[next] [prev] [prev-tail] [tail] [up]

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

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

[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]  time = 0.95, 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} \[ -\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \cos \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \cos \left (5 a-\frac {5 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}-\frac {3 \sqrt {\frac {\pi }{10}} d^{5/2} \sin \left (5 a-\frac {5 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{1600 b^{7/2}}+\frac {5 \sqrt {\frac {\pi }{6}} d^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{576 b^{7/2}}+\frac {15 \sqrt {\frac {\pi }{2}} d^{5/2} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{32 b^{7/2}}+\frac {15 d^2 \sqrt {c+d x} \cos (a+b x)}{32 b^3}+\frac {5 d^2 \sqrt {c+d x} \cos (3 a+3 b x)}{576 b^3}-\frac {3 d^2 \sqrt {c+d x} \cos (5 a+5 b x)}{1600 b^3}+\frac {5 d (c+d x)^{3/2} \sin (a+b x)}{16 b^2}+\frac {5 d (c+d x)^{3/2} \sin (3 a+3 b x)}{288 b^2}-\frac {d (c+d x)^{3/2} \sin (5 a+5 b x)}{160 b^2}-\frac {(c+d x)^{5/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{5/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{5/2} \cos (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[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 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 ^2(a+b x) \sin ^3(a+b x) \, dx &=\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 ) \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 \cos \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 \cos \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 {\left (3 d^2 \sin \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 \sin \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 \sin \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 {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 ) C\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 ) C\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 ) C\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}} S\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}} S\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}} S\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]  time = 22.97, size = 3348, normalized size = 5.44 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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]))/(16*b*E^((I*(b*c + a*d))/d)) + (c^2*(2*Sqrt[5
]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[5*(a + b*x)] - Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sq
rt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]*Sin[5*a - (5*b*c)/d]))/(160*Sqrt[5]*b*
Sqrt[b/d]) - (c^2*(2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*Fresne
lC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*
b*c)/d]))/(96*Sqrt[3]*b*Sqrt[b/d]) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[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[2*Pi]*FresnelC[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*Sqrt[b/d]*d*Sqrt[c + d*x]*(2*b*x*Cos[a + b*x] - 3*Sin[a + b*x])
))/(16*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC[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]*FresnelS[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]*(d*(-15 + 4*b^2*x
^2)*Cos[a + b*x] + 2*b*(c - 5*d*x)*Sin[a + b*x])))/(64*b^5) - (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sq
rt[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[2*Pi]*FresnelC[Sqrt[b/d]*
Sqrt[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]*Sqrt[b/d]*d*Sqrt[c
 + d*x]*(2*b*x*Cos[3*(a + b*x)] - Sin[3*(a + b*x)])))/(96*Sqrt[3]*b^3) + ((b/d)^(3/2)*d^2*(Sqrt[2*Pi]*FresnelC
[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]*FresnelS[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]*(d*(5 - 12*b^2*x^2)*Cos[3*(a + b*x)] - 2*b*(
c - 5*d*x)*Sin[3*(a + b*x)])))/(1152*Sqrt[3]*b^5) + (c*Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[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[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[1
0/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]*Sqrt[b/d]*d*Sqrt[c +
 d*x]*(10*b*x*Cos[5*(a + b*x)] - 3*Sin[5*(a + b*x)])))/(800*Sqrt[5]*b^3) - (d^2*(Sin[5*a]*((c^2*(-(Sqrt[5]*Sqr
t[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]])*Sin[(
5*b*c)/d])/(5*Sqrt[5]*(b/d)^(3/2)*d^3) + (c^2*Cos[(5*b*c)/d]*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt
[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (2*c*Cos[
(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqr
t[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/(25*Sqrt[5]*(b/d)
^(5/2)*d^3) - (2*c*Sin[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[
Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d])
)/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Sin[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Cos[(5*b*(c + d*
x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sq
rt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(c + d*x))/d]))/2))/(125*Sqrt[5]
*(b/d)^(7/2)*d^3) + (Cos[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Sin[(5*b*(c + d*x))/d] - (5*(-5*Sq
rt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqr
t[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3))
 + Cos[5*a]*((c^2*Cos[(5*b*c)/d]*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + Sqrt[Pi/2]*Fresn
elC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/(5*Sqrt[5]*(b/d)^(3/2)*d^3) - (c^2*Sin[(5*b*c)/d]*(-(Sqrt[Pi/2]*Fre
snelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/(5*Sqrt
[5]*(b/d)^(3/2)*d^3) + (2*c*Sin[(5*b*c)/d]*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) + S
qrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*(
c + d*x))/d]))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) - (2*c*Cos[(5*b*c)/d]*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[
(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*FresnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[
c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/(25*Sqrt[5]*(b/d)^(5/2)*d^3) + (Cos[(5*b*c)/d]*(-25*Sqrt[5]*(b/d)^(5/2)*
(c + d*x)^(5/2)*Cos[(5*b*(c + d*x))/d] + (5*((-3*(-(Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[(5*b*(c + d*x))/d]) +
Sqrt[Pi/2]*FresnelC[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]))/2 + 5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Sin[(5*b*
(c + d*x))/d]))/2))/(125*Sqrt[5]*(b/d)^(7/2)*d^3) - (Sin[(5*b*c)/d]*(25*Sqrt[5]*(b/d)^(5/2)*(c + d*x)^(5/2)*Si
n[(5*b*(c + d*x))/d] - (5*(-5*Sqrt[5]*(b/d)^(3/2)*(c + d*x)^(3/2)*Cos[(5*b*(c + d*x))/d] + (3*(-(Sqrt[Pi/2]*Fr
esnelS[Sqrt[b/d]*Sqrt[10/Pi]*Sqrt[c + d*x]]) + Sqrt[5]*Sqrt[b/d]*Sqrt[c + d*x]*Sin[(5*b*(c + d*x))/d]))/2))/2)
)/(125*Sqrt[5]*(b/d)^(7/2)*d^3))))/16

