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

Optimal. Leaf size=513 \[ \frac{3 \sqrt{\frac{\pi }{2}} \cos (a) (d e-c f)^2 \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) (d e-c f)^2 S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac{6 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac{6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}+\frac{315 \sqrt{\frac{\pi }{2}} f^2 \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac{315 \sqrt{\frac{\pi }{2}} f^2 \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac{21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac{315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac{105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac{3 f (c+d x)^{4/3} (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac{3 \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac{3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3} \]

[Out]

(6*f*(d*e - c*f)*Cos[a + b*(c + d*x)^(2/3)])/(b^3*d^3) - (3*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^
(2/3)])/(2*b*d^3) + (105*f^2*(c + d*x)*Cos[a + b*(c + d*x)^(2/3)])/(8*b^3*d^3) - (3*f*(d*e - c*f)*(c + d*x)^(4
/3)*Cos[a + b*(c + d*x)^(2/3)])/(b*d^3) - (3*f^2*(c + d*x)^(7/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d^3) + (3*(d
*e - c*f)^2*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(2*b^(3/2)*d^3) + (315*f^2*Sqrt[Pi
/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(16*b^(9/2)*d^3) + (315*f^2*Sqrt[Pi/2]*FresnelC[Sqrt[
b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(16*b^(9/2)*d^3) - (3*(d*e - c*f)^2*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/
Pi]*(c + d*x)^(1/3)]*Sin[a])/(2*b^(3/2)*d^3) - (315*f^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(2/3)])/(16*b^4*d^
3) + (6*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/(b^2*d^3) + (21*f^2*(c + d*x)^(5/3)*Sin[a +
b*(c + d*x)^(2/3)])/(4*b^2*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.535216, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {3433, 3385, 3354, 3352, 3351, 3379, 3296, 2638, 3386, 3353} \[ \frac{3 \sqrt{\frac{\pi }{2}} \cos (a) (d e-c f)^2 \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}-\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) (d e-c f)^2 S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^3}+\frac{6 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^3}+\frac{6 f (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^3}+\frac{315 \sqrt{\frac{\pi }{2}} f^2 \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac{315 \sqrt{\frac{\pi }{2}} f^2 \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{16 b^{9/2} d^3}+\frac{21 f^2 (c+d x)^{5/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d^3}-\frac{315 f^2 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{16 b^4 d^3}+\frac{105 f^2 (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{8 b^3 d^3}-\frac{3 f (c+d x)^{4/3} (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{b d^3}-\frac{3 \sqrt [3]{c+d x} (d e-c f)^2 \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3}-\frac{3 f^2 (c+d x)^{7/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*f*(d*e - c*f)*Cos[a + b*(c + d*x)^(2/3)])/(b^3*d^3) - (3*(d*e - c*f)^2*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^
(2/3)])/(2*b*d^3) + (105*f^2*(c + d*x)*Cos[a + b*(c + d*x)^(2/3)])/(8*b^3*d^3) - (3*f*(d*e - c*f)*(c + d*x)^(4
/3)*Cos[a + b*(c + d*x)^(2/3)])/(b*d^3) - (3*f^2*(c + d*x)^(7/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d^3) + (3*(d
*e - c*f)^2*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(2*b^(3/2)*d^3) + (315*f^2*Sqrt[Pi
/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(16*b^(9/2)*d^3) + (315*f^2*Sqrt[Pi/2]*FresnelC[Sqrt[
b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(16*b^(9/2)*d^3) - (3*(d*e - c*f)^2*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/
Pi]*(c + d*x)^(1/3)]*Sin[a])/(2*b^(3/2)*d^3) - (315*f^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(2/3)])/(16*b^4*d^
3) + (6*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/(b^2*d^3) + (21*f^2*(c + d*x)^(5/3)*Sin[a +
b*(c + d*x)^(2/3)])/(4*b^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 3385

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

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]

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 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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3386

