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

Optimal. Leaf size=615 \[ -\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \sin \left (5 a-\frac {5 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}-\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \cos \left (5 a-\frac {5 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{32 b^3}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{576 b^3}+\frac {3 d^2 \sqrt {c+d x} \sin (5 a+5 b x)}{1600 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b} \]

[Out]

5/16*d*(d*x+c)^(3/2)*cos(b*x+a)/b^2-5/288*d*(d*x+c)^(3/2)*cos(3*b*x+3*a)/b^2-1/160*d*(d*x+c)^(3/2)*cos(5*b*x+5
*a)/b^2+1/8*(d*x+c)^(5/2)*sin(b*x+a)/b-1/48*(d*x+c)^(5/2)*sin(3*b*x+3*a)/b-1/80*(d*x+c)^(5/2)*sin(5*b*x+5*a)/b
-3/16000*d^(5/2)*cos(5*a-5*b*c/d)*FresnelS(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)*FresnelC(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^(7/2)-5/3456*d^(5/2)*cos(3*a-3*b*c/d)*FresnelS(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*6^(1/2
)*Pi^(1/2)/b^(7/2)-5/3456*d^(5/2)*FresnelC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)*6^
(1/2)*Pi^(1/2)/b^(7/2)+15/64*d^(5/2)*cos(a-b*c/d)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(
1/2)*Pi^(1/2)/b^(7/2)+15/64*d^(5/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*2^(1
/2)*Pi^(1/2)/b^(7/2)-15/32*d^2*sin(b*x+a)*(d*x+c)^(1/2)/b^3+5/576*d^2*sin(3*b*x+3*a)*(d*x+c)^(1/2)/b^3+3/1600*
d^2*sin(5*b*x+5*a)*(d*x+c)^(1/2)/b^3

________________________________________________________________________________________

Rubi [A]  time = 1.14, antiderivative size = 615, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \sin \left (5 a-\frac {5 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}-\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \cos \left (5 a-\frac {5 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{32 b^3}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{576 b^3}+\frac {3 d^2 \sqrt {c+d x} \sin (5 a+5 b x)}{1600 b^3}+\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(16*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(288*b^2) - (d*(c + d*x)^
(3/2)*Cos[5*a + 5*b*x])/(160*b^2) + (15*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[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]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqr
t[c + d*x])/Sqrt[d]])/(576*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi
]*Sqrt[c + d*x])/Sqrt[d]])/(1600*b^(7/2)) - (3*d^(5/2)*Sqrt[Pi/10]*FresnelC[(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]*FresnelC[(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]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[
c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(32*b^(7/2)) - (15*d^2*Sqrt[c + d*x]*Sin[a + b*x])/(32*b^3) + ((c + d*x)^
(5/2)*Sin[a + b*x])/(8*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(576*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*
x])/(48*b) + (3*d^2*Sqrt[c + d*x]*Sin[5*a + 5*b*x])/(1600*b^3) - ((c + d*x)^(5/2)*Sin[5*a + 5*b*x])/(80*b)

Rule 3296

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 3304

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 3305

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 3306

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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4406

