3.222 \(\int (e+f x)^2 \sin (a+\frac{b}{(c+d x)^{2/3}}) \, dx\)

Optimal. Leaf size=630 \[ \frac{b^3 f \cos (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 \sqrt{2 \pi } b^{3/2} \sin (a) (d e-c f)^2 \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{2 \sqrt{2 \pi } b^{3/2} \cos (a) (d e-c f)^2 S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^3 f \sin (a) (d e-c f) \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{b^2 f (c+d x)^{2/3} (d e-c f) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{16 \sqrt{2 \pi } b^{9/2} f^2 \cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac{16 \sqrt{2 \pi } b^{9/2} f^2 \sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac{16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac{4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{105 d^3}-\frac{8 b^3 f^2 (c+d x) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac{f (c+d x)^2 (d e-c f) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{(c+d x) (d e-c f)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{b f (c+d x)^{4/3} (d e-c f) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3} \]

[Out]

(2*b*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/d^3 - (8*b^3*f^2*(c + d*x)*Cos[a + b/(c + d*x)^
(2/3)])/(315*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(2/3)])/(2*d^3) + (2*b*f^2*(c + d*x)^
(7/3)*Cos[a + b/(c + d*x)^(2/3)])/(21*d^3) + (b^3*f*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^(2/3)])/(2*d^3)
- (16*b^(9/2)*f^2*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(315*d^3) + (2*b^(3/2)*(d
*e - c*f)^2*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/d^3 + (2*b^(3/2)*(d*e - c*f)^2*S
qrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/d^3 + (16*b^(9/2)*f^2*Sqrt[2*Pi]*FresnelS[(Sq
rt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(315*d^3) + (16*b^4*f^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)])
/(315*d^3) - (b^2*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/(2*d^3) + ((d*e - c*f)^2*(c + d*x)
*Sin[a + b/(c + d*x)^(2/3)])/d^3 - (4*b^2*f^2*(c + d*x)^(5/3)*Sin[a + b/(c + d*x)^(2/3)])/(105*d^3) + (f*(d*e
- c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^(2/3)])/d^3 + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(2/3)])/(3*d^3) - (b
^3*f*(d*e - c*f)*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)])/(2*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.747672, antiderivative size = 630, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {3433, 3409, 3387, 3388, 3353, 3352, 3351, 3379, 3297, 3303, 3299, 3302, 3354} \[ \frac{b^3 f \cos (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 \sqrt{2 \pi } b^{3/2} \sin (a) (d e-c f)^2 \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{2 \sqrt{2 \pi } b^{3/2} \cos (a) (d e-c f)^2 S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^3 f \sin (a) (d e-c f) \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{b^2 f (c+d x)^{2/3} (d e-c f) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{16 \sqrt{2 \pi } b^{9/2} f^2 \cos (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac{16 \sqrt{2 \pi } b^{9/2} f^2 \sin (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac{16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac{4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{105 d^3}-\frac{8 b^3 f^2 (c+d x) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac{f (c+d x)^2 (d e-c f) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{(c+d x) (d e-c f)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{b f (c+d x)^{4/3} (d e-c f) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/d^3 - (8*b^3*f^2*(c + d*x)*Cos[a + b/(c + d*x)^
(2/3)])/(315*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(2/3)])/(2*d^3) + (2*b*f^2*(c + d*x)^
(7/3)*Cos[a + b/(c + d*x)^(2/3)])/(21*d^3) + (b^3*f*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^(2/3)])/(2*d^3)
- (16*b^(9/2)*f^2*Sqrt[2*Pi]*Cos[a]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/(315*d^3) + (2*b^(3/2)*(d
*e - c*f)^2*Sqrt[2*Pi]*Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)])/d^3 + (2*b^(3/2)*(d*e - c*f)^2*S
qrt[2*Pi]*FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/d^3 + (16*b^(9/2)*f^2*Sqrt[2*Pi]*FresnelS[(Sq
rt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(315*d^3) + (16*b^4*f^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(2/3)])
/(315*d^3) - (b^2*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)])/(2*d^3) + ((d*e - c*f)^2*(c + d*x)
*Sin[a + b/(c + d*x)^(2/3)])/d^3 - (4*b^2*f^2*(c + d*x)^(5/3)*Sin[a + b/(c + d*x)^(2/3)])/(105*d^3) + (f*(d*e
- c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^(2/3)])/d^3 + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(2/3)])/(3*d^3) - (b
^3*f*(d*e - c*f)*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)])/(2*d^3)

