\(\int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx\) [135]

Optimal result

Integrand size = 26, antiderivative size = 615 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{32 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 \sqrt {c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}-\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {3 d^{5/2} \sqrt {\frac {\pi }{10}} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}-\frac {3 d^{5/2} \sqrt {\frac {\pi }{10}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{1600 b^{7/2}}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{576 b^{7/2}}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{32 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2} \]

[Out]

-1/8*(d*x+c)^(5/2)*cos(b*x+a)/b-1/48*(d*x+c)^(5/2)*cos(3*b*x+3*a)/b+1/80*(d*x+c)^(5/2)*cos(5*b*x+5*a)/b+5/16*d
*(d*x+c)^(3/2)*sin(b*x+a)/b^2+5/288*d*(d*x+c)^(3/2)*sin(3*b*x+3*a)/b^2-1/160*d*(d*x+c)^(3/2)*sin(5*b*x+5*a)/b^
2+3/16000*d^(5/2)*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^(7/2)-3/16000*d^(5/2)*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)*P
i^(1/2)/b^(7/2)-5/3456*d^(5/2)*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^(7/2)+5/3456*d^(5/2)*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^(7/2)-15/64*d^(5/2)*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^(7/2)+15/64*d^(5/2)*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^(7/2)+15/32*d^2*cos(b*x+a)*(d*x+c)^(1/2)/b^3+5/576*d^2*cos(3*b*x+3*a)*(d*x+c)^(1/2)/b^3-3/1600
*d^2*cos(5*b*x+5*a)*(d*x+c)^(1/2)/b^3

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4491, 3377, 3387, 3386, 3432, 3385, 3433} \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}-\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}+\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{32 b^3}+\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac {3 d^2 \sqrt {c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b} \]

[In]

Int[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]

[Out]

(15*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(32*b^3) - ((c + d*x)^(5/2)*Cos[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Cos
[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*Sqrt[c + d*x]*Cos[5*a + 5*b*x])/
(1600*b^3) + ((c + d*x)^(5/2)*Cos[5*a + 5*b*x])/(80*b) - (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqr
t[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(32*b^(7/2)) - (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(S
qrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(576*b^(7/2)) + (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*Fresnel
C[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelS[(Sqrt[b]*Sqrt
[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(1600*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*FresnelS[(Sqrt[b]
*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(576*b^(7/2)) + (15*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqr
t[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(1
6*b^2) + (5*d*(c + d*x)^(3/2)*Sin[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^(3/2)*Sin[5*a + 5*b*x])/(160*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 {1}{8} (c+d x)^{5/2} \sin (a+b x)+\frac {1}{16} (c+d x)^{5/2} \sin (3 a+3 b x)-\frac {1}{16} (c+d x)^{5/2} \sin (5 a+5 b x)\right ) \, dx \\ & = \frac {1}{16} \int (c+d x)^{5/2} \sin (3 a+3 b x) \, dx-\frac {1}{16} \int (c+d x)^{5/2} \sin (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^{5/2} \sin (a+b x) \, dx \\ & = -\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}-\frac {d \int (c+d x)^{3/2} \cos (5 a+5 b x) \, dx}{32 b}+\frac {(5 d) \int (c+d x)^{3/2} \cos (3 a+3 b x) \, dx}{96 b}+\frac {(5 d) \int (c+d x)^{3/2} \cos (a+b x) \, dx}{16 b} \\ & = -\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac {\left (3 d^2\right ) \int \sqrt {c+d x} \sin (5 a+5 b x) \, dx}{320 b^2}-\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \sin (3 a+3 b x) \, dx}{192 b^2}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \sin (a+b x) \, dx}{32 b^2} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{32 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 \sqrt {c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac {\left (3 d^3\right ) \int \frac {\cos (5 a+5 b x)}{\sqrt {c+d x}} \, dx}{3200 b^3}-\frac {\left (5 d^3\right ) \int \frac {\cos (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{1152 b^3}-\frac {\left (15 d^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{64 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{32 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 \sqrt {c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac {\left (3 d^3 \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}{3200 b^3}-\frac {\left (5 d^3 \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}{1152 b^3}-\frac {\left (15 d^3 \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}{64 b^3}-\frac {\left (3 d^3 \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}{3200 b^3}+\frac {\left (5 d^3 \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}{1152 b^3}+\frac {\left (15 d^3 \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}{64 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{32 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 \sqrt {c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}+\frac {\left (3 d^2 \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 )}{1600 b^3}-\frac {\left (5 d^2 \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 )}{576 b^3}-\frac {\left (15 d^2 \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 )}{32 b^3}-\frac {\left (3 d^2 \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 )}{1600 b^3}+\frac {\left (5 d^2 \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 )}{576 b^3}+\frac {\left (15 d^2 \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 )}{32 b^3} \\ & = \frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{32 b^3}-\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 \sqrt {c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b}-\frac {15 d^{5/2} \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 )}{32 b^{7/2}}-\frac {5 d^{5/2} \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 )}{576 b^{7/2}}+\frac {3 d^{5/2} \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 )}{1600 b^{7/2}}-\frac {3 d^{5/2} \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 )}{1600 b^{7/2}}+\frac {5 d^{5/2} \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 )}{576 b^{7/2}}+\frac {15 d^{5/2} \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 )}{32 b^{7/2}}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.02 (sec) , antiderivative size = 2177, normalized size of antiderivative = 3.54 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Result too large to show} \]

