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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.879348, antiderivative size = 534, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [C]  time = 12.2837, 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} \text{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}} \text{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 ) \text{FresnelC}\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 ) \text{FresnelC}\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 } \text{FresnelC}\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 } \text{FresnelC}\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 } \text{FresnelC}\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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Maple [A]  time = 0.042, size = 580, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}

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 = 2.90146, size = 2790, normalized size = 5.22 \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)^(3/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/288000*sqrt(5)*sqrt(3)*(240*sqrt(5)*sqrt(3)*(d*x + c)^(3/2)*b*d*sqrt(abs(b)/abs(d))*abs(b)*cos(5*((d*x + c)*
b - b*c + a*d)/d)/abs(d) - 400*sqrt(5)*sqrt(3)*(d*x + c)^(3/2)*b*d*sqrt(abs(b)/abs(d))*abs(b)*cos(3*((d*x + c)
*b - b*c + a*d)/d)/abs(d) - 2400*sqrt(5)*sqrt(3)*(d*x + c)^(3/2)*b*d*sqrt(abs(b)/abs(d))*abs(b)*cos(((d*x + c)
*b - b*c + a*d)/d)/abs(d) - 72*sqrt(5)*sqrt(3)*sqrt(d*x + c)*d^2*sqrt(abs(b)/abs(d))*abs(b)*sin(5*((d*x + c)*b
- b*c + a*d)/d)/abs(d) + 200*sqrt(5)*sqrt(3)*sqrt(d*x + c)*d^2*sqrt(abs(b)/abs(d))*abs(b)*sin(3*((d*x + c)*b
- b*c + a*d)/d)/abs(d) + 3600*sqrt(5)*sqrt(3)*sqrt(d*x + c)*d^2*sqrt(abs(b)/abs(d))*abs(b)*sin(((d*x + c)*b -
b*c + a*d)/d)/abs(d) - (sqrt(3)*(-9*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) -
9*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 9*sqrt(pi)*sin(1/4*pi + 1/2*arc
tan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt
(d^2))))*d^2*abs(b)*cos(-5*(b*c - a*d)/d)/abs(d) - sqrt(3)*(9*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*ar
ctan2(0, d/sqrt(d^2))) + 9*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 9*I*sqrt(
pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0,
b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(b)*sin(-5*(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(5*I*b/d)) -
(sqrt(5)*(25*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 25*I*sqrt(pi)*cos(-1/
4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 25*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arc
tan2(0, d/sqrt(d^2))) - 25*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(b)
*cos(-3*(b*c - a*d)/d)/abs(d) + sqrt(5)*(25*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^
2))) + 25*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 25*I*sqrt(pi)*sin(1/4*pi +
1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 25*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan
2(0, d/sqrt(d^2))))*d^2*abs(b)*sin(-3*(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) - (sqrt(5)*sqrt(
3)*(450*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 450*I*sqrt(pi)*cos(-1/4*pi
+ 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 450*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2
(0, d/sqrt(d^2))) - 450*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(b)*co
s(-(b*c - a*d)/d)/abs(d) + sqrt(5)*sqrt(3)*(450*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqr
t(d^2))) + 450*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 450*I*sqrt(pi)*sin(1/
4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 450*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2
*arctan2(0, d/sqrt(d^2))))*d^2*abs(b)*sin(-(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) - (sqrt(5)*sq
rt(3)*(-450*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 450*I*sqrt(pi)*cos(-1/4
*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 450*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arc
tan2(0, d/sqrt(d^2))) - 450*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(b
)*cos(-(b*c - a*d)/d)/abs(d) + sqrt(5)*sqrt(3)*(450*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d
/sqrt(d^2))) + 450*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 450*I*sqrt(pi)*si
n(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 450*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) +
1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(b)*sin(-(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) - (sqrt(
5)*(-25*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 25*I*sqrt(pi)*cos(-1/4*pi +
1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 25*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0
, d/sqrt(d^2))) - 25*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(b)*cos(-
3*(b*c - a*d)/d)/abs(d) + sqrt(5)*(25*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) +
25*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 25*I*sqrt(pi)*sin(1/4*pi + 1/2*a
rctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 25*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d
/sqrt(d^2))))*d^2*abs(b)*sin(-3*(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d)) - (sqrt(3)*(9*I*sqrt(
pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0,
b) + 1/2*arctan2(0, d/sqrt(d^2))) - 9*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) +
9*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(b)*cos(-5*(b*c - a*d)/d)/a
bs(d) - sqrt(3)*(9*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*sqrt(pi)*cos(-1/
4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*ar
ctan2(0, d/sqrt(d^2))) - 9*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*abs(
b)*sin(-5*(b*c - a*d)/d)/abs(d))*erf(sqrt(d*x + c)*sqrt(-5*I*b/d)))*abs(d)/(b^2*d*sqrt(abs(b)/abs(d))*abs(b))

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Fricas [A]  time = 0.780473, size = 1118, normalized size = 2.09 \begin{align*} \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}} \end{align*}

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

[Out]

Timed out

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Giac [C]  time = 1.80713, size = 2269, normalized size = 4.25 \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)^(3/2)*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/144000*(10*(9*sqrt(10)*sqrt(pi)*d^2*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)*d^2*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)*b) - 450*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) - 450*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) -
25*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) + 9*sqrt(10)*sqrt(pi)*d^2*erf(-1/2*sqrt(10)*sqrt(b*d)*sqr
t(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 - 9*I*sqrt(10)*sqrt(pi)*(-10*I*b*c*d + 3*
d^2)*d*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((5*I*b*c - 5*I*a*d)/d)/(sqrt(
b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b^2) - 125*I*sqrt(6)*sqrt(pi)*(2*I*b*c*d - d^2)*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^2) -
2250*I*sqrt(2)*sqrt(pi)*(2*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) - 2250*I*sqrt(2)*sqrt(pi)*(2*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) - 125*I*sqrt(6)*sqrt(pi)*(2*I*b*c*d + d^2)*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^2
) - 9*I*sqrt(10)*sqrt(pi)*(-10*I*b*c*d - 3*d^2)*d*erf(-1/2*sqrt(10)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d
^2) + 1)/d)*e^((-5*I*b*c + 5*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b^2) + 90*I*(-10*I*(d*x + c)^(3/2
)*b*d + 10*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 + 750*
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 + 4500*I*(2*I*(d*x + c)^(3/2)*b*d - 2*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 + 4500*I*(2*I*(d*x + c)^(3/2)*b*d - 2*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 + 750*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 + 90*I*(-10*I*(d*x + c)^(3/2)*b*d + 10*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