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*} \int (c+d x)^{5/2} \cos ^3(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac {1}{8} (c+d x)^{5/2} \cos (a+b x)-\frac {1}{16} (c+d x)^{5/2} \cos (3 a+3 b x)-\frac {1}{16} (c+d x)^{5/2} \cos (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac {1}{16} \int (c+d x)^{5/2} \cos (3 a+3 b x) \, dx\right )-\frac {1}{16} \int (c+d x)^{5/2} \cos (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^{5/2} \cos (a+b x) \, dx\\ &=\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}+\frac {d \int (c+d x)^{3/2} \sin (5 a+5 b x) \, dx}{32 b}+\frac {(5 d) \int (c+d x)^{3/2} \sin (3 a+3 b x) \, dx}{96 b}-\frac {(5 d) \int (c+d x)^{3/2} \sin (a+b x) \, dx}{16 b}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}+\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}+\frac {\left (3 d^2\right ) \int \sqrt {c+d x} \cos (5 a+5 b x) \, dx}{320 b^2}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \cos (3 a+3 b x) \, dx}{192 b^2}-\frac {\left (15 d^2\right ) \int \sqrt {c+d x} \cos (a+b x) \, dx}{32 b^2}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{32 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}+\frac {3 d^2 \sqrt {c+d x} \sin (5 a+5 b x)}{1600 b^3}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}-\frac {\left (3 d^3\right ) \int \frac {\sin (5 a+5 b x)}{\sqrt {c+d x}} \, dx}{3200 b^3}-\frac {\left (5 d^3\right ) \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{1152 b^3}+\frac {\left (15 d^3\right ) \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{64 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{32 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}+\frac {3 d^2 \sqrt {c+d x} \sin (5 a+5 b x)}{1600 b^3}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}-\frac {\left (3 d^3 \cos \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 \cos \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 \cos \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 {\left (3 d^3 \sin \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 \sin \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 \sin \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 {5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}-\frac {15 d^2 \sqrt {c+d x} \sin (a+b x)}{32 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}+\frac {3 d^2 \sqrt {c+d x} \sin (5 a+5 b x)}{1600 b^3}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}-\frac {\left (3 d^2 \cos \left (5 a-\frac {5 b c}{d}\right )\right ) \operatorname {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 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {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 \cos \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \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 ) \operatorname {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 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \operatorname {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 \sin \left (a-\frac {b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{32 b^3}\\ &=\frac {5 d (c+d x)^{3/2} \cos (a+b x)}{16 b^2}-\frac {5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \cos (5 a+5 b x)}{160 b^2}+\frac {15 d^{5/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) S\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 ) S\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 ) S\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}} C\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}} C\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}} C\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 {15 d^2 \sqrt {c+d x} \sin (a+b x)}{32 b^3}+\frac {(c+d x)^{5/2} \sin (a+b x)}{8 b}+\frac {5 d^2 \sqrt {c+d x} \sin (3 a+3 b x)}{576 b^3}-\frac {(c+d x)^{5/2} \sin (3 a+3 b x)}{48 b}+\frac {3 d^2 \sqrt {c+d x} \sin (5 a+5 b x)}{1600 b^3}-\frac {(c+d x)^{5/2} \sin (5 a+5 b x)}{80 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 22.42, size = 1795, normalized size = 2.92 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1/16*I)*c^2*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(
((2*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (c*d*(Sqrt
[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[a - (b*c)/d] + 2*b*c*Sin[a - (b*c)/d])
+ Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[a - (b*c)/d] + 3*d*Sin[a - (b*
c)/d]) + 2*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(16*b^3) + ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*F
resnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d]
)) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2
)*Sin[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[a + b*x] + d*(-15 + 4*b^2*x^2)*Sin[a +
b*x])))/(64*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sq
rt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]
*Sin[3*(a + b*x)]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c
+ d*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sq
rt[6/Pi]*Sqrt[c + d*x]]*(2*b*c*Cos[3*a - (3*b*c)/d] + d*Sin[3*a - (3*b*c)/d]) + 2*Sqrt[3]*b*Sqrt[c + d*x]*(Cos
[3*(a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]
*Sqrt[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d])) - Sqrt
[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin
[3*a - (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[3*(a + b*x)] + d*(-5 + 12*b^2*x
^2)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[1
0/Pi]*Sqrt[c + d*x]]) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d] + 2*Sqrt
[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[5*(a + b*x)]))/(160*Sqrt[5]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC
[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-3*d*Cos[5*a - (5*b*c)/d] + 10*b*c*Sin[5*a - (5*b*c)/d]) + Sqrt[b/d]*Sq
rt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(10*b*c*Cos[5*a - (5*b*c)/d] + 3*d*Sin[5*a - (5*b*c)/d]
) + 2*Sqrt[5]*b*Sqrt[c + d*x]*(3*Cos[5*(a + b*x)] + 10*b*x*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - ((b/d)^(3/2
)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*((20*b^2*c^2 - 3*d^2)*Cos[5*a - (5*b*c)/d] +
12*b*c*d*Sin[5*a - (5*b*c)/d])) - Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[5*a
- (5*b*c)/d] + (20*b^2*c^2 - 3*d^2)*Sin[5*a - (5*b*c)/d]) + 2*Sqrt[5]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*
d*x)*Cos[5*(a + b*x)] + d*(-3 + 20*b^2*x^2)*Sin[5*(a + b*x)])))/(3200*Sqrt[5]*b^5)