Rule 3433

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[k/f^(m + 1), Subst[Int[ExpandIntegrand[(a + b*Sin[c +
d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f,
g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]

Rule 3409

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subst[Int[(a + b*Sin[c + d/x^
n])^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m] && EqQ[n, -2
]

Rule 3387

Int[((e_.)*(x_))^(m_)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[((e*x)^(m + 1)*Sin[c + d*x^n])/(e*(m + 1
)), x] - Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n,
0] && LtQ[m, -1]

Rule 3388

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_), x_Symbol] :> Simp[((e*x)^(m + 1)*Cos[c + d*x^n])/(e*(m + 1
)), x] + Dist[(d*n)/(e^n*(m + 1)), Int[(e*x)^(m + n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n,
0] && LtQ[m, -1]

Rule 3353

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, 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 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 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3297

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

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3354

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rubi steps

\begin{align*} \int (e+f x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right ) \, dx &=\frac{3 \operatorname{Subst}\left (\int \left ((d e-c f)^2 x^2 \sin \left (a+\frac{b}{x^2}\right )-2 f (-d e+c f) x^5 \sin \left (a+\frac{b}{x^2}\right )+f^2 x^8 \sin \left (a+\frac{b}{x^2}\right )\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int x^8 \sin \left (a+\frac{b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac{(6 f (d e-c f)) \operatorname{Subst}\left (\int x^5 \sin \left (a+\frac{b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac{\left (3 (d e-c f)^2\right ) \operatorname{Subst}\left (\int x^2 \sin \left (a+\frac{b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=-\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^{10}} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{(3 f (d e-c f)) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^4} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{d^3}-\frac{\left (3 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (a+b x^2\right )}{x^8} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac{(b f (d e-c f)) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^3} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{d^3}-\frac{\left (2 b (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (a+b x^2\right )}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac{2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac{\left (4 b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^6} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{21 d^3}+\frac{\left (b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{\left (4 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac{2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3}-\frac{b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}+\frac{\left (8 b^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos \left (a+b x^2\right )}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{105 d^3}+\frac{\left (b^3 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{\left (4 b^2 (d e-c f)^2 \cos (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{\left (4 b^2 (d e-c f)^2 \sin (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac{2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{8 b^3 f^2 (c+d x) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac{b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac{\left (16 b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac{\left (b^3 f (d e-c f) \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{\left (b^3 f (d e-c f) \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d^3}\\ &=\frac{2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{8 b^3 f^2 (c+d x) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac{b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac{b^3 f (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac{16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac{b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac{b^3 f (d e-c f) \sin (a) \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{\left (32 b^5 f^2\right ) \operatorname{Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{315 d^3}\\ &=\frac{2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{8 b^3 f^2 (c+d x) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac{b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac{b^3 f (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac{16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac{b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac{b^3 f (d e-c f) \sin (a) \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{\left (32 b^5 f^2 \cos (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac{\left (32 b^5 f^2 \sin (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{315 d^3}\\ &=\frac{2 b (d e-c f)^2 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{8 b^3 f^2 (c+d x) \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}+\frac{b f (d e-c f) (c+d x)^{4/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{2 b f^2 (c+d x)^{7/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{21 d^3}+\frac{b^3 f (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}-\frac{16 b^{9/2} f^2 \sqrt{2 \pi } \cos (a) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{315 d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{2 b^{3/2} (d e-c f)^2 \sqrt{2 \pi } C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d^3}+\frac{16 b^{9/2} f^2 \sqrt{2 \pi } S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{315 d^3}+\frac{16 b^4 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{315 d^3}-\frac{b^2 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}-\frac{4 b^2 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{105 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{3 d^3}-\frac{b^3 f (d e-c f) \sin (a) \text{Si}\left (\frac{b}{(c+d x)^{2/3}}\right )}{2 d^3}\\ \end{align*}

