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

Optimal. Leaf size=406 \[ -\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \]

[Out]

(5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(8*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(72*b^2) + (15*d^(5/2)*Sqr
t[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(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]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) - (5*d^(5/2)*Sq
rt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2
)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) - (15*d^2*Sqr
t[c + d*x]*Sin[a + b*x])/(16*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*
x])/(144*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(12*b)

________________________________________________________________________________________

Rubi [A]  time = 0.667682, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \sin \left (3 a-\frac{3 b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \sin \left (a-\frac{b c}{d}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}+\frac{15 \sqrt{\frac{\pi }{2}} d^{5/2} \cos \left (a-\frac{b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{16 b^{7/2}}-\frac{5 \sqrt{\frac{\pi }{6}} d^{5/2} \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\frac{\sqrt{b} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{144 b^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}+\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*d*(c + d*x)^(3/2)*Cos[a + b*x])/(8*b^2) - (5*d*(c + d*x)^(3/2)*Cos[3*a + 3*b*x])/(72*b^2) + (15*d^(5/2)*Sqr
t[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(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]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(144*b^(7/2)) - (5*d^(5/2)*Sq
rt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(144*b^(7/2)) + (15*d^(5/2
)*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(16*b^(7/2)) - (15*d^2*Sqr
t[c + d*x]*Sin[a + b*x])/(16*b^3) + ((c + d*x)^(5/2)*Sin[a + b*x])/(4*b) + (5*d^2*Sqrt[c + d*x]*Sin[3*a + 3*b*
x])/(144*b^3) - ((c + d*x)^(5/2)*Sin[3*a + 3*b*x])/(12*b)

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)^{5/2} \cos (a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac{1}{4} (c+d x)^{5/2} \cos (a+b x)-\frac{1}{4} (c+d x)^{5/2} \cos (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int (c+d x)^{5/2} \cos (a+b x) \, dx-\frac{1}{4} \int (c+d x)^{5/2} \cos (3 a+3 b x) \, dx\\ &=\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}+\frac{(5 d) \int (c+d x)^{3/2} \sin (3 a+3 b x) \, dx}{24 b}-\frac{(5 d) \int (c+d x)^{3/2} \sin (a+b x) \, dx}{8 b}\\ &=\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}+\frac{\left (5 d^2\right ) \int \sqrt{c+d x} \cos (3 a+3 b x) \, dx}{48 b^2}-\frac{\left (15 d^2\right ) \int \sqrt{c+d x} \cos (a+b x) \, dx}{16 b^2}\\ &=\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\frac{\left (5 d^3\right ) \int \frac{\sin (3 a+3 b x)}{\sqrt{c+d x}} \, dx}{288 b^3}+\frac{\left (15 d^3\right ) \int \frac{\sin (a+b x)}{\sqrt{c+d x}} \, dx}{32 b^3}\\ &=\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\frac{\left (5 d^3 \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}{288 b^3}+\frac{\left (15 d^3 \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^3}-\frac{\left (5 d^3 \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}{288 b^3}+\frac{\left (15 d^3 \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^3}\\ &=\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}-\frac{\left (5 d^2 \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 )}{144 b^3}+\frac{\left (15 d^2 \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^3}-\frac{\left (5 d^2 \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 )}{144 b^3}+\frac{\left (15 d^2 \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^3}\\ &=\frac{5 d (c+d x)^{3/2} \cos (a+b x)}{8 b^2}-\frac{5 d (c+d x)^{3/2} \cos (3 a+3 b x)}{72 b^2}+\frac{15 d^{5/2} \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^{7/2}}-\frac{5 d^{5/2} \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 )}{144 b^{7/2}}-\frac{5 d^{5/2} \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 )}{144 b^{7/2}}+\frac{15 d^{5/2} \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^{7/2}}-\frac{15 d^2 \sqrt{c+d x} \sin (a+b x)}{16 b^3}+\frac{(c+d x)^{5/2} \sin (a+b x)}{4 b}+\frac{5 d^2 \sqrt{c+d x} \sin (3 a+3 b x)}{144 b^3}-\frac{(c+d x)^{5/2} \sin (3 a+3 b x)}{12 b}\\ \end{align*}

