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

Optimal. Leaf size=410 \[ -\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}+\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}+\frac{45 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}-\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b} \]

[Out]

(45*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^3) - (2*(c + d*x)^(5/2)*Cos[a + b*x])/(3*b) - (5*d^2*Sqrt[c + d*x]*C
os[3*a + 3*b*x])/(144*b^3) - (45*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x
])/Sqrt[d]])/(16*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d
*x])/Sqrt[d]])/(144*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])/(144*b^(7/2)) + (45*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*
Sin[a - (b*c)/d])/(16*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(3*b^2) - ((c + d*x)^(5/2)*Cos[a + b*x]*Si
n[a + b*x]^2)/(3*b) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x]^3)/(18*b^2)

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Rubi [A]  time = 1.12658, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3311, 3296, 3306, 3305, 3351, 3304, 3352, 3312} \[ -\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}+\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}+\frac{45 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}+\frac{45 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}-\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{(c+d x)^{5/2} \sin ^2(a+b x) \cos (a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)*Sin[a + b*x]^3,x]

[Out]

(45*d^2*Sqrt[c + d*x]*Cos[a + b*x])/(16*b^3) - (2*(c + d*x)^(5/2)*Cos[a + b*x])/(3*b) - (5*d^2*Sqrt[c + d*x]*C
os[3*a + 3*b*x])/(144*b^3) - (45*d^(5/2)*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x
])/Sqrt[d]])/(16*b^(7/2)) + (5*d^(5/2)*Sqrt[Pi/6]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d
*x])/Sqrt[d]])/(144*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])/(144*b^(7/2)) + (45*d^(5/2)*Sqrt[Pi/2]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*
Sin[a - (b*c)/d])/(16*b^(7/2)) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x])/(3*b^2) - ((c + d*x)^(5/2)*Cos[a + b*x]*Si
n[a + b*x]^2)/(3*b) + (5*d*(c + d*x)^(3/2)*Sin[a + b*x]^3)/(18*b^2)

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

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 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 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 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 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 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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int (c+d x)^{5/2} \sin ^3(a+b x) \, dx &=-\frac{(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac{2}{3} \int (c+d x)^{5/2} \sin (a+b x) \, dx-\frac{\left (5 d^2\right ) \int \sqrt{c+d x} \sin ^3(a+b x) \, dx}{12 b^2}\\ &=-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac{(5 d) \int (c+d x)^{3/2} \cos (a+b x) \, dx}{3 b}-\frac{\left (5 d^2\right ) \int \left (\frac{3}{4} \sqrt{c+d x} \sin (a+b x)-\frac{1}{4} \sqrt{c+d x} \sin (3 a+3 b x)\right ) \, dx}{12 b^2}\\ &=-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac{\left (5 d^2\right ) \int \sqrt{c+d x} \sin (3 a+3 b x) \, dx}{48 b^2}-\frac{\left (5 d^2\right ) \int \sqrt{c+d x} \sin (a+b x) \, dx}{16 b^2}-\frac{\left (5 d^2\right ) \int \sqrt{c+d x} \sin (a+b x) \, dx}{2 b^2}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\frac{\left (5 d^3\right ) \int \frac{\cos (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{288 b^3}-\frac{\left (5 d^3\right ) \int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx}{32 b^3}-\frac{\left (5 d^3\right ) \int \frac{\cos (a+b x)}{\sqrt{c+d x}} \, dx}{4 b^3}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\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}{288 b^3}-\frac{\left (5 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}{32 b^3}-\frac{\left (5 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}{4 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}{288 b^3}+\frac{\left (5 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}{32 b^3}+\frac{\left (5 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}{4 b^3}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}+\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 )}{144 b^3}-\frac{\left (5 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 )}{16 b^3}-\frac{\left (5 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 )}{2 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 )}{144 b^3}+\frac{\left (5 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 )}{16 b^3}+\frac{\left (5 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 )}{2 b^3}\\ &=\frac{45 d^2 \sqrt{c+d x} \cos (a+b x)}{16 b^3}-\frac{2 (c+d x)^{5/2} \cos (a+b x)}{3 b}-\frac{5 d^2 \sqrt{c+d x} \cos (3 a+3 b x)}{144 b^3}-\frac{45 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 )}{16 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 )}{144 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 )}{144 b^{7/2}}+\frac{45 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 )}{16 b^{7/2}}+\frac{5 d (c+d x)^{3/2} \sin (a+b x)}{3 b^2}-\frac{(c+d x)^{5/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac{5 d (c+d x)^{3/2} \sin ^3(a+b x)}{18 b^2}\\ \end{align*}