Mathematica [C]  time = 2.93087, size = 613, normalized size = 0.97 \[ \frac{i e^{-i a} \left (4 \sqrt [4]{-1} \sqrt{\pi } e^{2 i a} b^{3/2} \left (f^2 \left (8 b^3+315 i c^2\right )-630 i c d e f+315 i d^2 e^2\right ) \text{Erfi}\left (\frac{\sqrt [4]{-1} \sqrt{b}}{\sqrt [3]{c+d x}}\right )-\sqrt [3]{c+d x} e^{i \left (2 a+\frac{b}{(c+d x)^{2/3}}\right )} \left (3 b^2 f \sqrt [3]{c+d x} (97 c f-105 d e-8 d f x)-16 i b^3 f^2 (c+d x)^{2/3}+32 b^4 f^2+15 i b \left (f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )+21 d e f (d x-7 c)+84 d^2 e^2\right )+210 (c+d x)^{2/3} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )+315 i e^{2 i a} b^3 f (c f-d e) \text{Ei}\left (\frac{i b}{(c+d x)^{2/3}}\right )-4 \sqrt [4]{-1} \sqrt{\pi } b^{3/2} \left (f^2 \left (315 c^2+8 i b^3\right )-630 c d e f+315 d^2 e^2\right ) \text{Erfi}\left (\frac{(-1)^{3/4} \sqrt{b}}{\sqrt [3]{c+d x}}\right )+\sqrt [3]{c+d x} e^{-\frac{i b}{(c+d x)^{2/3}}} \left (3 b^2 f \sqrt [3]{c+d x} (97 c f-105 d e-8 d f x)+16 i b^3 f^2 (c+d x)^{2/3}+32 b^4 f^2-15 i b \left (f^2 \left (67 c^2-13 c d x+4 d^2 x^2\right )+21 d e f (d x-7 c)+84 d^2 e^2\right )+210 (c+d x)^{2/3} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )+315 i b^3 f (c f-d e) \text{Ei}\left (-\frac{i b}{(c+d x)^{2/3}}\right )\right )}{1260 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/1260)*(((c + d*x)^(1/3)*(32*b^4*f^2 + (16*I)*b^3*f^2*(c + d*x)^(2/3) + 3*b^2*f*(c + d*x)^(1/3)*(-105*d*e +
97*c*f - 8*d*f*x) - (15*I)*b*(84*d^2*e^2 + 21*d*e*f*(-7*c + d*x) + f^2*(67*c^2 - 13*c*d*x + 4*d^2*x^2)) + 210
*(c + d*x)^(2/3)*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))))/E^((I*b)/(c + d*x)^(2/3)) -
E^(I*(2*a + b/(c + d*x)^(2/3)))*(c + d*x)^(1/3)*(32*b^4*f^2 - (16*I)*b^3*f^2*(c + d*x)^(2/3) + 3*b^2*f*(c + d
*x)^(1/3)*(-105*d*e + 97*c*f - 8*d*f*x) + (15*I)*b*(84*d^2*e^2 + 21*d*e*f*(-7*c + d*x) + f^2*(67*c^2 - 13*c*d*
x + 4*d^2*x^2)) + 210*(c + d*x)^(2/3)*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))) + 4*(-1
)^(1/4)*b^(3/2)*E^((2*I)*a)*((315*I)*d^2*e^2 - (630*I)*c*d*e*f + (8*b^3 + (315*I)*c^2)*f^2)*Sqrt[Pi]*Erfi[((-1
)^(1/4)*Sqrt[b])/(c + d*x)^(1/3)] - 4*(-1)^(1/4)*b^(3/2)*(315*d^2*e^2 - 630*c*d*e*f + ((8*I)*b^3 + 315*c^2)*f^
2)*Sqrt[Pi]*Erfi[((-1)^(3/4)*Sqrt[b])/(c + d*x)^(1/3)] + (315*I)*b^3*f*(-(d*e) + c*f)*ExpIntegralEi[((-I)*b)/(
c + d*x)^(2/3)] + (315*I)*b^3*E^((2*I)*a)*f*(-(d*e) + c*f)*ExpIntegralEi[(I*b)/(c + d*x)^(2/3)]))/(d^3*E^(I*a)
)