[In]

Integrate[(c + d*x)^(5/2)*Cos[a + b*x]^2*Sin[a + b*x]^3,x]

[Out]

(c*Sqrt[d]*(-12*Sqrt[b]*Sqrt[d]*E^(((3*I)*b*c)/d)*Sqrt[c + d*x]*(-I + 2*b*x + E^((6*I)*(a + b*x))*(I + 2*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*I)*b*c + 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]]))/(1152*b^(5/2)*E^(((3*I)*(a*d + b*(c + d*x)))/d)) - (c*Sqrt[d]*(-20*Sqrt[b]*Sqrt[d]*E^(((5*I)*b
*c)/d)*Sqrt[c + d*x]*(-3*I + 10*b*x + E^((10*I)*(a + b*x))*(3*I + 10*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*I)*b*c
+ 3*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]]))/(16000*b^(
5/2)*E^(((5*I)*(a*d + b*(c + d*x)))/d)) + ((I/16)*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]))/(b^2*E^((I*(
b*c + a*d))/d)*Sqrt[c + d*x]) + (c^2*(-1/6*(E^((3*I)*(a - (b*c)/d))*Sqrt[c + d*x]*Gamma[3/2, ((-3*I)*b*(c + d*
x))/d])/(Sqrt[3]*b*Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma[3/2, ((3*I)*b*(c + d*x))/d])/(6*Sqrt[3]*
b*E^((3*I)*(a - (b*c)/d))*Sqrt[(I*b*(c + d*x))/d])))/16 - (c^2*(-1/10*(E^((5*I)*(a - (b*c)/d))*Sqrt[c + d*x]*G
amma[3/2, ((-5*I)*b*(c + d*x))/d])/(Sqrt[5]*b*Sqrt[((-I)*b*(c + d*x))/d]) - (Sqrt[c + d*x]*Gamma[3/2, ((5*I)*b
*(c + d*x))/d])/(10*Sqrt[5]*b*E^((5*I)*(a - (b*c)/d))*Sqrt[(I*b*(c + d*x))/d])))/16 + (c*Sqrt[d]*(E^(I*(a - (b
*c)/d))*(-2*Sqrt[b]*Sqrt[d]*E^((I*b*(c + d*x))/d)*(3*I + 2*b*x)*Sqrt[c + d*x] + (-1)^(1/4)*((2*I)*b*c + 3*d)*S
qrt[Pi]*Erfi[((-1)^(1/4)*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]) + I*(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]*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])))/(32*b^(5/2)) - ((I/128)*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)*b^2*d*x^2 - 2*b*(c - 5*d*x))*(Cos
[b*(c/d + x)] + I*Sin[b*(c/d + x)])) - (2*Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((-15*I)*d + (4*I)*b^2*d*x^2 - 2*b*(c
- 5*d*x)) - (1 + I)*(4*b^2*c^2 + (12*I)*b*c*d - 15*d^2)*Sqrt[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])))/b^(7/2) - ((I/6912)*Sqr
t[d]*((Cos[3*a - (3*b*c)/d] + I*Sin[3*a - (3*b*c)/d])*((1 + I)*(12*b^2*c^2 - (12*I)*b*c*d - 5*d^2)*Sqrt[(3*Pi)
/2]*Erfi[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]] + 6*Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((5*I)*d - (12*I
)*b^2*d*x^2 - 2*b*(c - 5*d*x))*(Cos[(3*b*(c + d*x))/d] + I*Sin[(3*b*(c + d*x))/d])) - (Cos[3*(a + b*x)] - I*Si
n[3*(a + b*x)])*(6*Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((-5*I)*d + (12*I)*b^2*d*x^2 - 2*b*(c - 5*d*x)) - (1 + I)*(12
*b^2*c^2 + (12*I)*b*c*d - 5*d^2)*Sqrt[(3*Pi)/2]*Erf[((1 + I)*Sqrt[3/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]*(Cos[(3
*b*(c + d*x))/d] + I*Sin[(3*b*(c + d*x))/d]))))/b^(7/2) + ((I/32000)*Sqrt[d]*((Cos[5*a - (5*b*c)/d] + I*Sin[5*
a - (5*b*c)/d])*((1 + I)*(20*b^2*c^2 - (12*I)*b*c*d - 3*d^2)*Sqrt[(5*Pi)/2]*Erfi[((1 + I)*Sqrt[5/2]*Sqrt[b]*Sq
rt[c + d*x])/Sqrt[d]] + 10*Sqrt[b]*Sqrt[d]*Sqrt[c + d*x]*((3*I)*d - (20*I)*b^2*d*x^2 - 2*b*(c - 5*d*x))*(Cos[(
5*b*(c + d*x))/d] + I*Sin[(5*b*(c + d*x))/d])) - (Cos[5*(a + b*x)] - I*Sin[5*(a + b*x)])*(10*Sqrt[b]*Sqrt[d]*S
qrt[c + d*x]*((-3*I)*d + (20*I)*b^2*d*x^2 - 2*b*(c - 5*d*x)) - (1 + I)*(20*b^2*c^2 + (12*I)*b*c*d - 3*d^2)*Sqr
t[(5*Pi)/2]*Erf[((1 + I)*Sqrt[5/2]*Sqrt[b]*Sqrt[c + d*x])/Sqrt[d]]*(Cos[(5*b*(c + d*x))/d] + I*Sin[(5*b*(c + d
*x))/d]))))/b^(7/2)