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

Rubi steps

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

Mathematica [C]  time = 2.31289, size = 432, normalized size = 0.84 \[ -\frac{3 i \left (\left (\cos \left (a+b (c+d x)^{2/3}\right )-i \sin \left (a+b (c+d x)^{2/3}\right )\right ) \left (2 \sqrt{b} \left (-8 i b^3 d^2 \sqrt [3]{c+d x} (e+f x)^2+4 b^2 f (c+d x)^{2/3} (c f-8 d e-7 d f x)+2 i b f (19 c f+16 d e+35 d f x)+105 f^2 \sqrt [3]{c+d x}\right )+(1+i) \sqrt{\frac{\pi }{2}} \left (8 b^3 (d e-c f)^2+105 i f^2\right ) \text{Erf}\left (\frac{(1+i) \sqrt{b} \sqrt [3]{c+d x}}{\sqrt{2}}\right ) \left (\cos \left (b (c+d x)^{2/3}\right )+i \sin \left (b (c+d x)^{2/3}\right )\right )\right )+(\cos (a)+i \sin (a)) \left (2 \sqrt{b} \left (-8 i b^3 d^2 \sqrt [3]{c+d x} (e+f x)^2+4 b^2 f (c+d x)^{2/3} (-c f+8 d e+7 d f x)+2 i b f (19 c f+16 d e+35 d f x)-105 f^2 \sqrt [3]{c+d x}\right ) \left (\cos \left (b (c+d x)^{2/3}\right )+i \sin \left (b (c+d x)^{2/3}\right )\right )+(1+i) \sqrt{\frac{\pi }{2}} \left (8 b^3 (d e-c f)^2-105 i f^2\right ) \text{Erfi}\left (\frac{(1+i) \sqrt{b} \sqrt [3]{c+d x}}{\sqrt{2}}\right )\right )\right )}{64 b^{9/2} d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)/64)*((Cos[a] + I*Sin[a])*((1 + I)*((-105*I)*f^2 + 8*b^3*(d*e - c*f)^2)*Sqrt[Pi/2]*Erfi[((1 + I)*Sqrt[
b]*(c + d*x)^(1/3))/Sqrt[2]] + 2*Sqrt[b]*(-105*f^2*(c + d*x)^(1/3) - (8*I)*b^3*d^2*(c + d*x)^(1/3)*(e + f*x)^2
+ 4*b^2*f*(c + d*x)^(2/3)*(8*d*e - c*f + 7*d*f*x) + (2*I)*b*f*(16*d*e + 19*c*f + 35*d*f*x))*(Cos[b*(c + d*x)^
(2/3)] + I*Sin[b*(c + d*x)^(2/3)])) + (2*Sqrt[b]*(105*f^2*(c + d*x)^(1/3) - (8*I)*b^3*d^2*(c + d*x)^(1/3)*(e +
f*x)^2 + 4*b^2*f*(c + d*x)^(2/3)*(-8*d*e + c*f - 7*d*f*x) + (2*I)*b*f*(16*d*e + 19*c*f + 35*d*f*x)) + (1 + I)
*((105*I)*f^2 + 8*b^3*(d*e - c*f)^2)*Sqrt[Pi/2]*Erf[((1 + I)*Sqrt[b]*(c + d*x)^(1/3))/Sqrt[2]]*(Cos[b*(c + d*x
)^(2/3)] + I*Sin[b*(c + d*x)^(2/3)]))*(Cos[a + b*(c + d*x)^(2/3)] - I*Sin[a + b*(c + d*x)^(2/3)])))/(b^(9/2)*d
^3)

