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

Optimal. Leaf size=534 \[ \frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \sin \left (5 a-\frac {5 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \cos \left (5 a-\frac {5 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b} \]

[Out]

-1/8*(d*x+c)^(3/2)*cos(b*x+a)/b-1/48*(d*x+c)^(3/2)*cos(3*b*x+3*a)/b+1/80*(d*x+c)^(3/2)*cos(5*b*x+5*a)/b+3/8000
*d^(3/2)*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^(5/2)+
3/8000*d^(3/2)*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^
(5/2)-1/576*d^(3/2)*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^(5/2)-1/576*d^(3/2)*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^(5/2)-3/32*d^(3/2)*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^(5/2)-3/32*d^(3/2)*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^(5/2)+3/16*d*sin(b*x+a)*(d*x+c)^(1/2)/b^2+1/96*d*sin(3*b*x+3*a)*(d*x+c)^(1/2)/b^2-3/800*d*sin(5*b*x+5*a)*(d*
x+c)^(1/2)/b^2

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Rubi [A]  time = 0.80, antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 23, 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} \[ \frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \sin \left (5 a-\frac {5 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \sin \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{16 b^{5/2}}-\frac {\sqrt {\frac {\pi }{6}} d^{3/2} \cos \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{96 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{10}} d^{3/2} \cos \left (5 a-\frac {5 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{800 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (a+b x)}{16 b^2}+\frac {d \sqrt {c+d x} \sin (3 a+3 b x)}{96 b^2}-\frac {3 d \sqrt {c+d x} \sin (5 a+5 b x)}{800 b^2}-\frac {(c+d x)^{3/2} \cos (a+b x)}{8 b}-\frac {(c+d x)^{3/2} \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^{3/2} \cos (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C]  time = 11.45, size = 1041, normalized size = 1.95 \[ \frac {c e^{-\frac {i (b c+a d)}{d}} \sqrt {c+d x} \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}+\frac {c \left (2 \sqrt {5} \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (5 (a+b x))-\sqrt {2 \pi } \cos \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 } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )\right )}{160 \sqrt {5} b \sqrt {\frac {b}{d}}}-\frac {c \left (2 \sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (3 (a+b x))-\sqrt {2 \pi } \cos \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 } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )\right )}{96 \sqrt {3} b \sqrt {\frac {b}{d}}}-\frac {\sqrt {\frac {b}{d}} d \left (\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (3 d \cos \left (a-\frac {b c}{d}\right )-2 b c \sin \left (a-\frac {b c}{d}\right )\right )+\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \cos \left (a-\frac {b c}{d}\right )+3 d \sin \left (a-\frac {b c}{d}\right )\right )+2 \sqrt {\frac {b}{d}} d \sqrt {c+d x} (2 b x \cos (a+b x)-3 \sin (a+b x))\right )}{32 b^3}-\frac {\sqrt {\frac {b}{d}} d \left (\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (d \cos \left (3 a-\frac {3 b c}{d}\right )-2 b c \sin \left (3 a-\frac {3 b c}{d}\right )\right )+\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right ) \left (2 b c \cos \left (3 a-\frac {3 b c}{d}\right )+d \sin \left (3 a-\frac {3 b c}{d}\right )\right )+2 \sqrt {3} \sqrt {\frac {b}{d}} d \sqrt {c+d x} (2 b x \cos (3 (a+b x))-\sin (3 (a+b x)))\right )}{192 \sqrt {3} b^3}+\frac {\sqrt {\frac {b}{d}} d \left (\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}\right ) \left (3 d \cos \left (5 a-\frac {5 b c}{d}\right )-10 b c \sin \left (5 a-\frac {5 b c}{d}\right )\right )+\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {10}{\pi }} \sqrt {c+d x}\right ) \left (10 b c \cos \left (5 a-\frac {5 b c}{d}\right )+3 d \sin \left (5 a-\frac {5 b c}{d}\right )\right )+2 \sqrt {5} \sqrt {\frac {b}{d}} d \sqrt {c+d x} (10 b x \cos (5 (a+b x))-3 \sin (5 (a+b x)))\right )}{1600 \sqrt {5} b^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*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*Sqrt[5]*Sq
rt[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]*Sqrt[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*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Cos[3*(a + b*x)] - Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqr
t[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]) - (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])))/(32*b
^3) - (Sqrt[b/d]*d*(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(d*Cos[3*a - (3*b*c)/d] - 2*b*c*Si
n[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)])))/(192
*Sqrt[3]*b^3) + (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[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]*Sqrt[b/d]*d*Sqrt[c + d*x]*(10*b*x*Cos[5*(a + b*x)] - 3*Sin
[5*(a + b*x)])))/(1600*Sqrt[5]*b^3)