Mathematica [A]  time = 3.1518, size = 542, normalized size = 1.32 \[ \frac{-648 b^3 c^2 \sqrt{c+d x} \cos (a+b x)+72 b^3 c^2 \sqrt{c+d x} \cos (3 (a+b x))-648 b^3 d^2 x^2 \sqrt{c+d x} \cos (a+b x)+72 b^3 d^2 x^2 \sqrt{c+d x} \cos (3 (a+b x))+1620 b^2 d^2 x \sqrt{c+d x} \sin (a+b x)-60 b^2 d^2 x \sqrt{c+d x} \sin (3 (a+b x))+1620 b^2 c d \sqrt{c+d x} \sin (a+b x)-60 b^2 c d \sqrt{c+d x} \sin (3 (a+b x))-1296 b^3 c d x \sqrt{c+d x} \cos (a+b x)+144 b^3 c d x \sqrt{c+d x} \cos (3 (a+b x))-1215 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )+5 \sqrt{6 \pi } d^3 \sqrt{\frac{b}{d}} \cos \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-5 \sqrt{6 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left (3 a-\frac{3 b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right )+1215 \sqrt{2 \pi } d^3 \sqrt{\frac{b}{d}} \sin \left (a-\frac{b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right )+2430 b d^2 \sqrt{c+d x} \cos (a+b x)-30 b d^2 \sqrt{c+d x} \cos (3 (a+b x))}{864 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-648*b^3*c^2*Sqrt[c + d*x]*Cos[a + b*x] + 2430*b*d^2*Sqrt[c + d*x]*Cos[a + b*x] - 1296*b^3*c*d*x*Sqrt[c + d*x
]*Cos[a + b*x] - 648*b^3*d^2*x^2*Sqrt[c + d*x]*Cos[a + b*x] + 72*b^3*c^2*Sqrt[c + d*x]*Cos[3*(a + b*x)] - 30*b
*d^2*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 144*b^3*c*d*x*Sqrt[c + d*x]*Cos[3*(a + b*x)] + 72*b^3*d^2*x^2*Sqrt[c + d
*x]*Cos[3*(a + b*x)] - 1215*Sqrt[b/d]*d^3*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d
*x]] + 5*Sqrt[b/d]*d^3*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]] - 5*Sqrt[b
/d]*d^3*Sqrt[6*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 1215*Sqrt[b/d]*d^3*Sqrt
[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*Sin[a - (b*c)/d] + 1620*b^2*c*d*Sqrt[c + d*x]*Sin[a + b*x]
+ 1620*b^2*d^2*x*Sqrt[c + d*x]*Sin[a + b*x] - 60*b^2*c*d*Sqrt[c + d*x]*Sin[3*(a + b*x)] - 60*b^2*d^2*x*Sqrt[c
+ d*x]*Sin[3*(a + b*x)])/(864*b^4)

