3.193 \(\int \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx\)

Optimal. Leaf size=459 \[ \frac {\sqrt {\frac {\pi }{10}} \sqrt {d} \sin \left (5 a-\frac {5 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{80 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {\sqrt {\frac {\pi }{10}} \sqrt {d} \cos \left (5 a-\frac {5 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{80 b^{3/2}}+\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b} \]

[Out]

1/800*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))*d^(1/2)*10^(1/2)*Pi^(1/2)/b^(
3/2)+1/800*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)*d^(1/2)*10^(1/2)*Pi^(1/2
)/b^(3/2)+1/288*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))*d^(1/2)*6^(1/2)*Pi^(
1/2)/b^(3/2)+1/288*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)*d^(1/2)*6^(1/2)*P
i^(1/2)/b^(3/2)-1/16*cos(a-b*c/d)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*d^(1/2)*2^(1/2)*Pi^
(1/2)/b^(3/2)-1/16*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*d^(1/2)*2^(1/2)*Pi^(1
/2)/b^(3/2)+1/8*sin(b*x+a)*(d*x+c)^(1/2)/b-1/48*sin(3*b*x+3*a)*(d*x+c)^(1/2)/b-1/80*sin(5*b*x+5*a)*(d*x+c)^(1/
2)/b

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Rubi [A]  time = 0.67, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 20, 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} \[ \frac {\sqrt {\frac {\pi }{10}} \sqrt {d} \sin \left (5 a-\frac {5 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{80 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {d} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {d} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{48 b^{3/2}}+\frac {\sqrt {\frac {\pi }{10}} \sqrt {d} \cos \left (5 a-\frac {5 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{80 b^{3/2}}+\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d]*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(8*b^(3/2)) + (Sqr
t[d]*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(48*b^(3/2)) + (Sqr
t[d]*Sqrt[Pi/10]*Cos[5*a - (5*b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(80*b^(3/2)) + (S
qrt[d]*Sqrt[Pi/10]*FresnelC[(Sqrt[b]*Sqrt[10/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[5*a - (5*b*c)/d])/(80*b^(3/2)) +
(Sqrt[d]*Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(48*b^(3/2)) -
(Sqrt[d]*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(8*b^(3/2)) + (Sqrt
[c + d*x]*Sin[a + b*x])/(8*b) - (Sqrt[c + d*x]*Sin[3*a + 3*b*x])/(48*b) - (Sqrt[c + d*x]*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 \sqrt {c+d x} \cos ^3(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac {1}{8} \sqrt {c+d x} \cos (a+b x)-\frac {1}{16} \sqrt {c+d x} \cos (3 a+3 b x)-\frac {1}{16} \sqrt {c+d x} \cos (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac {1}{16} \int \sqrt {c+d x} \cos (3 a+3 b x) \, dx\right )-\frac {1}{16} \int \sqrt {c+d x} \cos (5 a+5 b x) \, dx+\frac {1}{8} \int \sqrt {c+d x} \cos (a+b x) \, dx\\ &=\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b}+\frac {d \int \frac {\sin (5 a+5 b x)}{\sqrt {c+d x}} \, dx}{160 b}+\frac {d \int \frac {\sin (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{96 b}-\frac {d \int \frac {\sin (a+b x)}{\sqrt {c+d x}} \, dx}{16 b}\\ &=\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b}+\frac {\left (d \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}{160 b}+\frac {\left (d \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}{96 b}-\frac {\left (d \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}{16 b}+\frac {\left (d \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}{160 b}+\frac {\left (d \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}{96 b}-\frac {\left (d \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}{16 b}\\ &=\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b}+\frac {\cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {5 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{80 b}+\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{48 b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{8 b}+\frac {\sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {5 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{80 b}+\frac {\sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{48 b}-\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{8 b}\\ &=-\frac {\sqrt {d} \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 )}{8 b^{3/2}}+\frac {\sqrt {d} \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 )}{48 b^{3/2}}+\frac {\sqrt {d} \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 )}{80 b^{3/2}}+\frac {\sqrt {d} \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 )}{80 b^{3/2}}+\frac {\sqrt {d} \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 )}{48 b^{3/2}}-\frac {\sqrt {d} \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 )}{8 b^{3/2}}+\frac {\sqrt {c+d x} \sin (a+b x)}{8 b}-\frac {\sqrt {c+d x} \sin (3 a+3 b x)}{48 b}-\frac {\sqrt {c+d x} \sin (5 a+5 b x)}{80 b}\\ \end {align*}

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Mathematica [C]  time = 6.87, size = 435, normalized size = 0.95 \[ -\frac {-\sqrt {2 \pi } \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )-\sqrt {2 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+2 \sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (3 (a+b x))}{96 \sqrt {3} b \sqrt {\frac {b}{d}}}-\frac {-\sqrt {2 \pi } \sin \left (5 a-\frac {5 b c}{d}\right ) C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}\right )-\sqrt {2 \pi } \cos \left (5 a-\frac {5 b c}{d}\right ) S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}\right )+2 \sqrt {5} \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (5 (a+b x))}{160 \sqrt {5} b \sqrt {\frac {b}{d}}}-\frac {i \sqrt {c+d x} e^{-\frac {i (a d+b c)}{d}} \left (\frac {e^{2 i a} \Gamma \left (\frac {3}{2},-\frac {i b (c+d x)}{d}\right )}{\sqrt {-\frac {i b (c+d x)}{d}}}-\frac {e^{\frac {2 i b c}{d}} \Gamma \left (\frac {3}{2},\frac {i b (c+d x)}{d}\right )}{\sqrt {\frac {i b (c+d x)}{d}}}\right )}{16 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/16*I)*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)) - (-(Sqrt[2*Pi]*
Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[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]) - (-(Sqrt[2*Pi]*Cos[5*a - (5*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[10/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])

