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

Optimal. Leaf size=407 \[ \frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 \sqrt {\pi } d^{5/2} \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}+\frac {45 \sqrt {\pi } d^{5/2} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b} \]

[Out]

-3/64*(d*x+c)^(5/2)*cos(2*b*x+2*a)/b+1/192*(d*x+c)^(5/2)*cos(6*b*x+6*a)/b+15/256*d*(d*x+c)^(3/2)*sin(2*b*x+2*a
)/b^2-5/2304*d*(d*x+c)^(3/2)*sin(6*b*x+6*a)/b^2+5/55296*d^(5/2)*cos(6*a-6*b*c/d)*FresnelC(2*b^(1/2)*3^(1/2)/Pi
^(1/2)*(d*x+c)^(1/2)/d^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)-5/55296*d^(5/2)*FresnelS(2*b^(1/2)*3^(1/2)/Pi^(1/2)*(d*
x+c)^(1/2)/d^(1/2))*sin(6*a-6*b*c/d)*3^(1/2)*Pi^(1/2)/b^(7/2)-45/2048*d^(5/2)*cos(2*a-2*b*c/d)*FresnelC(2*b^(1
/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(7/2)+45/2048*d^(5/2)*FresnelS(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/
Pi^(1/2))*sin(2*a-2*b*c/d)*Pi^(1/2)/b^(7/2)+45/1024*d^2*cos(2*b*x+2*a)*(d*x+c)^(1/2)/b^3-5/9216*d^2*cos(6*b*x+
6*a)*(d*x+c)^(1/2)/b^3

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Rubi [A]  time = 0.90, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 18, 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} \[ \frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \cos \left (6 a-\frac {6 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 \sqrt {\pi } d^{5/2} \cos \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {\pi } \sqrt {d}}\right )}{2048 b^{7/2}}-\frac {5 \sqrt {\frac {\pi }{3}} d^{5/2} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}+\frac {45 \sqrt {\pi } d^{5/2} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}+\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*d^2*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(1024*b^3) - (3*(c + d*x)^(5/2)*Cos[2*a + 2*b*x])/(64*b) - (5*d^2*Sqrt
[c + d*x]*Cos[6*a + 6*b*x])/(9216*b^3) + ((c + d*x)^(5/2)*Cos[6*a + 6*b*x])/(192*b) + (5*d^(5/2)*Sqrt[Pi/3]*Co
s[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(18432*b^(7/2)) - (45*d^(5/2)*Sqrt[
Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(2048*b^(7/2)) - (5*d^(5/2)*S
qrt[Pi/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(18432*b^(7/2)) + (45*d
^(5/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2048*b^(7/2)) +
(15*d*(c + d*x)^(3/2)*Sin[2*a + 2*b*x])/(256*b^2) - (5*d*(c + d*x)^(3/2)*Sin[6*a + 6*b*x])/(2304*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 ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac {3}{32} (c+d x)^{5/2} \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^{5/2} \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac {1}{32} \int (c+d x)^{5/2} \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^{5/2} \sin (2 a+2 b x) \, dx\\ &=-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}-\frac {(5 d) \int (c+d x)^{3/2} \cos (6 a+6 b x) \, dx}{384 b}+\frac {(15 d) \int (c+d x)^{3/2} \cos (2 a+2 b x) \, dx}{128 b}\\ &=-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^2\right ) \int \sqrt {c+d x} \sin (6 a+6 b x) \, dx}{1536 b^2}-\frac {\left (45 d^2\right ) \int \sqrt {c+d x} \sin (2 a+2 b x) \, dx}{512 b^2}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^3\right ) \int \frac {\cos (6 a+6 b x)}{\sqrt {c+d x}} \, dx}{18432 b^3}-\frac {\left (45 d^3\right ) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{2048 b^3}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^3 \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{18432 b^3}-\frac {\left (45 d^3 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{2048 b^3}-\frac {\left (5 d^3 \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{18432 b^3}+\frac {\left (45 d^3 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{2048 b^3}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}+\frac {\left (5 d^2 \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{9216 b^3}-\frac {\left (45 d^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{1024 b^3}-\frac {\left (5 d^2 \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{9216 b^3}+\frac {\left (45 d^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{1024 b^3}\\ &=\frac {45 d^2 \sqrt {c+d x} \cos (2 a+2 b x)}{1024 b^3}-\frac {3 (c+d x)^{5/2} \cos (2 a+2 b x)}{64 b}-\frac {5 d^2 \sqrt {c+d x} \cos (6 a+6 b x)}{9216 b^3}+\frac {(c+d x)^{5/2} \cos (6 a+6 b x)}{192 b}+\frac {5 d^{5/2} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{18432 b^{7/2}}-\frac {45 d^{5/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2048 b^{7/2}}-\frac {5 d^{5/2} \sqrt {\frac {\pi }{3}} S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{18432 b^{7/2}}+\frac {45 d^{5/2} \sqrt {\pi } S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2048 b^{7/2}}+\frac {15 d (c+d x)^{3/2} \sin (2 a+2 b x)}{256 b^2}-\frac {5 d (c+d x)^{3/2} \sin (6 a+6 b x)}{2304 b^2}\\ \end {align*}