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Maple [A]  time = 0.013, size = 476, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d} \left ( -3/8\,{\frac{d \left ( dx+c \right ) ^{5/2}}{b}\cos \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }+{\frac{15\,d}{8\,b} \left ( 1/2\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }-3/2\,{\frac{d}{b} \left ( -1/2\,{\frac{d\sqrt{dx+c}}{b}\cos \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{da-cb}{d}} \right ) }+1/4\,{\frac{d\sqrt{2}\sqrt{\pi }}{b} \left ( \cos \left ({\frac{da-cb}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ({\frac{da-cb}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) }+1/24\,{\frac{d \left ( dx+c \right ) ^{5/2}}{b}\cos \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{da-cb}{d}} \right ) }-{\frac{5\,d}{24\,b} \left ( 1/6\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\sin \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{da-cb}{d}} \right ) }-1/2\,{\frac{d}{b} \left ( -1/6\,{\frac{d\sqrt{dx+c}}{b}\cos \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{da-cb}{d}} \right ) }+1/36\,{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{3}}{b} \left ( \cos \left ( 3\,{\frac{da-cb}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 3\,{\frac{da-cb}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-3/8/b*d*(d*x+c)^(5/2)*cos(1/d*(d*x+c)*b+(a*d-b*c)/d)+15/8/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))))+1/24/b*d*(d*x+c)^(5/2)*cos(3/d*(d*x+c)*b+3*(a*d-b*c)/d)-5/24/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)))))

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Maxima [C]  time = 2.20947, size = 1868, normalized size = 4.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3456*sqrt(3)*(80*sqrt(3)*(d*x + c)^(3/2)*b*d^2*abs(b)*sin(3*((d*x + c)*b - b*c + a*d)/d)/abs(d) - 2160*sqrt
(3)*(d*x + c)^(3/2)*b*d^2*abs(b)*sin(((d*x + c)*b - b*c + a*d)/d)/abs(d) - 8*(12*sqrt(3)*(d*x + c)^(5/2)*b^2*d
*abs(b)/abs(d) - 5*sqrt(3)*sqrt(d*x + c)*d^3*abs(b)/abs(d))*cos(3*((d*x + c)*b - b*c + a*d)/d) + 216*(4*sqrt(3
)*(d*x + c)^(5/2)*b^2*d*abs(b)/abs(d) - 15*sqrt(3)*sqrt(d*x + c)*d^3*abs(b)/abs(d))*cos(((d*x + c)*b - b*c + a
*d)/d) - ((5*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*cos(-1/4*pi +
1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 5*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(
0, d/sqrt(d^2))) + 5*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b
)/abs(d))*cos(-3*(b*c - a*d)/d) - (5*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2)))
+ 5*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*sin(1/4*pi + 1/2*ar
ctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 5*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqr
t(d^2))))*d^3*sqrt(abs(b)/abs(d))*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) + (sqrt(3)*(405*sqrt
(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0,
b) + 1/2*arctan2(0, d/sqrt(d^2))) - 405*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2
))) + 405*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*c
os(-(b*c - a*d)/d) + sqrt(3)*(-405*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) -
405*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 405*sqrt(pi)*sin(1/4*pi + 1/2*
arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d
/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) + (sqrt(3)*(405*sqrt
(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0,
b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2
))) - 405*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*c
os(-(b*c - a*d)/d) + sqrt(3)*(405*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 4
05*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 405*sqrt(pi)*sin(1/4*pi + 1/2*a
rctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 405*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/
sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) - ((5*sqrt(pi)*cos(1
/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*ar
ctan2(0, d/sqrt(d^2))) + 5*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 5*I*sqrt
(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*cos(-3*(b*c - a*d
)/d) - (-5*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 5*I*sqrt(pi)*cos(-1/4*pi
+ 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 5*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(
0, d/sqrt(d^2))) - 5*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/
abs(d))*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)))*abs(d)/(b^3*d*abs(b))