Mathematica [C]  time = 14.6632, size = 1171, normalized size = 2.88 \[ -\frac{i 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 ) c^2}{8 b}-\frac{\left (-\sqrt{2 \pi } \cos \left (3 a-\frac{3 b c}{d}\right ) S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right )-\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right ) \sin \left (3 a-\frac{3 b c}{d}\right )+2 \sqrt{3} \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (3 (a+b x))\right ) c^2}{24 \sqrt{3} b \sqrt{\frac{b}{d}}}+\frac{d \left (\sqrt{\frac{b}{d}} \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right ) \left (2 b c \sin \left (a-\frac{b c}{d}\right )-3 d \cos \left (a-\frac{b c}{d}\right )\right )+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\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 b \sqrt{c+d x} (3 \cos (a+b x)+2 b x \sin (a+b x))\right ) c}{8 b^3}-\frac{d \left (\sqrt{\frac{b}{d}} \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right ) \left (2 b c \sin \left (3 a-\frac{3 b c}{d}\right )-d \cos \left (3 a-\frac{3 b c}{d}\right )\right )+\sqrt{\frac{b}{d}} \sqrt{2 \pi } S\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} b \sqrt{c+d x} (\cos (3 (a+b x))+2 b x \sin (3 (a+b x)))\right ) c}{24 \sqrt{3} b^3}+\frac{\left (\frac{b}{d}\right )^{3/2} d^2 \left (-\sqrt{2 \pi } S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{2}{\pi }} \sqrt{c+d x}\right ) \left (\left (4 b^2 c^2-15 d^2\right ) \cos \left (a-\frac{b c}{d}\right )+12 b c d \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 (\left (4 b^2 c^2-15 d^2\right ) \sin \left (a-\frac{b c}{d}\right )-12 b c d \cos \left (a-\frac{b c}{d}\right )\right )+2 \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left (d \left (4 b^2 x^2-15\right ) \sin (a+b x)-2 b (c-5 d x) \cos (a+b x)\right )\right )}{32 b^5}-\frac{\left (\frac{b}{d}\right )^{3/2} d^2 \left (-\sqrt{2 \pi } S\left (\sqrt{\frac{b}{d}} \sqrt{\frac{6}{\pi }} \sqrt{c+d x}\right ) \left (\left (12 b^2 c^2-5 d^2\right ) \cos \left (3 a-\frac{3 b c}{d}\right )+12 b c d \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 (\left (12 b^2 c^2-5 d^2\right ) \sin \left (3 a-\frac{3 b c}{d}\right )-12 b c d \cos \left (3 a-\frac{3 b c}{d}\right )\right )+2 \sqrt{3} \sqrt{\frac{b}{d}} d \sqrt{c+d x} \left (d \left (12 b^2 x^2-5\right ) \sin (3 (a+b x))-2 b (c-5 d x) \cos (3 (a+b x))\right )\right )}{288 \sqrt{3} b^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/8)*c^2*Sqrt[c + d*x]*((E^((2*I)*a)*Gamma[3/2, ((-I)*b*(c + d*x))/d])/Sqrt[((-I)*b*(c + d*x))/d] - (E^(((2
*I)*b*c)/d)*Gamma[3/2, (I*b*(c + d*x))/d])/Sqrt[(I*b*(c + d*x))/d]))/(b*E^((I*(b*c + a*d))/d)) + (c*d*(Sqrt[b/
d]*Sqrt[2*Pi]*FresnelC[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[b/d]*Sqrt[2*Pi]*FresnelS[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*b*Sqrt[c + d*x]*(3*Cos[a + b*x] + 2*b*x*Sin[a + b*x])))/(8*b^3) + ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*Fresn
elS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*((4*b^2*c^2 - 15*d^2)*Cos[a - (b*c)/d] + 12*b*c*d*Sin[a - (b*c)/d])) -
Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[a - (b*c)/d] + (4*b^2*c^2 - 15*d^2)*Si
n[a - (b*c)/d]) + 2*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[a + b*x] + d*(-15 + 4*b^2*x^2)*Sin[a + b*x
])))/(32*b^5) - (c^2*(-(Sqrt[2*Pi]*Cos[3*a - (3*b*c)/d]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) - Sqrt[2
*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*Sin[3*a - (3*b*c)/d] + 2*Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]*Sin
[3*(a + b*x)]))/(24*Sqrt[3]*b*Sqrt[b/d]) - (c*d*(Sqrt[b/d]*Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d
*x]]*(-(d*Cos[3*a - (3*b*c)/d]) + 2*b*c*Sin[3*a - (3*b*c)/d]) + Sqrt[b/d]*Sqrt[2*Pi]*FresnelS[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]*b*Sqrt[c + d*x]*(Cos[3*(
a + b*x)] + 2*b*x*Sin[3*(a + b*x)])))/(24*Sqrt[3]*b^3) - ((b/d)^(3/2)*d^2*(-(Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqr
t[6/Pi]*Sqrt[c + d*x]]*((12*b^2*c^2 - 5*d^2)*Cos[3*a - (3*b*c)/d] + 12*b*c*d*Sin[3*a - (3*b*c)/d])) - Sqrt[2*P
i]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]*(-12*b*c*d*Cos[3*a - (3*b*c)/d] + (12*b^2*c^2 - 5*d^2)*Sin[3*a
- (3*b*c)/d]) + 2*Sqrt[3]*Sqrt[b/d]*d*Sqrt[c + d*x]*(-2*b*(c - 5*d*x)*Cos[3*(a + b*x)] + d*(-5 + 12*b^2*x^2)*
Sin[3*(a + b*x)])))/(288*Sqrt[3]*b^5)

