\(\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx\) [191]

Optimal result

Integrand size = 26, antiderivative size = 534 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {3 d \sqrt {c+d x} \cos (a+b x)}{16 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \cos (5 a+5 b x)}{800 b^2}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{800 b^{5/2}}-\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{96 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{16 b^{5/2}}+\frac {(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b} \]

[Out]

1/8*(d*x+c)^(3/2)*sin(b*x+a)/b-1/48*(d*x+c)^(3/2)*sin(3*b*x+3*a)/b-1/80*(d*x+c)^(3/2)*sin(5*b*x+5*a)/b+3/8000*
d^(3/2)*cos(5*a-5*b*c/d)*FresnelC(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)*FresnelS(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)*FresnelC(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)*FresnelS(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)*FresnelC(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)*FresnelS(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*cos(b*x+a)*(d*x+c)^(1/2)/b^2-1/96*d*cos(3*b*x+3*a)*(d*x+c)^(1/2)/b^2-3/800*d*cos(5*b*x+5*a)*(d*x
+c)^(1/2)/b^2

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}+\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}+\frac {3 d \sqrt {c+d x} \cos (a+b x)}{16 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \cos (5 a+5 b x)}{800 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b} \]

[In]

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

[Out]

(3*d*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^2) - (d*Sqrt[c + d*x]*Cos[3*a + 3*b*x])/(96*b^2) - (3*d*Sqrt[c + d*x]*C
os[5*a + 5*b*x])/(800*b^2) - (3*d^(3/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(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]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x]
)/Sqrt[d]])/(96*b^(5/2)) + (3*d^(3/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c +
d*x])/Sqrt[d]])/(800*b^(5/2)) - (3*d^(3/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*S
in[5*a - (5*b*c)/d])/(800*b^(5/2)) - (d^(3/2)*Sqrt[Pi/6]*FresnelS[(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]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]
]*Sin[a - (b*c)/d])/(16*b^(5/2)) + ((c + d*x)^(3/2)*Sin[a + b*x])/(8*b) - ((c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(
48*b) - ((c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(80*b)

