x + c)^(3/2)*b*d + 8*I*sqrt(d*x + c)*b*c*d - sqrt(d*x + c)*d^2)*e^(-6*(I*(d*x + c)
*b - I*b*c + I*a*d)/d)/b^2)/d^2 - 27*(sqrt(pi)*(16*b^2*c^2 - 8*I*b*c*d - 3*d^2)*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^2) + 2*I*(4*I*
(d*x + c)^(3/2)*b*d - 8*I*sqrt(d*x + c)*b*c*d - 3*sqrt(d*x + c)*d^2)*e^(-2*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)
/b^2)/d^2 - 27*(sqrt(pi)*(16*b^2*c^2 + 8*I*b*c*d - 3*d^2)*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^2) + 2*I*(4*I*(d*x + c)^(3/2)*b*d
- 8*I*sqrt(d*x + c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^(-2*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2)/d^2) - 8*(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) - 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 + c)*b +
I*b*c - I*a*d)/d)/b)*c)/d

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]

[In]

int(cos(a + b*x)^3*sin(a + b*x)^3*(c + d*x)^(3/2),x)

[Out]

int(cos(a + b*x)^3*sin(a + b*x)^3*(c + d*x)^(3/2), x)