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Maple [A]  time = 0.017, size = 452, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/d^3*(1/3*(c^2*f^2-2*c*d*e*f+d^2*e^2)*(d*x+c)*sin(a+b/(d*x+c)^(2/3))-2/3*(c^2*f^2-2*c*d*e*f+d^2*e^2)*b*(-(d*x
+c)^(1/3)*cos(a+b/(d*x+c)^(2/3))-b^(1/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1
/3))+sin(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))))+1/6*(-2*c*f^2+2*d*e*f)*(d*x+c)^2*sin(a+b/(d*x+c
)^(2/3))-1/3*(-2*c*f^2+2*d*e*f)*b*(-1/4*(d*x+c)^(4/3)*cos(a+b/(d*x+c)^(2/3))-1/2*b*(-1/2*(d*x+c)^(2/3)*sin(a+b
/(d*x+c)^(2/3))+b*(1/2*cos(a)*Ci(b/(d*x+c)^(2/3))-1/2*sin(a)*Si(b/(d*x+c)^(2/3)))))+1/9*f^2*(d*x+c)^3*sin(a+b/
(d*x+c)^(2/3))-2/9*f^2*b*(-1/7*(d*x+c)^(7/3)*cos(a+b/(d*x+c)^(2/3))-2/7*b*(-1/5*(d*x+c)^(5/3)*sin(a+b/(d*x+c)^
(2/3))+2/5*b*(-1/3*(d*x+c)*cos(a+b/(d*x+c)^(2/3))-2/3*b*(-(d*x+c)^(1/3)*sin(a+b/(d*x+c)^(2/3))+b^(1/2)*2^(1/2)
*Pi^(1/2)*(cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)/(d*x+c)^(1/3))-sin(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)/(d
*x+c)^(1/3))))))))