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Maple [A]  time = 0.01, size = 395, normalized size = 0.8 \begin{align*} 3\,{\frac{1}{{d}^{3}} \left ( -1/2\,{\frac{{f}^{2} \left ( dx+c \right ) ^{7/3}\cos \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{b}}+7/2\,{\frac{{f}^{2}}{b} \left ( 1/2\,{\frac{ \left ( dx+c \right ) ^{5/3}\sin \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{b}}-5/2\,{\frac{1}{b} \left ( -1/2\,{\frac{ \left ( dx+c \right ) \cos \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{b}}+3/2\,{\frac{1}{b} \left ( 1/2\,{\frac{\sqrt [3]{dx+c}\sin \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{b}}-1/4\,{\frac{\sqrt{2}\sqrt{\pi }}{{b}^{3/2}} \left ( \cos \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt [3]{dx+c}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) +\sin \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt [3]{dx+c}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) \right ) } \right ) } \right ) } \right ) }-1/2\,{\frac{ \left ( -2\,c{f}^{2}+2\,def \right ) \left ( dx+c \right ) ^{4/3}\cos \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{b}}+2\,{\frac{-2\,c{f}^{2}+2\,def}{b} \left ( 1/2\,{\frac{ \left ( dx+c \right ) ^{2/3}\sin \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{b}}+1/2\,{\frac{\cos \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{{b}^{2}}} \right ) }-1/2\,{\frac{ \left ({c}^{2}{f}^{2}-2\,cdef+{d}^{2}{e}^{2} \right ) \sqrt [3]{dx+c}\cos \left ( a+b \left ( dx+c \right ) ^{2/3} \right ) }{b}}+1/4\,{\frac{ \left ({c}^{2}{f}^{2}-2\,cdef+{d}^{2}{e}^{2} \right ) \sqrt{2}\sqrt{\pi }}{{b}^{3/2}} \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{\sqrt [3]{dx+c}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{\sqrt [3]{dx+c}\sqrt{b}\sqrt{2}}{\sqrt{\pi }}} \right ) \right ) } \right ) } \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*(-1/2*f^2/b*(d*x+c)^(7/3)*cos(a+b*(d*x+c)^(2/3))+7/2*f^2/b*(1/2/b*(d*x+c)^(5/3)*sin(a+b*(d*x+c)^(2/3))-5
/2/b*(-1/2/b*(d*x+c)*cos(a+b*(d*x+c)^(2/3))+3/2/b*(1/2/b*(d*x+c)^(1/3)*sin(a+b*(d*x+c)^(2/3))-1/4/b^(3/2)*2^(1
/2)*Pi^(1/2)*(cos(a)*FresnelS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))+sin(a)*FresnelC((d*x+c)^(1/3)*b^(1/2)*2^
(1/2)/Pi^(1/2))))))-1/2*(-2*c*f^2+2*d*e*f)/b*(d*x+c)^(4/3)*cos(a+b*(d*x+c)^(2/3))+2*(-2*c*f^2+2*d*e*f)/b*(1/2/
b*(d*x+c)^(2/3)*sin(a+b*(d*x+c)^(2/3))+1/2/b^2*cos(a+b*(d*x+c)^(2/3)))-1/2*(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))+1/4*(c^2*f^2-2*c*d*e*f+d^2*e^2)/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelC((d*x
+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*FresnelS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))))

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Maxima [C]  time = 2.38944, size = 1769, normalized size = 3.45 \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]