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Maple [A]  time = 0.036, size = 474, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{d} \left ( 1/8\,{\frac{d \left ( dx+c \right ) ^{5/2}}{b}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{ad-bc}{d}} \right ) }-5/8\,{\frac{d}{b} \left ( -1/2\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\cos \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{ad-bc}{d}} \right ) }+3/2\,{\frac{d}{b} \left ( 1/2\,{\frac{d\sqrt{dx+c}}{b}\sin \left ({\frac{ \left ( dx+c \right ) b}{d}}+{\frac{ad-bc}{d}} \right ) }-1/4\,{\frac{d\sqrt{2}\sqrt{\pi }}{b} \left ( \cos \left ({\frac{ad-bc}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ({\frac{ad-bc}{d}} \right ){\it FresnelC} \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}\sin \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{ad-bc}{d}} \right ) }+{\frac{5\,d}{24\,b} \left ( -1/6\,{\frac{d \left ( dx+c \right ) ^{3/2}}{b}\cos \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{ad-bc}{d}} \right ) }+1/2\,{\frac{d}{b} \left ( 1/6\,{\frac{d\sqrt{dx+c}}{b}\sin \left ( 3\,{\frac{ \left ( dx+c \right ) b}{d}}+3\,{\frac{ad-bc}{d}} \right ) }-1/36\,{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{3}}{b} \left ( \cos \left ( 3\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{dx+c}b}{\sqrt{\pi }d}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) +\sin \left ( 3\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \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)*cos(b*x+a)*sin(b*x+a)^2,x)

[Out]