Maple [A] (verified)

Time = 11.52 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}+\frac {5 d \left (\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 )}{2 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 )}{2 b}\right )}{8 b}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {5 d \left (\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 )}{6 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 )}{2 b}\right )}{48 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}-\frac {d \left (\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 )}{10 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 )}{10 b}\right )}{16 b}}{d}\)	\(719\)
default	\(\frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -c b}{d}\right )}{8 b}+\frac {5 d \left (\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 )}{2 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 )}{2 b}\right )}{8 b}-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 c b}{d}\right )}{48 b}+\frac {5 d \left (\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 )}{6 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 )}{2 b}\right )}{48 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {5 b \left (d x +c \right )}{d}+\frac {5 a d -5 c b}{d}\right )}{80 b}-\frac {d \left (\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 )}{10 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 )}{10 b}\right )}{16 b}}{d}\)	\(719\)

[In]

int((d*x+c)^(5/2)*cos(b*x+a)^2*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2/d*(-1/16/b*d*(d*x+c)^(5/2)*cos(b/d*(d*x+c)+(a*d-b*c)/d)+5/16/b*d*(1/2/b*d*(d*x+c)^(3/2)*sin(b/d*(d*x+c)+(a*d
-b*c)/d)-3/2/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)*(co
s((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)^(5/2)*cos(3*b/d*(d*x+c)+3*(a*d-b*c)/d)+5/96/b*d*(1/6/b
*d*(d*x+c)^(3/2)*sin(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)*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)^(5/2)*cos(5*b/d*(d*x+c)+5*(a*d-b*c)/d)-1/32/b*d*(1/10/b*d*(d*x+c)^(3/2)*sin(5*b/d*(d*x+c)
+5*(a*d-b*c)/d)-3/10/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 = 0.29 (sec) , antiderivative size = 521, normalized size of antiderivative = 0.85 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {81 \, \sqrt {10} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 625 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 101250 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 101250 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 625 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 81 \, \sqrt {10} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 480 \, {\left (9 \, {\left (20 \, b^{3} d^{2} x^{2} + 40 \, b^{3} c d x + 20 \, b^{3} c^{2} - 3 \, b d^{2}\right )} \cos \left (b x + a\right )^{5} + 390 \, b d^{2} \cos \left (b x + a\right ) - 5 \, {\left (60 \, b^{3} d^{2} x^{2} + 120 \, b^{3} c d x + 60 \, b^{3} c^{2} - 13 \, b d^{2}\right )} \cos \left (b x + a\right )^{3} + 10 \, {\left (26 \, b^{2} d^{2} x - 9 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{4} + 26 \, b^{2} c d + 13 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{432000 \, b^{4}} \]