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 548, normalized size = 0.89 \[ -\frac {81 \, \sqrt {10} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 625 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 101250 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 101250 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 625 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 81 \, \sqrt {10} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 480 \, {\left (90 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{5} - 50 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 300 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) - {\left (120 \, b^{3} d^{2} x^{2} + 240 \, b^{3} c d x + 120 \, b^{3} c^{2} - 9 \, {\left (20 \, b^{3} d^{2} x^{2} + 40 \, b^{3} c d x + 20 \, b^{3} c^{2} - 3 \, b d^{2}\right )} \cos \left (b x + a\right )^{4} - 428 \, b d^{2} + {\left (60 \, b^{3} d^{2} x^{2} + 120 \, b^{3} c d x + 60 \, b^{3} c^{2} + 11 \, b d^{2}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{432000 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/432000*(81*sqrt(10)*pi*d^3*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_sin(sqrt(10)*sqrt(d*x + c)*sqrt(b/(
pi*d))) + 625*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi
*d))) - 101250*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*
d))) - 101250*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)
/d) + 625*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d
) + 81*sqrt(10)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d)
+ 480*(90*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^5 - 50*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^3 - 300*(b^2*d^2*x + b
^2*c*d)*cos(b*x + a) - (120*b^3*d^2*x^2 + 240*b^3*c*d*x + 120*b^3*c^2 - 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)^4 - 428*b*d^2 + (60*b^3*d^2*x^2 + 120*b^3*c*d*x + 60*b^3*c^2 + 11*b*d^2)*cos(b
*x + a)^2)*sin(b*x + a))*sqrt(d*x + c))/b^4

________________________________________________________________________________________

giac [C]  time = 18.66, size = 3677, normalized size = 5.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864000*(1800*(3*sqrt(10)*sqrt(pi)*d*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e
^((5*I*b*c - 5*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*sqrt(6)*sqrt(b*
d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((3*I*b*c - 3*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*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)) - 30*sqrt(2)*sqrt(pi)*d*erf(-1/2*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*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c + 3*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*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(
b^2*d^2) + 1)/d)*e^((-5*I*b*c + 5*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)))*c^3 + 18*c*d^2*(9*(sqrt(10
)*sqrt(pi)*(100*b^2*c^2 + 20*I*b*c*d - 3*d^2)*d*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2)
+ 1)/d)*e^((5*I*b*c - 5*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 + 5*I*b*c - 5*I*a*d)/d)/b^2)/d^2 + 125*(s
qrt(6)*sqrt(pi)*(12*b^2*c^2 + 4*I*b*c*d - d^2)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2)
+ 1)/d)*e^((3*I*b*c - 3*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 + 3*I*b*c - 3*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*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 - 2250*(sqrt(2)*sqrt(pi
)*(4*b^2*c^2 - 4*I*b*c*d - 3*d^2)*d*erf(-1/2*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*(sqrt(6)*sqrt(pi)*(12*b^2*
c^2 - 4*I*b*c*d - d^2)*d*erf(-1/2*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c +
3*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 - 3*I*b*c + 3*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*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-5*I*b*c
+ 5*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 - 5*I*b*c + 5*I*a*d)/d)/b^2)/d^2) - d^3*(27*(sqrt(10)*sqrt(p
i)*(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*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sq
rt(b^2*d^2) + 1)/d)*e^((5*I*b*c - 5*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) + 10*(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 + 5*I*b*c - 5*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*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b
*d/sqrt(b^2*d^2) + 1)/d)*e^((3*I*b*c - 3*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^3) + 6*(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*s
qrt(d*x + c)*b*c*d^2 - 5*I*sqrt(d*x + c)*d^3)*e^((-3*I*(d*x + c)*b + 3*I*b*c - 3*I*a*d)/d)/b^3)/d^3 - 6750*(sq
rt(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*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*(-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 - 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*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*(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*sq
rt(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)*sq
rt(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*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d
/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c + 3*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) + 6*(-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 - 3*I*b*c + 3*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*sqrt(10)*sqrt(b*d)*sqrt(d*x + c
)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-5*I*b*c + 5*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) + 10*(-2
0*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 - 5*I*b*c + 5*I*a*d)/d)/b^3)/d^3)
- 180*(9*sqrt(10)*sqrt(pi)*(10*b*c + I*d)*d*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) +
1)/d)*e^((5*I*b*c - 5*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*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((3*I*b*c - 3*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*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) - 450*sqrt(2)*sqrt(pi)*
(2*b*c - I*d)*d*erf(-1/2*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*sqrt(6)*sqrt(b*d)*sqrt
(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-3*I*b*c + 3*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*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e
^((-5*I*b*c + 5*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 90*I*sqrt(d*x + c)*d*e^((5*I*(d*x + c)*b
- 5*I*b*c + 5*I*a*d)/d)/b - 150*I*sqrt(d*x + c)*d*e^((3*I*(d*x + c)*b - 3*I*b*c + 3*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 - 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 + 3*I*b*c - 3*I*a*d)/d)/b + 90*I*sqrt(d*x + c)*d*e^((-5*I*(d
*x + c)*b + 5*I*b*c - 5*I*a*d)/d)/b)*c^2)/d