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Mathematica [A]  time = 5.07, size = 550, normalized size = 1.35 \[ \frac {-2592 b^3 c^2 \sqrt {c+d x} \cos (2 (a+b x))+288 b^3 c^2 \sqrt {c+d x} \cos (6 (a+b x))-2592 b^3 d^2 x^2 \sqrt {c+d x} \cos (2 (a+b x))+288 b^3 d^2 x^2 \sqrt {c+d x} \cos (6 (a+b x))-5184 b^3 c d x \sqrt {c+d x} \cos (2 (a+b x))+576 b^3 c d x \sqrt {c+d x} \cos (6 (a+b x))+3240 b^2 d^2 x \sqrt {c+d x} \sin (2 (a+b x))-120 b^2 d^2 x \sqrt {c+d x} \sin (6 (a+b x))+3240 b^2 c d \sqrt {c+d x} \sin (2 (a+b x))-120 b^2 c d \sqrt {c+d x} \sin (6 (a+b x))+5 \sqrt {3 \pi } d^3 \sqrt {\frac {b}{d}} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}\right )-1215 \sqrt {\pi } d^3 \sqrt {\frac {b}{d}} \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )-5 \sqrt {3 \pi } d^3 \sqrt {\frac {b}{d}} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}\right )+1215 \sqrt {\pi } d^3 \sqrt {\frac {b}{d}} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )+2430 b d^2 \sqrt {c+d x} \cos (2 (a+b x))-30 b d^2 \sqrt {c+d x} \cos (6 (a+b x))}{55296 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2592*b^3*c^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 2430*b*d^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 5184*b^3*c*d*x*Sqr
t[c + d*x]*Cos[2*(a + b*x)] - 2592*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 288*b^3*c^2*Sqrt[c + d*x]*Cos[
6*(a + b*x)] - 30*b*d^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 576*b^3*c*d*x*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 288*b^
3*d^2*x^2*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*Cos[6*a - (6*b*c)/d]*FresnelC[2*Sqrt[b/d
]*Sqrt[3/Pi]*Sqrt[c + d*x]] - 1215*Sqrt[b/d]*d^3*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b/d]*Sqrt[c +
d*x])/Sqrt[Pi]] - 5*Sqrt[b/d]*d^3*Sqrt[3*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c)/
d] + 1215*Sqrt[b/d]*d^3*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 3240*b^
2*c*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] + 3240*b^2*d^2*x*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 120*b^2*c*d*Sqrt[c + d*
x]*Sin[6*(a + b*x)] - 120*b^2*d^2*x*Sqrt[c + d*x]*Sin[6*(a + b*x)])/(55296*b^4)