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

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Maxima [C]  time = 2.45579, size = 1866, normalized size = 4.6 \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)*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/3456*sqrt(3)*(80*sqrt(3)*(d*x + c)^(3/2)*b*d^2*abs(b)*cos(3*((d*x + c)*b - b*c + a*d)/d)/abs(d) - 720*sqrt(
3)*(d*x + c)^(3/2)*b*d^2*abs(b)*cos(((d*x + c)*b - b*c + a*d)/d)/abs(d) + ((5*I*sqrt(pi)*cos(1/4*pi + 1/2*arct
an2(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/sqr
t(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))*cos(-3*(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))*sin(-3*(b*c
- a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) + (sqrt(3)*(-135*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*
arctan2(0, d/sqrt(d^2))) - 135*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 135
*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 135*sqrt(pi)*sin(-1/4*pi + 1/2*arcta
n2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*cos(-(b*c - a*d)/d) - sqrt(3)*(135*sqrt(pi)*c
os(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 135*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) +
1/2*arctan2(0, d/sqrt(d^2))) - 135*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) +
135*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))*sin(-(b
*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(I*b/d)) + (sqrt(3)*(135*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*a
rctan2(0, d/sqrt(d^2))) + 135*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 135*
sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 135*sqrt(pi)*sin(-1/4*pi + 1/2*arctan
2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*cos(-(b*c - a*d)/d) - sqrt(3)*(135*sqrt(pi)*co
s(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 135*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1
/2*arctan2(0, d/sqrt(d^2))) + 135*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 1
35*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))*sin(-(b*
c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-I*b/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*a
rctan2(0, d/sqrt(d^2))))*d^3*sqrt(abs(b)/abs(d))*cos(-3*(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))*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x +
c)*sqrt(-3*I*b/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)/a
bs(d))*sin(3*((d*x + c)*b - b*c + a*d)/d) - 72*(4*sqrt(3)*(d*x + c)^(5/2)*b^2*d*abs(b)/abs(d) - 15*sqrt(3)*sqr
t(d*x + c)*d^3*abs(b)/abs(d))*sin(((d*x + c)*b - b*c + a*d)/d))*abs(d)/(b^3*d*abs(b))

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Fricas [A]  time = 0.638412, size = 918, normalized size = 2.26 \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{S}\left (\sqrt{6} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 405 \, \sqrt{2} \pi d^{3} \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 ) - 405 \, \sqrt{2} \pi d^{3} \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 ) + 5 \, \sqrt{6} \pi d^{3} \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 ) + 24 \,{\left (10 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right )^{3} - 30 \,{\left (b^{2} d^{2} x + b^{2} c d\right )} \cos \left (b x + a\right ) -{\left (12 \, b^{3} d^{2} x^{2} + 24 \, b^{3} c d x + 12 \, b^{3} c^{2} - 35 \, b d^{2} -{\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 )^{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)*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/864*(5*sqrt(6)*pi*d^3*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))
) - 405*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) -
405*sqrt(2)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) + 5*sq
rt(6)*pi*d^3*sqrt(b/(pi*d))*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 24*(10*(
b^2*d^2*x + b^2*c*d)*cos(b*x + a)^3 - 30*(b^2*d^2*x + b^2*c*d)*cos(b*x + a) - (12*b^3*d^2*x^2 + 24*b^3*c*d*x +
12*b^3*c^2 - 35*b*d^2 - (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)^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)*cos(b*x+a)*sin(b*x+a)**2,x)

[Out]

Timed out

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Giac [C]  time = 1.5621, size = 2722, normalized size = 6.7 \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)*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="giac")

[Out]

-1/1728*(12*(I*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*I*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)
+ 9*I*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) - I*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
*I*sqrt(d*x + c)*d*e^((3*I*(d*x + c)*b - 3*I*b*c + 3*I*a*d)/d)/b + 18*I*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*
b*c + I*a*d)/d)/b - 18*I*sqrt(d*x + c)*d*e^((-I*(d*x + c)*b + I*b*c - I*a*d)/d)/b + 6*I*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*((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*(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*(sqrt(2)*sqrt(pi)*(4*I*b^2*c^2*d - 12*b*c*d^2 - 15*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*(-4*I*(d*x + c)^(5/2)*b^2*d + 8*I*(d*x + c)^(3/2)*b^2*c*d - 4*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 + 15*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*(sqrt(2)*sqrt(pi)*(-4*I*b^2*c^2*d - 12*b*c*d^2 + 15*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*(4*I*(d*x + c)^(5/2)*b^2*d - 8*I*(d*x + c)^(3/2)*b^2*c*d + 4*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 - 15*I*sqrt(d*x + c)*d^3)*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b^3)/d^2 +
(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*(-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*s
qrt(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*(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*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) + 9*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) + 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) - 6*(-2*I*(d*x + c)^(3/2)*b*d + 2*I*sqrt(d*x + c)*b*c*d + sq
rt(d*x + c)*d^2)*e^((3*I*(d*x + c)*b - 3*I*b*c + 3*I*a*d)/d)/b^2 - 18*(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 - 18*(-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 - 6*(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