-3/128*(8*(8*(d*x + c)^(1/3)*abs(b)*cos((d*x + c)^(2/3)*b + a) - sqrt(pi)*(((cos(1/4*pi + 1/2*arctan2(0, b)) +
cos(-1/4*pi + 1/2*arctan2(0, b)) - I*sin(1/4*pi + 1/2*arctan2(0, b)) + I*sin(-1/4*pi + 1/2*arctan2(0, b)))*co
s(a) - (I*cos(1/4*pi + 1/2*arctan2(0, b)) + I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi + 1/2*arctan2(0, b
)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x + c)^(1/3)*sqrt(I*b)) + ((cos(1/4*pi + 1/2*arctan2(0,
b)) + cos(-1/4*pi + 1/2*arctan2(0, b)) + I*sin(1/4*pi + 1/2*arctan2(0, b)) - I*sin(-1/4*pi + 1/2*arctan2(0, b)
))*cos(a) - (-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi + 1/2*arctan
2(0, b)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x + c)^(1/3)*sqrt(-I*b)))*sqrt(abs(b)))*e^2/(b*abs
(b)) - 16*(8*(d*x + c)^(1/3)*abs(b)*cos((d*x + c)^(2/3)*b + a) - sqrt(pi)*(((cos(1/4*pi + 1/2*arctan2(0, b)) +
cos(-1/4*pi + 1/2*arctan2(0, b)) - I*sin(1/4*pi + 1/2*arctan2(0, b)) + I*sin(-1/4*pi + 1/2*arctan2(0, b)))*co
s(a) - (I*cos(1/4*pi + 1/2*arctan2(0, b)) + I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi + 1/2*arctan2(0, b
)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x + c)^(1/3)*sqrt(I*b)) + ((cos(1/4*pi + 1/2*arctan2(0,
b)) + cos(-1/4*pi + 1/2*arctan2(0, b)) + I*sin(1/4*pi + 1/2*arctan2(0, b)) - I*sin(-1/4*pi + 1/2*arctan2(0, b)
))*cos(a) - (-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi + 1/2*arctan
2(0, b)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x + c)^(1/3)*sqrt(-I*b)))*sqrt(abs(b)))*c*e*f/(b*d
*abs(b)) + 8*(8*(d*x + c)^(1/3)*abs(b)*cos((d*x + c)^(2/3)*b + a) - sqrt(pi)*(((cos(1/4*pi + 1/2*arctan2(0, b)
) + cos(-1/4*pi + 1/2*arctan2(0, b)) - I*sin(1/4*pi + 1/2*arctan2(0, b)) + I*sin(-1/4*pi + 1/2*arctan2(0, b)))
*cos(a) - (I*cos(1/4*pi + 1/2*arctan2(0, b)) + I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi + 1/2*arctan2(0
, b)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x + c)^(1/3)*sqrt(I*b)) + ((cos(1/4*pi + 1/2*arctan2(
0, b)) + cos(-1/4*pi + 1/2*arctan2(0, b)) + I*sin(1/4*pi + 1/2*arctan2(0, b)) - I*sin(-1/4*pi + 1/2*arctan2(0,
b)))*cos(a) - (-I*cos(1/4*pi + 1/2*arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) + sin(1/4*pi + 1/2*arc
tan2(0, b)) - sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x + c)^(1/3)*sqrt(-I*b)))*sqrt(abs(b)))*c^2*f^2
/(b*d^2*abs(b)) - 128*(2*(d*x + c)^(2/3)*b*sin((d*x + c)^(2/3)*b + a) - ((d*x + c)^(4/3)*b^2 - 2)*cos((d*x + c
)^(2/3)*b + a))*e*f/(b^3*d) + 128*(2*(d*x + c)^(2/3)*b*sin((d*x + c)^(2/3)*b + a) - ((d*x + c)^(4/3)*b^2 - 2)*
cos((d*x + c)^(2/3)*b + a))*c*f^2/(b^3*d^2) + (sqrt(pi)*(((-105*I*cos(1/4*pi + 1/2*arctan2(0, b)) - 105*I*cos(
-1/4*pi + 1/2*arctan2(0, b)) - 105*sin(1/4*pi + 1/2*arctan2(0, b)) + 105*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos
(a) - (105*cos(1/4*pi + 1/2*arctan2(0, b)) + 105*cos(-1/4*pi + 1/2*arctan2(0, b)) - 105*I*sin(1/4*pi + 1/2*arc
tan2(0, b)) + 105*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x + c)^(1/3)*sqrt(I*b)) + ((105*I*cos(1/4
*pi + 1/2*arctan2(0, b)) + 105*I*cos(-1/4*pi + 1/2*arctan2(0, b)) - 105*sin(1/4*pi + 1/2*arctan2(0, b)) + 105*
sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (105*cos(1/4*pi + 1/2*arctan2(0, b)) + 105*cos(-1/4*pi + 1/2*arctan
2(0, b)) + 105*I*sin(1/4*pi + 1/2*arctan2(0, b)) - 105*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf((d*x +
c)^(1/3)*sqrt(-I*b)))*sqrt(abs(b)) + 16*(4*(d*x + c)^(7/3)*b^3*abs(b) - 35*(d*x + c)*b*abs(b))*cos((d*x + c)^(
2/3)*b + a) - 56*(4*(d*x + c)^(5/3)*b^2*abs(b) - 15*(d*x + c)^(1/3)*abs(b))*sin((d*x + c)^(2/3)*b + a))*f^2/(b
^4*d^2*abs(b)))/d

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Fricas [A]  time = 2.07091, size = 782, normalized size = 1.52 \begin{align*} \frac{3 \,{\left (\sqrt{2}{\left (105 \, \pi f^{2} \sin \left (a\right ) + 8 \, \pi{\left (b^{3} d^{2} e^{2} - 2 \, b^{3} c d e f + b^{3} c^{2} f^{2}\right )} \cos \left (a\right )\right )} \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\sqrt{2}{\left (d x + c\right )}^{\frac{1}{3}} \sqrt{\frac{b}{\pi }}\right ) + \sqrt{2}{\left (105 \, \pi f^{2} \cos \left (a\right ) - 8 \, \pi{\left (b^{3} d^{2} e^{2} - 2 \, b^{3} c d e f + b^{3} c^{2} f^{2}\right )} \sin \left (a\right )\right )} \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2}{\left (d x + c\right )}^{\frac{1}{3}} \sqrt{\frac{b}{\pi }}\right ) + 4 \,{\left (35 \, b^{2} d f^{2} x + 16 \, b^{2} d e f + 19 \, b^{2} c f^{2} - 4 \,{\left (b^{4} d^{2} f^{2} x^{2} + 2 \, b^{4} d^{2} e f x + b^{4} d^{2} e^{2}\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right ) - 2 \,{\left (105 \,{\left (d x + c\right )}^{\frac{1}{3}} b f^{2} - 4 \,{\left (7 \, b^{3} d f^{2} x + 8 \, b^{3} d e f - b^{3} c f^{2}\right )}{\left (d x + c\right )}^{\frac{2}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )\right )}}{32 \, b^{5} 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]