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fricas [A]  time = 0.59, size = 445, normalized size = 1.09 \[ \frac {5 \, \sqrt {3} \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 5 \, \sqrt {3} \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 1215 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 1215 \, \pi d^{3} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 96 \, {\left (24 \, b^{3} d^{2} x^{2} + 2 \, {\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{6} + 48 \, b^{3} c d x + 24 \, b^{3} c^{2} + 45 \, b d^{2} \cos \left (b x + a\right )^{2} - 3 \, {\left (48 \, b^{3} d^{2} x^{2} + 96 \, b^{3} c d x + 48 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{4} - 25 \, b d^{2} - 20 \, {\left (2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{5} - 2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{55296 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/55296*(5*sqrt(3)*pi*d^3*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_cos(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*
d))) - 5*sqrt(3)*pi*d^3*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/
d) - 1215*pi*d^3*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d))) + 1215*pi*d^
3*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) + 96*(24*b^3*d^2*x^2 + 2*(4
8*b^3*d^2*x^2 + 96*b^3*c*d*x + 48*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^6 + 48*b^3*c*d*x + 24*b^3*c^2 + 45*b*d^2*cos
(b*x + a)^2 - 3*(48*b^3*d^2*x^2 + 96*b^3*c*d*x + 48*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^4 - 25*b*d^2 - 20*(2*(b^2*
d^2*x + b^2*c*d)*cos(b*x + a)^5 - 2*(b^2*d^2*x + b^2*c*d)*cos(b*x + a)^3 - 3*(b^2*d^2*x + b^2*c*d)*cos(b*x + a
))*sin(b*x + a))*sqrt(d*x + c))/b^4