[In]

integrate((d*x+c)^(5/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

1/432000*(81*sqrt(10)*pi*d^3*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(p
i*d))) - 625*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*
d))) - 101250*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d
))) + 101250*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/
d) + 625*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d)
- 81*sqrt(10)*pi*d^3*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*(20*b^3*d^2*x^2 + 40*b^3*c*d*x + 20*b^3*c^2 - 3*b*d^2)*cos(b*x + a)^5 + 390*b*d^2*cos(b*x + a) - 5*(6
0*b^3*d^2*x^2 + 120*b^3*c*d*x + 60*b^3*c^2 - 13*b*d^2)*cos(b*x + a)^3 + 10*(26*b^2*d^2*x - 9*(b^2*d^2*x + b^2*
c*d)*cos(b*x + a)^4 + 26*b^2*c*d + 13*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^2)*sin(b*x + a))*sqrt(d*x + c))/b^4

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**(5/2)*cos(b*x+a)**2*sin(b*x+a)**3,x)

[Out]

Timed out

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.34 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(5/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/1728000*sqrt(2)*(5400*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(5*((d*x + c)*b - b*c + a*d)/d)/d - 15000*sqrt(2)*(d*x
+ c)^(3/2)*b^4*sin(3*((d*x + c)*b - b*c + a*d)/d)/d - 270000*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(((d*x + c)*b - b
*c + a*d)/d)/d - 540*(20*sqrt(2)*(d*x + c)^(5/2)*b^5/d^2 - 3*sqrt(2)*sqrt(d*x + c)*b^3)*cos(5*((d*x + c)*b - b
*c + a*d)/d) + 1500*(12*sqrt(2)*(d*x + c)^(5/2)*b^5/d^2 - 5*sqrt(2)*sqrt(d*x + c)*b^3)*cos(3*((d*x + c)*b - b*
c + a*d)/d) + 27000*(4*sqrt(2)*(d*x + c)^(5/2)*b^5/d^2 - 15*sqrt(2)*sqrt(d*x + c)*b^3)*cos(((d*x + c)*b - b*c
+ a*d)/d) - 81*(-(I - 1)*25^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) - (I + 1)*25^(1/4)*sqrt
(pi)*b^2*d*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(5*I*b/d)) - 625*((I - 1)*9^(1/4)*sqrt
(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) + (I + 1)*9^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*sin(-3*(b*c
- a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) - 101250*((I - 1)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/
d) + (I + 1)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) - 101250*(-(I
+ 1)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (I - 1)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*sin(-(b*c - a
*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) - 625*(-(I + 1)*9^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-3*(b*c - a
*d)/d) - (I - 1)*9^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d
)) - 81*((I + 1)*25^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) + (I - 1)*25^(1/4)*sqrt(pi)*b^2
*d*(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^6