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 {1}{8} (c+d x)^{3/2} \cos (a+b x)-\frac {1}{16} (c+d x)^{3/2} \cos (3 a+3 b x)-\frac {1}{16} (c+d x)^{3/2} \cos (5 a+5 b x)\right ) \, dx \\ & = -\left (\frac {1}{16} \int (c+d x)^{3/2} \cos (3 a+3 b x) \, dx\right )-\frac {1}{16} \int (c+d x)^{3/2} \cos (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^{3/2} \cos (a+b x) \, dx \\ & = \frac {(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b}+\frac {(3 d) \int \sqrt {c+d x} \sin (5 a+5 b x) \, dx}{160 b}+\frac {d \int \sqrt {c+d x} \sin (3 a+3 b x) \, dx}{32 b}-\frac {(3 d) \int \sqrt {c+d x} \sin (a+b x) \, dx}{16 b} \\ & = \frac {3 d \sqrt {c+d x} \cos (a+b x)}{16 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \cos (5 a+5 b x)}{800 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b}+\frac {\left (3 d^2\right ) \int \frac {\cos (5 a+5 b x)}{\sqrt {c+d x}} \, dx}{1600 b^2}+\frac {d^2 \int \frac {\cos (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{192 b^2}-\frac {\left (3 d^2\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{32 b^2} \\ & = \frac {3 d \sqrt {c+d x} \cos (a+b x)}{16 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \cos (5 a+5 b x)}{800 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b}+\frac {\left (3 d^2 \cos \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {5 b c}{d}+5 b x\right )}{\sqrt {c+d x}} \, dx}{1600 b^2}+\frac {\left (d^2 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{192 b^2}-\frac {\left (3 d^2 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{32 b^2}-\frac {\left (3 d^2 \sin \left (5 a-\frac {5 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {5 b c}{d}+5 b x\right )}{\sqrt {c+d x}} \, dx}{1600 b^2}-\frac {\left (d^2 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{192 b^2}+\frac {\left (3 d^2 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{32 b^2} \\ & = \frac {3 d \sqrt {c+d x} \cos (a+b x)}{16 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \cos (5 a+5 b x)}{800 b^2}+\frac {(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b}+\frac {\left (3 d \cos \left (5 a-\frac {5 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {5 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{800 b^2}+\frac {\left (d \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{96 b^2}-\frac {\left (3 d \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^2}-\frac {\left (3 d \sin \left (5 a-\frac {5 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {5 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{800 b^2}-\frac {\left (d \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{96 b^2}+\frac {\left (3 d \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{16 b^2} \\ & = \frac {3 d \sqrt {c+d x} \cos (a+b x)}{16 b^2}-\frac {d \sqrt {c+d x} \cos (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \cos (5 a+5 b x)}{800 b^2}-\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \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 {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \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 d^{3/2} \sqrt {\frac {\pi }{10}} \cos \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 {3 d^{3/2} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\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 {FresnelS}\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 {FresnelS}\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 {(c+d x)^{3/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{3/2} \sin (5 a+5 b x)}{80 b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.76 (sec) , antiderivative size = 1088, normalized size of antiderivative = 2.04 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(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 (1+2 i b x+e^{6 i (a+b x)} (1-2 i 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 b c-i 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+10 i b x+e^{10 i (a+b x)} (3-10 i 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 b c-3 i 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 {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 {c e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^{3/2} \left (-\frac {e^{6 i a} \Gamma \left (\frac {3}{2},-\frac {3 i b (c+d x)}{d}\right )}{\left (-\frac {i b (c+d x)}{d}\right )^{3/2}}-\frac {e^{\frac {6 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {3 i b (c+d x)}{d}\right )}{\left (\frac {i b (c+d x)}{d}\right )^{3/2}}\right )}{96 \sqrt {3} d}-\frac {c e^{-\frac {5 i (b c+a d)}{d}} (c+d x)^{3/2} \left (-\frac {e^{10 i a} \Gamma \left (\frac {3}{2},-\frac {5 i b (c+d x)}{d}\right )}{\left (-\frac {i b (c+d x)}{d}\right )^{3/2}}-\frac {e^{\frac {10 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {5 i b (c+d x)}{d}\right )}{\left (\frac {i b (c+d x)}{d}\right )^{3/2}}\right )}{160 \sqrt {5} d}+\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}} (3-2 i b x) \sqrt {c+d x}+\sqrt [4]{-1} (-2 b c+3 i d) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt [4]{-1} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )\right )+\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}} \]

[In]

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

[Out]

-1/2304*(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]*Sqr
t[c + d*x])/Sqrt[d]] - (1 + I)*(2*b*c - I*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]]))/(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)*Sqrt[c + d*x]*(3 + (10*I)*b*x + E^((10*I)*(a + b*x))*(3 - (10*I)*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*b*
c - (3*I)*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]]))/(3200
0*b^(5/2)*E^(((5*I)*(a*d + b*(c + d*x)))/d)) + (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]))/(16*b^2*E^((I*(b*c
+ a*d))/d)*Sqrt[c + d*x]) - (c*(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)))/(96*Sqr
t[3]*d*E^(((3*I)*(b*c + a*d))/d)) - (c*(c + d*x)^(3/2)*(-((E^((10*I)*a)*Gamma[3/2, ((-5*I)*b*(c + d*x))/d])/((
(-I)*b*(c + d*x))/d)^(3/2)) - (E^(((10*I)*b*c)/d)*Gamma[3/2, ((5*I)*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^(3/2)
))/(160*Sqrt[5]*d*E^(((5*I)*(b*c + a*d))/d)) + (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*Sqrt[b]*Sqrt[d]*(3 + (2*I)*b*x)*Sqrt[c + d*x] + (1 + I)*(2*b*c + (3*I)*d)*Sqrt[Pi/2]*E
rf[((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])))/(64*b^(5/2))