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giac [C]  time = 13.77, size = 2417, normalized size = 5.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/110592*(576*(-I*sqrt(3)*sqrt(pi)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*
I*b*c - 6*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) + I*sqrt(3)*sqrt(pi)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d
*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-6*I*b*c + 6*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)) + 9*I*
sqrt(pi)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*d)*(I*b
*d/sqrt(b^2*d^2) + 1)) - 9*I*sqrt(pi)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-2*I*b*
c + 2*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)))*c^3 + 36*c*d^2*((-I*sqrt(3)*sqrt(pi)*(48*b^2*c^2 + 8*I
*b*c*d - d^2)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*I*b*c - 6*I*a*d)/d)/(s
qrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^2) - 6*I*(-4*I*(d*x + c)^(3/2)*b*d + 8*I*sqrt(d*x + c)*b*c*d - sqrt(d*x +
c)*d^2)*e^((-6*I*(d*x + c)*b + 6*I*b*c - 6*I*a*d)/d)/b^2)/d^2 + (I*sqrt(3)*sqrt(pi)*(48*b^2*c^2 - 8*I*b*c*d -
d^2)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-6*I*b*c + 6*I*a*d)/d)/(sqrt(b*
d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) - 6*I*(-4*I*(d*x + c)^(3/2)*b*d + 8*I*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*d
^2)*e^((6*I*(d*x + c)*b - 6*I*b*c + 6*I*a*d)/d)/b^2)/d^2 + 9*(I*sqrt(pi)*(48*b^2*c^2 + 24*I*b*c*d - 9*d^2)*d*e
rf(-sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*
d^2) + 1)*b^2) - 2*I*(12*I*(d*x + c)^(3/2)*b*d - 24*I*sqrt(d*x + c)*b*c*d + 9*sqrt(d*x + c)*d^2)*e^((-2*I*(d*x
+ c)*b + 2*I*b*c - 2*I*a*d)/d)/b^2)/d^2 + 9*(-I*sqrt(pi)*(48*b^2*c^2 - 24*I*b*c*d - 9*d^2)*d*erf(-sqrt(b*d)*s
qrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-2*I*b*c + 2*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^
2) - 2*I*(12*I*(d*x + c)^(3/2)*b*d - 24*I*sqrt(d*x + c)*b*c*d - 9*sqrt(d*x + c)*d^2)*e^((2*I*(d*x + c)*b - 2*I
*b*c + 2*I*a*d)/d)/b^2)/d^2) + d^3*((I*sqrt(3)*sqrt(pi)*(576*b^3*c^3 + 144*I*b^2*c^2*d - 36*b*c*d^2 - 5*I*d^3)
*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*I*b*c - 6*I*a*d)/d)/(sqrt(b*d)*(I*b
*d/sqrt(b^2*d^2) + 1)*b^3) - 6*I*(-48*I*(d*x + c)^(5/2)*b^2*d + 144*I*(d*x + c)^(3/2)*b^2*c*d - 144*I*sqrt(d*x
+ c)*b^2*c^2*d - 20*(d*x + c)^(3/2)*b*d^2 + 36*sqrt(d*x + c)*b*c*d^2 + 5*I*sqrt(d*x + c)*d^3)*e^((-6*I*(d*x +
c)*b + 6*I*b*c - 6*I*a*d)/d)/b^3)/d^3 + (-I*sqrt(3)*sqrt(pi)*(576*b^3*c^3 - 144*I*b^2*c^2*d - 36*b*c*d^2 + 5*
I*d^3)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-6*I*b*c + 6*I*a*d)/d)/(sqrt(b
*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 6*I*(-48*I*(d*x + c)^(5/2)*b^2*d + 144*I*(d*x + c)^(3/2)*b^2*c*d - 144*I
*sqrt(d*x + c)*b^2*c^2*d + 20*(d*x + c)^(3/2)*b*d^2 - 36*sqrt(d*x + c)*b*c*d^2 + 5*I*sqrt(d*x + c)*d^3)*e^((6*
I*(d*x + c)*b - 6*I*b*c + 6*I*a*d)/d)/b^3)/d^3 + 27*(-I*sqrt(pi)*(192*b^3*c^3 + 144*I*b^2*c^2*d - 108*b*c*d^2
- 45*I*d^3)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*d)*(
I*b*d/sqrt(b^2*d^2) + 1)*b^3) - 2*I*(48*I*(d*x + c)^(5/2)*b^2*d - 144*I*(d*x + c)^(3/2)*b^2*c*d + 144*I*sqrt(d
*x + c)*b^2*c^2*d + 60*(d*x + c)^(3/2)*b*d^2 - 108*sqrt(d*x + c)*b*c*d^2 - 45*I*sqrt(d*x + c)*d^3)*e^((-2*I*(d
*x + c)*b + 2*I*b*c - 2*I*a*d)/d)/b^3)/d^3 + 27*(I*sqrt(pi)*(192*b^3*c^3 - 144*I*b^2*c^2*d - 108*b*c*d^2 + 45*
I*d^3)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-2*I*b*c + 2*I*a*d)/d)/(sqrt(b*d)*(-I*
b*d/sqrt(b^2*d^2) + 1)*b^3) - 2*I*(48*I*(d*x + c)^(5/2)*b^2*d - 144*I*(d*x + c)^(3/2)*b^2*c*d + 144*I*sqrt(d*x
+ c)*b^2*c^2*d - 60*(d*x + c)^(3/2)*b*d^2 + 108*sqrt(d*x + c)*b*c*d^2 - 45*I*sqrt(d*x + c)*d^3)*e^((2*I*(d*x
+ c)*b - 2*I*b*c + 2*I*a*d)/d)/b^3)/d^3) + 144*(I*sqrt(3)*sqrt(pi)*(12*b*c + I*d)*d*erf(-sqrt(3)*sqrt(b*d)*sqr
t(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*I*b*c - 6*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) - I*
sqrt(3)*sqrt(pi)*(12*b*c - I*d)*d*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-6*I*
b*c + 6*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 9*I*sqrt(pi)*(12*b*c + 3*I*d)*d*erf(-sqrt(b*d)*sq
rt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) + 9
*I*sqrt(pi)*(12*b*c - 3*I*d)*d*erf(-sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-2*I*b*c + 2*I*a
*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 6*sqrt(d*x + c)*d*e^((6*I*(d*x + c)*b - 6*I*b*c + 6*I*a*d)/d
)/b + 54*sqrt(d*x + c)*d*e^((2*I*(d*x + c)*b - 2*I*b*c + 2*I*a*d)/d)/b + 54*sqrt(d*x + c)*d*e^((-2*I*(d*x + c)
*b + 2*I*b*c - 2*I*a*d)/d)/b - 6*sqrt(d*x + c)*d*e^((-6*I*(d*x + c)*b + 6*I*b*c - 6*I*a*d)/d)/b)*c^2)/d