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.71 (sec) , antiderivative size = 3706, normalized size of antiderivative = 6.03 \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^(5/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/864000*(1800*(30*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*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*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*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*sqrt(6)*sq
rt(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)/(sq
rt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) - 3*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^3 + 18*c*d^2*(2250*
(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*I*(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*(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*I*(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*(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*I*(-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 + 2250*(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*I*(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*(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*I*(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*(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*I*(-10*I*(d*x + c)^(3/2)*b*d + 20*I*s
qrt(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) - d^3*(6750*(sqrt
(2)*sqrt(pi)*(8*b^3*c^3 + 12*I*b^2*c^2*d - 18*b*c*d^2 - 15*I*d^3)*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^3) - 2*I*(4*I*(d*x
+ c)^(5/2)*b^2*d - 12*I*(d*x + c)^(3/2)*b^2*c*d + 12*I*sqrt(d*x + c)*b^2*c^2*d + 10*(d*x + c)^(3/2)*b*d^2 - 1
8*sqrt(d*x + c)*b*c*d^2 - 15*I*sqrt(d*x + c)*d^3)*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^3)/d^3 + 125*(sqrt(
6)*sqrt(pi)*(72*b^3*c^3 - 36*I*b^2*c^2*d - 18*b*c*d^2 + 5*I*d^3)*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^3) - 6*I*(12*I*(d
*x + c)^(5/2)*b^2*d - 36*I*(d*x + c)^(3/2)*b^2*c*d + 36*I*sqrt(d*x + c)*b^2*c^2*d - 10*(d*x + c)^(3/2)*b*d^2 +
18*sqrt(d*x + c)*b*c*d^2 - 5*I*sqrt(d*x + c)*d^3)*e^(-3*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^3)/d^3 - 27*(sq
rt(10)*sqrt(pi)*(200*b^3*c^3 - 60*I*b^2*c^2*d - 18*b*c*d^2 + 3*I*d^3)*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^3) + 10*I*(
-20*I*(d*x + c)^(5/2)*b^2*d + 60*I*(d*x + c)^(3/2)*b^2*c*d - 60*I*sqrt(d*x + c)*b^2*c^2*d + 10*(d*x + c)^(3/2)
*b*d^2 - 18*sqrt(d*x + c)*b*c*d^2 + 3*I*sqrt(d*x + c)*d^3)*e^(-5*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^3)/d^3
+ 6750*(sqrt(2)*sqrt(pi)*(8*b^3*c^3 - 12*I*b^2*c^2*d - 18*b*c*d^2 + 15*I*d^3)*d*erf(-1/2*I*sqrt(2)*sqrt(b*d)*s
qrt(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^3) - 2
*I*(4*I*(d*x + c)^(5/2)*b^2*d - 12*I*(d*x + c)^(3/2)*b^2*c*d + 12*I*sqrt(d*x + c)*b^2*c^2*d - 10*(d*x + c)^(3/
2)*b*d^2 + 18*sqrt(d*x + c)*b*c*d^2 - 15*I*sqrt(d*x + c)*d^3)*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^3)/d^3 +
125*(sqrt(6)*sqrt(pi)*(72*b^3*c^3 + 36*I*b^2*c^2*d - 18*b*c*d^2 - 5*I*d^3)*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^3) -
6*I*(12*I*(d*x + c)^(5/2)*b^2*d - 36*I*(d*x + c)^(3/2)*b^2*c*d + 36*I*sqrt(d*x + c)*b^2*c^2*d + 10*(d*x + c)^
(3/2)*b*d^2 - 18*sqrt(d*x + c)*b*c*d^2 - 5*I*sqrt(d*x + c)*d^3)*e^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^3)/
d^3 - 27*(sqrt(10)*sqrt(pi)*(200*b^3*c^3 + 60*I*b^2*c^2*d - 18*b*c*d^2 - 3*I*d^3)*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^3) + 10*I*(-20*I*(d*x + c)^(5/2)*b^2*d + 60*I*(d*x + c)^(3/2)*b^2*c*d - 60*I*sqrt(d*x + c)*b^2*c^2*d - 10*(
d*x + c)^(3/2)*b*d^2 + 18*sqrt(d*x + c)*b*c*d^2 + 3*I*sqrt(d*x + c)*d^3)*e^(-5*(I*(d*x + c)*b - I*b*c + I*a*d)
/d)/b^3)/d^3) - 180*(450*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/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) + 25*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)/(s
qrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) - 9*sqrt(10)*sqrt(pi)*(10*b*c - I*d)*d*erf(-1/2*I*sqrt(10)*sqrt(b*d)*sqr
t(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
*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*sqrt(6)*sqrt(pi)*(6*b*c + I*d)*d*erf(1/2*I*sqr
t(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*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*sqrt(d*x + c)*d*e^(
(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 150*sqrt(d*x + c)*d*e^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 90*sqr
t(d*x + c)*d*e^(-5*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 900*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a
*d)/d)/b + 150*sqrt(d*x + c)*d*e^(-3*(-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b - 90*sqrt(d*x + c)*d*e^(-5*(-I*(d*x
+ c)*b + I*b*c - I*a*d)/d)/b)*c^2)/d

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^{5/2} \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{5/2} \,d x \]

[In]

int(cos(a + b*x)^2*sin(a + b*x)^3*(c + d*x)^(5/2),x)

[Out]

int(cos(a + b*x)^2*sin(a + b*x)^3*(c + d*x)^(5/2), x)