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Maxima [C]  time = 2.91142, size = 3386, normalized size = 5.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(630*(4*(d*x + c)^(2/3)*b*sqrt(abs(b)/(d*x + c)^(2/3))*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) + (
((I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(1/4*
pi + 1/2*arctan2(0, b)) + (I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*b/(d*x +
c)^(2/3))) - 1))*cos(-1/4*pi + 1/2*arctan2(0, b)) + (sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*
(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0, b)) - (sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/
3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) + ((sqrt(
pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(1/4*pi + 1/2*ar
ctan2(0, b)) + (sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1)
)*cos(-1/4*pi + 1/2*arctan2(0, b)) + (-I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + I*sqrt(pi)*(erf(sqrt(
-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0, b)) + (I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1
) - I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*b^2 + 2*(d*x +
c)^(4/3)*sqrt(abs(b)/(d*x + c)^(2/3))*sin(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)))*e^2/((d*x + c)^(1/3)*sqrt
(abs(b)/(d*x + c)^(2/3))) - 1260*(4*(d*x + c)^(2/3)*b*sqrt(abs(b)/(d*x + c)^(2/3))*cos(((d*x + c)^(2/3)*a + b)
/(d*x + c)^(2/3)) + (((I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(
2/3))) - 1))*cos(1/4*pi + 1/2*arctan2(0, b)) + (I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - I*sqrt(pi)*(
erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(-1/4*pi + 1/2*arctan2(0, b)) + (sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/
3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0, b)) - (sqrt(pi)*(erf(s
qrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(-1/4*pi + 1/2*arctan2(0,
b)))*cos(a) + ((sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1)
)*cos(1/4*pi + 1/2*arctan2(0, b)) + (sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(
d*x + c)^(2/3))) - 1))*cos(-1/4*pi + 1/2*arctan2(0, b)) + (-I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) +
I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0, b)) + (I*sqrt(pi)*(erf(sqrt(I*b/
(d*x + c)^(2/3))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(-1/4*pi + 1/2*arctan2(0, b)))*s
in(a))*b^2 + 2*(d*x + c)^(4/3)*sqrt(abs(b)/(d*x + c)^(2/3))*sin(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)))*c*e*
f/((d*x + c)^(1/3)*d*sqrt(abs(b)/(d*x + c)^(2/3))) + 630*(4*(d*x + c)^(2/3)*b*sqrt(abs(b)/(d*x + c)^(2/3))*cos
(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) + (((I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - I*sqrt(pi)*(e
rf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(1/4*pi + 1/2*arctan2(0, b)) + (I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/
3))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(-1/4*pi + 1/2*arctan2(0, b)) + (sqrt(pi)*(er
f(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0
, b)) - (sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(-
1/4*pi + 1/2*arctan2(0, b)))*cos(a) + ((sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*
b/(d*x + c)^(2/3))) - 1))*cos(1/4*pi + 1/2*arctan2(0, b)) + (sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + s
qrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(-1/4*pi + 1/2*arctan2(0, b)) + (-I*sqrt(pi)*(erf(sqrt(I*b/(
d*x + c)^(2/3))) - 1) + I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0, b)) + (I
*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(-1/4*pi
+ 1/2*arctan2(0, b)))*sin(a))*b^2 + 2*(d*x + c)^(4/3)*sqrt(abs(b)/(d*x + c)^(2/3))*sin(((d*x + c)^(2/3)*a + b
)/(d*x + c)^(2/3)))*c^2*f^2/((d*x + c)^(1/3)*d^2*sqrt(abs(b)/(d*x + c)^(2/3))) + 315*(((Ei(I*b/(d*x + c)^(2/3)
) + Ei(-I*b/(d*x + c)^(2/3)))*cos(a) + (I*Ei(I*b/(d*x + c)^(2/3)) - I*Ei(-I*b/(d*x + c)^(2/3)))*sin(a))*b^3 +
2*(d*x + c)^(4/3)*b*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) - 2*((d*x + c)^(2/3)*b^2 - 2*(d*x + c)^2)*sin
(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)))*e*f/d - 315*(((Ei(I*b/(d*x + c)^(2/3)) + Ei(-I*b/(d*x + c)^(2/3)))*
cos(a) + (I*Ei(I*b/(d*x + c)^(2/3)) - I*Ei(-I*b/(d*x + c)^(2/3)))*sin(a))*b^3 + 2*(d*x + c)^(4/3)*b*cos(((d*x
+ c)^(2/3)*a + b)/(d*x + c)^(2/3)) - 2*((d*x + c)^(2/3)*b^2 - 2*(d*x + c)^2)*sin(((d*x + c)^(2/3)*a + b)/(d*x
+ c)^(2/3)))*c*f^2/d^2 - 4*(((4*(sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x
+ c)^(2/3))) - 1))*cos(1/4*pi + 1/2*arctan2(0, b)) + 4*(sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(p
i)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(-1/4*pi + 1/2*arctan2(0, b)) - (4*I*sqrt(pi)*(erf(sqrt(I*b/(d*x
+ c)^(2/3))) - 1) - 4*I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0, b)) - (-4*
I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + 4*I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(-1/4
*pi + 1/2*arctan2(0, b)))*cos(a) - ((4*I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) - 4*I*sqrt(pi)*(erf(sqr
t(-I*b/(d*x + c)^(2/3))) - 1))*cos(1/4*pi + 1/2*arctan2(0, b)) + (4*I*sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3)))
- 1) - 4*I*sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*cos(-1/4*pi + 1/2*arctan2(0, b)) + 4*(sqrt(pi)*(er
f(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin(1/4*pi + 1/2*arctan2(0
, b)) - 4*(sqrt(pi)*(erf(sqrt(I*b/(d*x + c)^(2/3))) - 1) + sqrt(pi)*(erf(sqrt(-I*b/(d*x + c)^(2/3))) - 1))*sin
(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*b^5 + 2*(4*(d*x + c)^(4/3)*b^3*sqrt(abs(b)/(d*x + c)^(2/3)) - 15*(d*x +
c)^(8/3)*b*sqrt(abs(b)/(d*x + c)^(2/3)))*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)) - (16*(d*x + c)^(2/3)*b
^4*sqrt(abs(b)/(d*x + c)^(2/3)) - 12*(d*x + c)^2*b^2*sqrt(abs(b)/(d*x + c)^(2/3)) + 105*(d*x + c)^(10/3)*sqrt(
abs(b)/(d*x + c)^(2/3)))*sin(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3)))*f^2/((d*x + c)^(1/3)*d^2*sqrt(abs(b)/(d
*x + c)^(2/3))))/d