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fricas [A]  time = 0.63, size = 427, normalized size = 0.80 \[ \frac {27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 6750 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 125 \, \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 27 \, \sqrt {10} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {10} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 480 \, {\left (30 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{5} - 50 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{3} - {\left (9 \, b d \cos \left (b x + a\right )^{4} - 13 \, b d \cos \left (b x + a\right )^{2} - 26 \, b d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{72000 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72000*(27*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*cos(-5*(b*c - a*d)/d)*fresnel_sin(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi
*d))) - 125*sqrt(6)*pi*d^2*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d
))) - 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))
- 6750*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) -
125*sqrt(6)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 27
*sqrt(10)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(10)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-5*(b*c - a*d)/d) + 480
*(30*(b^2*d*x + b^2*c)*cos(b*x + a)^5 - 50*(b^2*d*x + b^2*c)*cos(b*x + a)^3 - (9*b*d*cos(b*x + a)^4 - 13*b*d*c
os(b*x + a)^2 - 26*b*d)*sin(b*x + a))*sqrt(d*x + c))/b^3

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giac [C]  time = 4.21, size = 2300, normalized size = 4.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144000*(300*(-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)*sq
rt(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*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*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^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/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^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/s
qrt(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/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 + 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/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^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) + 20*(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)/(sqr
t(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*sqr
t(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*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*sqrt(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)/d

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maple [A]  time = 0.00, size = 580, normalized size = 1.09 \[ \frac {-\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 )}{8 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 )}{8 b}-\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 )}{48 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 )}{16 b}+\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 )}{80 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 )}{80 b}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-1/16/b*d*(d*x+c)^(3/2)*cos(1/d*(d*x+c)*b+(a*d-b*c)/d)+3/16/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)^(3
/2)*cos(3/d*(d*x+c)*b+3*(a*d-b*c)/d)+1/32/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)^(3/2)*cos(5/d*(d*x+c)*b+5*(a*d-b*c)/d)-3/160/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)))
)

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maxima [C]  time = 0.80, size = 760, normalized size = 1.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576000*sqrt(2)*(3600*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(5*((d*x + c)*b - b*c + a*d)/d)/d^2 - 6000*sqrt(2)*(d*x
+ c)^(3/2)*b^4*cos(3*((d*x + c)*b - b*c + a*d)/d)/d^2 - 36000*sqrt(2)*(d*x + c)^(3/2)*b^4*cos(((d*x + c)*b - b
*c + a*d)/d)/d^2 - 1080*sqrt(2)*sqrt(d*x + c)*b^3*sin(5*((d*x + c)*b - b*c + a*d)/d)/d + 3000*sqrt(2)*sqrt(d*x
+ c)*b^3*sin(3*((d*x + c)*b - b*c + a*d)/d)/d + 54000*sqrt(2)*sqrt(d*x + c)*b^3*sin(((d*x + c)*b - b*c + a*d)
/d)/d - (-(54*I + 54)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) + (54*I - 54)*25^(1/4)*sqrt(
pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(5*I*b/d)) - ((250*I + 250)*9^(1/4)*sqrt(
pi)*b^2*(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) - (250*I - 250)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-3*(b*c
- a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) - ((13500*I + 13500)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d
)/d) - (13500*I - 13500)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) - (-
(13500*I - 13500)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (13500*I + 13500)*sqrt(pi)*b^2*(b^2/d^2)^
(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) - (-(250*I - 250)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1
/4)*cos(-3*(b*c - a*d)/d) + (250*I + 250)*9^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt
(d*x + c)*sqrt(-3*I*b/d)) - ((54*I - 54)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*cos(-5*(b*c - a*d)/d) - (54*I +
54)*25^(1/4)*sqrt(pi)*b^2*(b^2/d^2)^(1/4)*sin(-5*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-5*I*b/d)))*d^2/b^5

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

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)^(3/2),x)

[Out]

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

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

[Out]

Timed out

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