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Fricas [A]  time = 2.76394, size = 923, normalized size = 2.25 \begin{align*} \frac{5 \, \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 ) - 1215 \, \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 ) + 1215 \, \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 ) - 5 \, \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 ) + 24 \,{\left ({\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 5 \, b d^{2}\right )} \cos \left (b x + a\right )^{3} - 3 \,{\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 35 \, b d^{2}\right )} \cos \left (b x + a\right ) + 10 \,{\left (7 \, b^{2} d^{2} x + 7 \, b^{2} c d -{\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}}{864 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864*(5*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)))
- 1215*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))) +
1215*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) - 5*s
qrt(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) + 24*((12
*b^3*d^2*x^2 + 24*b^3*c*d*x + 12*b^3*c^2 - 5*b*d^2)*cos(b*x + a)^3 - 3*(12*b^3*d^2*x^2 + 24*b^3*c*d*x + 12*b^3
*c^2 - 35*b*d^2)*cos(b*x + a) + 10*(7*b^2*d^2*x + 7*b^2*c*d - (b^2*d^2*x + b^2*c*d)*cos(b*x + a)^2)*sin(b*x +
a))*sqrt(d*x + c))/b^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.62919, size = 2726, normalized size = 6.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1728*(12*(sqrt(6)*sqrt(pi)*d^2*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) - 27*sqrt(2)*sqrt(pi)*d^2*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
7*sqrt(2)*sqrt(pi)*d^2*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) + sqrt(6)*sqrt(pi)*d^2*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) + 6*sqrt(d
*x + c)*d*e^((3*I*(d*x + c)*b - 3*I*b*c + 3*I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d
)/d)/b - 54*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b + 6*sqrt(d*x + c)*d*e^((-3*I*(d*x + c)*b
+ 3*I*b*c - 3*I*a*d)/d)/b)*c^2 - d^2*((I*sqrt(6)*sqrt(pi)*(12*I*b^2*c^2*d - 12*b*c*d^2 - 5*I*d^3)*d*erf(-1/2*s
qrt(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 + 24*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 + 12*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^2 + 27*(I*sqrt(2)*sqrt(pi)*(-12*I*b^2*c^2*d + 36*b*c*d^2 + 45*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*(12*I*(d*x + c)^(5/2)*b^2*d - 24*I*(d*x + c)^(3/2)*b^2*c*d + 12*I*sqrt(d*x + c)*b^2*c^2*d + 3
0*(d*x + c)^(3/2)*b*d^2 - 36*sqrt(d*x + c)*b*c*d^2 - 45*I*sqrt(d*x + c)*d^3)*e^((-I*(d*x + c)*b + I*b*c - I*a*
d)/d)/b^3)/d^2 + 27*(I*sqrt(2)*sqrt(pi)*(-12*I*b^2*c^2*d - 36*b*c*d^2 + 45*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*(12*I*(d*x + c)^(5/2)*b^2*d - 24*I*(d*x + c)^(3/2)*b^2*c*d + 12*I*sqrt(d*x + c)*b^2*c^2*d - 30*(d*x + c
)^(3/2)*b*d^2 + 36*sqrt(d*x + c)*b*c*d^2 - 45*I*sqrt(d*x + c)*d^3)*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^3)/
d^2 + (I*sqrt(6)*sqrt(pi)*(12*I*b^2*c^2*d + 12*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 + 24*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 - 12*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^2) -
12*(I*sqrt(6)*sqrt(pi)*(-2*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) + 9*I*sqrt(2)*sqrt(pi)*(6*I*b*c*d - 9
*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) + 9*I*sqrt(2)*sqrt(pi)*(6*I*b*c*d + 9*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) + I*sq
rt(6)*sqrt(pi)*(-2*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 + 2*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 - 18*I*(6*I*(d*x + c)^(3/2
)*b*d - 6*I*sqrt(d*x + c)*b*c*d - 9*sqrt(d*x + c)*d^2)*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^2 - 18*I*(6*I*(
d*x + c)^(3/2)*b*d - 6*I*sqrt(d*x + c)*b*c*d + 9*sqrt(d*x + c)*d^2)*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b^2
- 6*I*(-2*I*(d*x + c)^(3/2)*b*d + 2*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)*c)/d