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Fricas [A]  time = 2.43705, size = 1303, normalized size = 2.07 \begin{align*} \frac{315 \,{\left (b^{3} d e f - b^{3} c f^{2}\right )} \cos \left (a\right ) \operatorname{Ci}\left (\frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right ) + 315 \,{\left (b^{3} d e f - b^{3} c f^{2}\right )} \cos \left (a\right ) \operatorname{Ci}\left (-\frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right ) - 8 \, \sqrt{2}{\left (8 \, \pi b^{4} f^{2} \cos \left (a\right ) - 315 \, \pi{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \sin \left (a\right )\right )} \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{{\left (d x + c\right )}^{\frac{1}{3}}}\right ) + 8 \, \sqrt{2}{\left (8 \, \pi b^{4} f^{2} \sin \left (a\right ) + 315 \, \pi{\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \cos \left (a\right )\right )} \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\frac{\sqrt{2} \sqrt{\frac{b}{\pi }}}{{\left (d x + c\right )}^{\frac{1}{3}}}\right ) - 630 \,{\left (b^{3} d e f - b^{3} c f^{2}\right )} \sin \left (a\right ) \operatorname{Si}\left (\frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right ) - 2 \,{\left (16 \, b^{3} d f^{2} x + 16 \, b^{3} c f^{2} - 15 \,{\left (4 \, b d^{2} f^{2} x^{2} + 84 \, b d^{2} e^{2} - 147 \, b c d e f + 67 \, b c^{2} f^{2} +{\left (21 \, b d^{2} e f - 13 \, b c d f^{2}\right )} x\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right ) + 2 \,{\left (210 \, d^{3} f^{2} x^{3} + 630 \, d^{3} e f x^{2} + 32 \,{\left (d x + c\right )}^{\frac{1}{3}} b^{4} f^{2} + 630 \, d^{3} e^{2} x + 630 \, c d^{2} e^{2} - 630 \, c^{2} d e f + 210 \, c^{3} f^{2} - 3 \,{\left (8 \, b^{2} d f^{2} x + 105 \, b^{2} d e f - 97 \, b^{2} c f^{2}\right )}{\left (d x + c\right )}^{\frac{2}{3}}\right )} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right )}{1260 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*(315*(b^3*d*e*f - b^3*c*f^2)*cos(a)*cos_integral(b/(d*x + c)^(2/3)) + 315*(b^3*d*e*f - b^3*c*f^2)*cos(a
)*cos_integral(-b/(d*x + c)^(2/3)) - 8*sqrt(2)*(8*pi*b^4*f^2*cos(a) - 315*pi*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*
f^2)*sin(a))*sqrt(b/pi)*fresnel_cos(sqrt(2)*sqrt(b/pi)/(d*x + c)^(1/3)) + 8*sqrt(2)*(8*pi*b^4*f^2*sin(a) + 315
*pi*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cos(a))*sqrt(b/pi)*fresnel_sin(sqrt(2)*sqrt(b/pi)/(d*x + c)^(1/3)) -
630*(b^3*d*e*f - b^3*c*f^2)*sin(a)*sin_integral(b/(d*x + c)^(2/3)) - 2*(16*b^3*d*f^2*x + 16*b^3*c*f^2 - 15*(4
*b*d^2*f^2*x^2 + 84*b*d^2*e^2 - 147*b*c*d*e*f + 67*b*c^2*f^2 + (21*b*d^2*e*f - 13*b*c*d*f^2)*x)*(d*x + c)^(1/3
))*cos((a*d*x + a*c + (d*x + c)^(1/3)*b)/(d*x + c)) + 2*(210*d^3*f^2*x^3 + 630*d^3*e*f*x^2 + 32*(d*x + c)^(1/3
)*b^4*f^2 + 630*d^3*e^2*x + 630*c*d^2*e^2 - 630*c^2*d*e*f + 210*c^3*f^2 - 3*(8*b^2*d*f^2*x + 105*b^2*d*e*f - 9
7*b^2*c*f^2)*(d*x + c)^(2/3))*sin((a*d*x + a*c + (d*x + c)^(1/3)*b)/(d*x + c)))/d^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e + f x\right )^{2} \sin{\left (a + \frac{b}{\left (c + d x\right )^{\frac{2}{3}}} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e + f*x)**2*sin(a + b/(c + d*x)**(2/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} \sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x + e)^2*sin(a + b/(d*x + c)^(2/3)), x)