________________________________________________________________________________________

maple [A]  time = 0.07, size = 716, normalized size = 1.16 \[ \frac {\frac {d \left (d x +c \right )^{\frac {5}{2}} \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{8 b}-\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\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}} \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{48 b}+\frac {5 d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}+\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\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}} \sin \left (\frac {5 \left (d x +c \right ) b}{d}+\frac {5 d a -5 c b}{d}\right )}{80 b}+\frac {d \left (-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {5 \left (d x +c \right ) b}{d}+\frac {5 d a -5 c b}{d}\right )}{10 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 \left (d x +c \right ) b}{d}+\frac {5 d a -5 c b}{d}\right )}{10 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 d a -5 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 d a -5 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{100 b \sqrt {\frac {b}{d}}}\right )}{10 b}\right )}{16 b}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(1/16/b*d*(d*x+c)^(5/2)*sin(1/d*(d*x+c)*b+(a*d-b*c)/d)-5/16/b*d*(-1/2/b*d*(d*x+c)^(3/2)*cos(1/d*(d*x+c)*b+
(a*d-b*c)/d)+3/2/b*d*(1/2/b*d*(d*x+c)^(1/2)*sin(1/d*(d*x+c)*b+(a*d-b*c)/d)-1/4/b*d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2
)*(cos((a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)+sin((a*d-b*c)/d)*FresnelC(2^(1/2)
/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d))))-1/96/b*d*(d*x+c)^(5/2)*sin(3/d*(d*x+c)*b+3*(a*d-b*c)/d)+5/96/b*d*(
-1/6/b*d*(d*x+c)^(3/2)*cos(3/d*(d*x+c)*b+3*(a*d-b*c)/d)+1/2/b*d*(1/6/b*d*(d*x+c)^(1/2)*sin(3/d*(d*x+c)*b+3*(a*
d-b*c)/d)-1/36/b*d*2^(1/2)*Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(cos(3*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/
(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)+sin(3*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*
b/d))))-1/160/b*d*(d*x+c)^(5/2)*sin(5/d*(d*x+c)*b+5*(a*d-b*c)/d)+1/32/b*d*(-1/10/b*d*(d*x+c)^(3/2)*cos(5/d*(d*
x+c)*b+5*(a*d-b*c)/d)+3/10/b*d*(1/10/b*d*(d*x+c)^(1/2)*sin(5/d*(d*x+c)*b+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)*FresnelS(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)+
sin(5*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*5^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)))))

________________________________________________________________________________________

maxima [C]  time = 0.57, size = 820, normalized size = 1.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3456000*sqrt(2)*(10800*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(5*((d*x + c)*b - b*c + a*d)/d)/d + 30000*sqrt(2)*(d*
x + c)^(3/2)*b^4*cos(3*((d*x + c)*b - b*c + a*d)/d)/d - 540000*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(((d*x + c)*b -
b*c + a*d)/d)/d + ((162*I + 162)*25^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) - (162*I - 162)
*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)) + ((1250*I +
1250)*9^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) - (1250*I - 1250)*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)) + (-(202500*I + 202500)*sqrt(pi)*b^2*d*(
b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (202500*I - 202500)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*e
rf(sqrt(d*x + c)*sqrt(I*b/d)) + ((202500*I - 202500)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (202
500*I + 202500)*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)) + (-(1250*
I - 1250)*9^(1/4)*sqrt(pi)*b^2*d*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) + (1250*I + 1250)*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)) + (-(162*I - 162)*25^(1/4)*sqrt(pi)
*b^2*d*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) + (162*I + 162)*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)) + 1080*(20*sqrt(2)*(d*x + c)^(5/2)*b^5/d^2 - 3*sqrt(2)*sqrt(d*
x + c)*b^3)*sin(5*((d*x + c)*b - b*c + a*d)/d) + 3000*(12*sqrt(2)*(d*x + c)^(5/2)*b^5/d^2 - 5*sqrt(2)*sqrt(d*x
+ c)*b^3)*sin(3*((d*x + c)*b - b*c + a*d)/d) - 54000*(4*sqrt(2)*(d*x + c)^(5/2)*b^5/d^2 - 15*sqrt(2)*sqrt(d*x
+ c)*b^3)*sin(((d*x + c)*b - b*c + a*d)/d))*d^2/b^6

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________