3/32*(sqrt(2)*(105*pi*f^2*sin(a) + 8*pi*(b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*cos(a))*sqrt(b/pi)*fresnel
_cos(sqrt(2)*(d*x + c)^(1/3)*sqrt(b/pi)) + sqrt(2)*(105*pi*f^2*cos(a) - 8*pi*(b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^
3*c^2*f^2)*sin(a))*sqrt(b/pi)*fresnel_sin(sqrt(2)*(d*x + c)^(1/3)*sqrt(b/pi)) + 4*(35*b^2*d*f^2*x + 16*b^2*d*e
*f + 19*b^2*c*f^2 - 4*(b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + b^4*d^2*e^2)*(d*x + c)^(1/3))*cos((d*x + c)^(2/3)*b
+ a) - 2*(105*(d*x + c)^(1/3)*b*f^2 - 4*(7*b^3*d*f^2*x + 8*b^3*d*e*f - b^3*c*f^2)*(d*x + c)^(2/3))*sin((d*x +
c)^(2/3)*b + a))/(b^5*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 + 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 [C]  time = 1.33173, size = 1049, normalized size = 2.04 \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="giac")

[Out]

-3/64*(f^2*((I*sqrt(2)*sqrt(pi)*(-8*I*b^3*c^2 - 105)*erf(-1/2*sqrt(2)*(d*x + c)^(1/3)*(-I*b/abs(b) + 1)*sqrt(a
bs(b)))*e^(I*a)/(b^4*(-I*b/abs(b) + 1)*sqrt(abs(b))) - 2*I*(8*I*(d*x + c)^(7/3)*b^3 - 16*I*(d*x + c)^(4/3)*b^3
*c + 8*I*(d*x + c)^(1/3)*b^3*c^2 - 28*(d*x + c)^(5/3)*b^2 + 32*(d*x + c)^(2/3)*b^2*c - (70*I*d*x + 70*I*c)*b +
32*I*b*c + 105*(d*x + c)^(1/3))*e^(I*(d*x + c)^(2/3)*b + I*a)/b^4)/d^2 + (I*sqrt(2)*sqrt(pi)*(-8*I*b^3*c^2 +
105)*erf(-1/2*sqrt(2)*(d*x + c)^(1/3)*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(-I*a)/(b^4*(I*b/abs(b) + 1)*sqrt(abs(b
))) - 2*I*(8*I*(d*x + c)^(7/3)*b^3 - 16*I*(d*x + c)^(4/3)*b^3*c + 8*I*(d*x + c)^(1/3)*b^3*c^2 + 28*(d*x + c)^(
5/3)*b^2 - 32*(d*x + c)^(2/3)*b^2*c - (70*I*d*x + 70*I*c)*b + 32*I*b*c - 105*(d*x + c)^(1/3))*e^(-I*(d*x + c)^
(2/3)*b - I*a)/b^4)/d^2) + 8*(sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*(d*x + c)^(1/3)*(-I*b/abs(b) + 1)*sqrt(abs(b))
)*e^(I*a)/(b*(-I*b/abs(b) + 1)*sqrt(abs(b))) + sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*(d*x + c)^(1/3)*(I*b/abs(b) +
1)*sqrt(abs(b)))*e^(-I*a)/(b*(I*b/abs(b) + 1)*sqrt(abs(b))) + 2*(d*x + c)^(1/3)*e^(I*(d*x + c)^(2/3)*b + I*a)
/b + 2*(d*x + c)^(1/3)*e^(-I*(d*x + c)^(2/3)*b - I*a)/b)*e^2 - 16*(sqrt(2)*sqrt(pi)*c*erf(-1/2*sqrt(2)*(d*x +
c)^(1/3)*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(b*(-I*b/abs(b) + 1)*sqrt(abs(b))) + sqrt(2)*sqrt(pi)*c*erf(-
1/2*sqrt(2)*(d*x + c)^(1/3)*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(-I*a)/(b*(I*b/abs(b) + 1)*sqrt(abs(b))) + 2*I*(I
*(d*x + c)^(4/3)*b^2 - I*(d*x + c)^(1/3)*b^2*c - 2*(d*x + c)^(2/3)*b - 2*I)*e^(I*(d*x + c)^(2/3)*b + I*a)/b^3
+ 2*I*(I*(d*x + c)^(4/3)*b^2 - I*(d*x + c)^(1/3)*b^2*c + 2*(d*x + c)^(2/3)*b - 2*I)*e^(-I*(d*x + c)^(2/3)*b -
I*a)/b^3)*f*e/d)/d