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 521, normalized size = 0.85 \[ \frac {81 \, \sqrt {10} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 625 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 101250 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 101250 \, \sqrt {2} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 625 \, \sqrt {6} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 81 \, \sqrt {10} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 480 \, {\left (9 \, {\left (20 \, b^{3} d^{2} x^{2} + 40 \, b^{3} c d x + 20 \, b^{3} c^{2} - 3 \, b d^{2}\right )} \cos \left (b x + a\right )^{5} + 390 \, b d^{2} \cos \left (b x + a\right ) - 5 \, {\left (60 \, b^{3} d^{2} x^{2} + 120 \, b^{3} c d x + 60 \, b^{3} c^{2} - 13 \, b d^{2}\right )} \cos \left (b x + a\right )^{3} + 10 \, {\left (26 \, b^{2} d^{2} x - 9 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{4} + 26 \, b^{2} c d + 13 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{432000 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [C]  time = 7.58, size = 3689, normalized size = 6.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864000*(1800*(-3*I*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*I*sqrt(6)*sqrt(pi)*d*erf(-1/2*sqrt(6)*s
qrt(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*I*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*I*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*I*sqr
t(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*I*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*(-I*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*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 + 5*I*b*c - 5*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*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*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 + 3*I*b*c - 3*I*a*d)/d
)/b^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*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 + 2250*(-I*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*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*(-I*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/s
qrt(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*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 - 3*I*b*c + 3*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*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*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 - 5*I*b*c + 5*I*a*d)
/d)/b^2)/d^2) + d^3*(27*(I*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*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 +
5*I*b*c - 5*I*a*d)/d)/b^3)/d^3 + 125*(-I*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*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 + 3*I*b*c - 3*I*a*d)/d)/b^3)/d^3 + 6750*(-I*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*I*(4*I*(d*x + c)^(5/2)*b^2*d - 12*I*(d*x + c)^(3/2)*b^2*c*d + 12*I*sq
rt(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*(I*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)/(sqr
t(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*(I*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*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 - 3*I*b*c + 3*I*a*d)/d)/b^3)/d^3 + 27*(-I*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*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 - 5*I*b*c + 5*I*a*d)/d)/b^3)/d^3) + 180*(9*I*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*I*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*I*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*I*sqrt(2)*sqrt(pi)*(2*b*c - I*d)*d*erf(-1/2*sqrt(2)*sqr
t(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*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*I*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*sqrt(d*x + c)*d*e^((5*I*(d*x + c)*b - 5*I*b*c + 5*I*a*d)/d)/b + 150*sq
rt(d*x + c)*d*e^((3*I*(d*x + c)*b - 3*I*b*c + 3*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 + 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 + 3*I*b*c - 3*I*a*d)/d)/b - 90*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 = 0.00, size = 719, normalized size = 1.17 \[ \frac {-\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \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}} \sin \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}\, \cos \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 ) \FresnelC \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 ) \mathrm {S}\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}} \cos \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}} \sin \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}\, \cos \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 ) \FresnelC \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 ) \mathrm {S}\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}} \cos \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}} \sin \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}\, \cos \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 ) \FresnelC \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 ) \mathrm {S}\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)^2*sin(b*x+a)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 1.79, 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)^2*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/3456000*sqrt(2)*(10800*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(5*((d*x + c)*b - b*c + a*d)/d)/d - 30000*sqrt(2)*(d*
x + c)^(3/2)*b^4*sin(3*((d*x + c)*b - b*c + a*d)/d)/d - 540000*sqrt(2)*(d*x + c)^(3/2)*b^4*sin(((d*x + c)*b -
b*c + a*d)/d)/d - 1080*(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) + 3000*(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) + 54000*(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) + ((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 - 125
0)*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))*erf(
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) + (202500
*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)))*d^2/b^6

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^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)

________________________________________________________________________________________

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)**2*sin(b*x+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]