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fricas [A]  time = 0.54, size = 365, normalized size = 0.80 \[ \frac {9 \, \sqrt {10} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 25 \, \sqrt {6} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 450 \, \sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 450 \, \sqrt {2} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 25 \, \sqrt {6} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 9 \, \sqrt {10} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) - 480 \, {\left (3 \, b \cos \left (b x + a\right )^{4} - b \cos \left (b x + a\right )^{2} - 2 \, b\right )} \sqrt {d x + c} \sin \left (b x + a\right )}{7200 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

1/7200*(9*sqrt(10)*pi*d*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_sin(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d))
) + 25*sqrt(6)*pi*d*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 4
50*sqrt(2)*pi*d*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 450*sqr
t(2)*pi*d*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) + 25*sqrt(6)*pi
*d*sqrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 9*sqrt(10)*pi*d*sq
rt(b/(pi*d))*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) - 480*(3*b*cos(b*x + a)^
4 - b*cos(b*x + a)^2 - 2*b)*sqrt(d*x + c)*sin(b*x + a))/b^2

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giac [C]  time = 4.48, size = 1258, normalized size = 2.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="giac")

[Out]

-1/14400*(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)*sq
rt(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) - 30*(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*sq
rt(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 - 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)/d

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maple [A]  time = 0.00, size = 444, normalized size = 0.97 \[ \frac {\frac {d \sqrt {d x +c}\, \sin \left (\frac {\left (d x +c \right ) b}{d}+\frac {d a -c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{16 b \sqrt {\frac {b}{d}}}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 \left (d x +c \right ) b}{d}+\frac {3 d a -3 c b}{d}\right )}{48 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{288 b \sqrt {\frac {b}{d}}}-\frac {d \sqrt {d x +c}\, \sin \left (\frac {5 \left (d x +c \right ) b}{d}+\frac {5 d a -5 c b}{d}\right )}{80 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \left (\cos \left (\frac {5 d a -5 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {5 d a -5 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{800 b \sqrt {\frac {b}{d}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^2,x)

[Out]

2/d*(1/16/b*d*(d*x+c)^(1/2)*sin(1/d*(d*x+c)*b+(a*d-b*c)/d)-1/32/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)^(1/2)*sin(3/d*(d*x+c)*b+3*(a*d-b*c)/d)+1/576/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)^(1/2)*sin(5/
d*(d*x+c)*b+5*(a*d-b*c)/d)+1/1600/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)))

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maxima [C]  time = 0.51, size = 674, normalized size = 1.47 \[ -\frac {\sqrt {2} {\left (\frac {360 \, \sqrt {2} \sqrt {d x + c} b^{3} \sin \left (\frac {5 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d^{2}} + \frac {600 \, \sqrt {2} \sqrt {d x + c} b^{3} \sin \left (\frac {3 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d^{2}} - \frac {3600 \, \sqrt {2} \sqrt {d x + c} b^{3} \sin \left (\frac {{\left (d x + c\right )} b - b c + a d}{d}\right )}{d^{2}} + {\left (-\frac {\left (18 i + 18\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d} + \frac {\left (18 i - 18\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {5 i \, b}{d}}\right ) + {\left (-\frac {\left (50 i + 50\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d} + \frac {\left (50 i - 50\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {3 i \, b}{d}}\right ) + {\left (\frac {\left (900 i + 900\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right )}{d} - \frac {\left (900 i - 900\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) + {\left (-\frac {\left (900 i - 900\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {b c - a d}{d}\right )}{d} + \frac {\left (900 i + 900\right ) \, \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {b c - a d}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) + {\left (\frac {\left (50 i - 50\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d} - \frac {\left (50 i + 50\right ) \cdot 9^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {3 i \, b}{d}}\right ) + {\left (\frac {\left (18 i - 18\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d} - \frac {\left (18 i + 18\right ) \cdot 25^{\frac {1}{4}} \sqrt {\pi } b^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{d}\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {5 i \, b}{d}}\right )\right )} d^{2}}{57600 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/57600*sqrt(2)*(360*sqrt(2)*sqrt(d*x + c)*b^3*sin(5*((d*x + c)*b - b*c + a*d)/d)/d^2 + 600*sqrt(2)*sqrt(d*x
+ c)*b^3*sin(3*((d*x + c)*b - b*c + a*d)/d)/d^2 - 3600*sqrt(2)*sqrt(d*x + c)*b^3*sin(((d*x + c)*b - b*c + a*d)
/d)/d^2 + (-(18*I + 18)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d)/d + (18*I - 18)*25^(1/4)*s
qrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt(5*I*b/d)) + (-(50*I + 50)*9^(1/4)*
sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d)/d + (50*I - 50)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-3
*(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) + ((900*I + 900)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c -
a*d)/d)/d - (900*I - 900)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt(I*b/d)) +
(-(900*I - 900)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d)/d + (900*I + 900)*sqrt(pi)*b^2*(b^2/d^2)^(1/
4)*sin(-(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt(-I*b/d)) + ((50*I - 50)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*
cos(-3*(b*c - a*d)/d)/d - (50*I + 50)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d)/d)*erf(sqrt(d
*x + c)*sqrt(-3*I*b/d)) + ((18*I - 18)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d)/d - (18*I +
18)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d)/d)*erf(sqrt(d*x + c)*sqrt(-5*I*b/d)))*d^2/b^4

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

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)^(1/2),x)

[Out]

int(cos(a + b*x)^3*sin(a + b*x)^2*(c + d*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c + d x} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)*cos(b*x+a)**3*sin(b*x+a)**2,x)

[Out]

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

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