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maple [A]  time = 0.05, size = 477, normalized size = 1.17 \[ \frac {-\frac {3 d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{64 b}+\frac {15 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{4 b}-\frac {3 d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{4 b}+\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \FresnelC \left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 d a -2 c b}{d}\right ) \mathrm {S}\left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{64 b}+\frac {d \left (d x +c \right )^{\frac {5}{2}} \cos \left (\frac {6 \left (d x +c \right ) b}{d}+\frac {6 d a -6 c b}{d}\right )}{192 b}-\frac {5 d \left (\frac {d \left (d x +c \right )^{\frac {3}{2}} \sin \left (\frac {6 \left (d x +c \right ) b}{d}+\frac {6 d a -6 c b}{d}\right )}{12 b}-\frac {d \left (-\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 \left (d x +c \right ) b}{d}+\frac {6 d a -6 c b}{d}\right )}{12 b}+\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 d a -6 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 d a -6 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{144 b \sqrt {\frac {b}{d}}}\right )}{4 b}\right )}{192 b}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-3/128/b*d*(d*x+c)^(5/2)*cos(2/d*(d*x+c)*b+2*(a*d-b*c)/d)+15/128/b*d*(1/4/b*d*(d*x+c)^(3/2)*sin(2/d*(d*x+
c)*b+2*(a*d-b*c)/d)-3/4/b*d*(-1/4/b*d*(d*x+c)^(1/2)*cos(2/d*(d*x+c)*b+2*(a*d-b*c)/d)+1/8/b*d*Pi^(1/2)/(b/d)^(1
/2)*(cos(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)-sin(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1
/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d))))+1/384/b*d*(d*x+c)^(5/2)*cos(6/d*(d*x+c)*b+6*(a*d-b*c)/d)-5/384/b*d*(1/12
/b*d*(d*x+c)^(3/2)*sin(6/d*(d*x+c)*b+6*(a*d-b*c)/d)-1/4/b*d*(-1/12/b*d*(d*x+c)^(1/2)*cos(6/d*(d*x+c)*b+6*(a*d-
b*c)/d)+1/144/b*d*2^(1/2)*Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(cos(6*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)/(
b/d)^(1/2)*(d*x+c)^(1/2)*b/d)-sin(6*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b
/d)))))

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maxima [C]  time = 0.53, size = 557, normalized size = 1.37 \[ -\frac {{\left (1920 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 51840 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - 96 \, {\left (\frac {48 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 5 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 2592 \, {\left (\frac {16 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4}}{d} - 15 \, \sqrt {d x + c} b^{2} d\right )} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + {\left (\left (10 i - 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + \left (10 i + 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {6 i \, b}{d}}\right ) + {\left (-\left (2430 i - 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (2430 i + 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) + {\left (\left (2430 i + 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (2430 i - 2430\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right ) + {\left (-\left (10 i + 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - \left (10 i - 10\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d^{2} \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {6 i \, b}{d}}\right )\right )} d}{884736 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/884736*(1920*(d*x + c)^(3/2)*b^3*sin(6*((d*x + c)*b - b*c + a*d)/d) - 51840*(d*x + c)^(3/2)*b^3*sin(2*((d*x
+ c)*b - b*c + a*d)/d) - 96*(48*(d*x + c)^(5/2)*b^4/d - 5*sqrt(d*x + c)*b^2*d)*cos(6*((d*x + c)*b - b*c + a*d
)/d) + 2592*(16*(d*x + c)^(5/2)*b^4/d - 15*sqrt(d*x + c)*b^2*d)*cos(2*((d*x + c)*b - b*c + a*d)/d) + ((10*I -
10)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d) + (10*I + 10)*36^(1/4)*sqrt(2)*sqrt(
pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d)) + (-(2430*I - 2430)*4^(1/4)*
sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) - (2430*I + 2430)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*
(b^2/d^2)^(1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) + ((2430*I + 2430)*4^(1/4)*sqrt(2)*sqr
t(pi)*b*d^2*(b^2/d^2)^(1/4)*cos(-2*(b*c - a*d)/d) + (2430*I - 2430)*4^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(
1/4)*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-2*I*b/d)) + (-(10*I + 10)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*
(b^2/d^2)^(1/4)*cos(-6*(b*c - a*d)/d) - (10*I - 10)*36^(1/4)*sqrt(2)*sqrt(pi)*b*d^2*(b^2/d^2)^(1/4)*sin(-6*(b*
c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-6*I*b/d)))*d/b^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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