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}\)	\(583\)
default	\(\frac {\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}} \sin \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}\, \cos \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 {FresnelC}\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 {FresnelS}\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}\)	\(583\)

[In]

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

[Out]

2/d*(1/16/b*d*(d*x+c)^(3/2)*sin(b/d*(d*x+c)+(a*d-b*c)/d)-3/16/b*d*(-1/2/b*d*(d*x+c)^(1/2)*cos(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)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c
)^(1/2)/d)-sin((a*d-b*c)/d)*FresnelS(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)*
sin(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)*cos(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)*FresnelC(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)*FresnelS(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)*sin(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)*cos(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)*FresnelC(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)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.84 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) - 480 \, {\left (9 \, b d \cos \left (b x + a\right )^{5} - 5 \, b d \cos \left (b x + a\right )^{3} - 30 \, b d \cos \left (b x + a\right ) + 10 \, {\left (3 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} - 2 \, b^{2} d x - 2 \, b^{2} c - {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{72000 \, b^{3}} \]

[In]

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

[Out]

1/72000*(27*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_cos(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_cos(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_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))
+ 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fresnel_sin(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_sin(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_sin(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) - 480
*(9*b*d*cos(b*x + a)^5 - 5*b*d*cos(b*x + a)^3 - 30*b*d*cos(b*x + a) + 10*(3*(b^2*d*x + b^2*c)*cos(b*x + a)^4 -
2*b^2*d*x - 2*b^2*c - (b^2*d*x + b^2*c)*cos(b*x + a)^2)*sin(b*x + a))*sqrt(d*x + c))/b^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 760, normalized size of antiderivative = 1.42 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/288000*sqrt(2)*(1800*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(5*((d*x + c)*b - b*c + a*d)/d)/d^2 + 3000*sqrt(2)*(d*x
+ c)^(3/2)*b^4*sin(3*((d*x + c)*b - b*c + a*d)/d)/d^2 - 18000*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(((d*x + c)*b -
b*c + a*d)/d)/d^2 + 540*sqrt(2)*sqrt(d*x + c)*b^3*cos(5*((d*x + c)*b - b*c + a*d)/d)/d + 1500*sqrt(2)*sqrt(d*x
+ c)*b^3*cos(3*((d*x + c)*b - b*c + a*d)/d)/d - 27000*sqrt(2)*sqrt(d*x + c)*b^3*cos(((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))*erf(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)*sq
rt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(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)) - 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)*s
qrt(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)*sq
rt(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/b^5

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 2319, normalized size of antiderivative = 4.34 \[ \int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/144000*(300*(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)*sqr
t(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/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*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/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*(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/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 + 125
*(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*sq
rt(10)*sqrt(pi)*(100*b^2*c^2 - 20*I*b*c*d - 3*d^2)*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^2) + 10*(-10*I*(d*x + c)^(3/2)
*b*d + 20*I*sqrt(d*x + c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^(-5*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2)/d^2 + 22
50*(-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/sq
rt(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*(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*(-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(
10)*sqrt(pi)*(100*b^2*c^2 + 20*I*b*c*d - 3*d^2)*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^2) + 10*(10*I*(d*x + c)^(3/2)*b
*d - 20*I*sqrt(d*x + c)*b*c*d + 3*sqrt(d*x + c)*d^2)*e^(-5*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2)/d^2) + 20*(
-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) + 450*I*sqrt(2)*sqr
t(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) + 900*I*sqrt(d*x + c)*d*e^((I*(d*
x + c)*b - I*b*c + I*a*d)/d)/b + 150*I*sqrt(d*x + c)*d*e^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 90*I*sqrt(
d*x + c)*d*e^(-5*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 900*I*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a
*d)/d)/b - 150*I*sqrt(d*x + c)*d*e^(-3*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b - 90*I*sqrt(d*x + c)*d*e^(-5*(-I*
(d*x + c)*b + I*b*c - I*a*d)/d)/b)*c)/d

Mupad [F(-1)]

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

[In]

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

[Out]

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