3.207 \(\int (e+f x)^2 \sin (a+b \sqrt [3]{c+d x}) \, dx\)

Optimal. Leaf size=633 \[ \frac{30 f (c+d x)^{4/3} (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{6 \sqrt [3]{c+d x} (d e-c f)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{120 f (c+d x) (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 f \sqrt [3]{c+d x} (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{120960 f^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac{6 f (c+d x)^{5/3} (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{3 (c+d x)^{2/3} (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

[Out]

(-120960*f^2*Cos[a + b*(c + d*x)^(1/3)])/(b^9*d^3) + (6*(d*e - c*f)^2*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) -
(720*f*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*f^2*(c + d*x)^(2/3)*Cos[a +
b*(c + d*x)^(1/3)])/(b^7*d^3) - (3*(d*e - c*f)^2*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (120*f*
(d*e - c*f)*(c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (5040*f^2*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1
/3)])/(b^5*d^3) - (6*f*(d*e - c*f)*(c + d*x)^(5/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (168*f^2*(c + d*x)^2*
Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (3*f^2*(c + d*x)^(8/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (720*f*(d
*e - c*f)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*f^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^8
*d^3) + (6*(d*e - c*f)^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (360*f*(d*e - c*f)*(c + d*x)^
(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (20160*f^2*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d^3) + (30
*f*(d*e - c*f)*(c + d*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*f^2*(c + d*x)^(5/3)*Sin[a + b*(c
+ d*x)^(1/3)])/(b^4*d^3) + (24*f^2*(c + d*x)^(7/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3)

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Rubi [A]  time = 0.647044, antiderivative size = 633, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3431, 3296, 2638, 2637} \[ \frac{30 f (c+d x)^{4/3} (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (c+d x)^{2/3} (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{6 \sqrt [3]{c+d x} (d e-c f)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{120 f (c+d x) (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 f \sqrt [3]{c+d x} (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{120960 f^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac{6 f (c+d x)^{5/3} (d e-c f) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{3 (c+d x)^{2/3} (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)^2*Sin[a + b*(c + d*x)^(1/3)],x]

[Out]

(-120960*f^2*Cos[a + b*(c + d*x)^(1/3)])/(b^9*d^3) + (6*(d*e - c*f)^2*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) -
(720*f*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d^3) + (60480*f^2*(c + d*x)^(2/3)*Cos[a +
b*(c + d*x)^(1/3)])/(b^7*d^3) - (3*(d*e - c*f)^2*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (120*f*
(d*e - c*f)*(c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (5040*f^2*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1
/3)])/(b^5*d^3) - (6*f*(d*e - c*f)*(c + d*x)^(5/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (168*f^2*(c + d*x)^2*
Cos[a + b*(c + d*x)^(1/3)])/(b^3*d^3) - (3*f^2*(c + d*x)^(8/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d^3) + (720*f*(d
*e - c*f)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*f^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^8
*d^3) + (6*(d*e - c*f)^2*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (360*f*(d*e - c*f)*(c + d*x)^
(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (20160*f^2*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d^3) + (30
*f*(d*e - c*f)*(c + d*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3) - (1008*f^2*(c + d*x)^(5/3)*Sin[a + b*(c
+ d*x)^(1/3)])/(b^4*d^3) + (24*f^2*(c + d*x)^(7/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d^3)

Rule 3431

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

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (e+f x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{3 \operatorname{Subst}\left (\int \left (\frac{(d e-c f)^2 x^2 \sin (a+b x)}{d^2}+\frac{2 f (d e-c f) x^5 \sin (a+b x)}{d^2}+\frac{f^2 x^8 \sin (a+b x)}{d^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int x^8 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac{(6 f (d e-c f)) \operatorname{Subst}\left (\int x^5 \sin (a+b x) \, 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 (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{\left (24 f^2\right ) \operatorname{Subst}\left (\int x^7 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac{(30 f (d e-c f)) \operatorname{Subst}\left (\int x^4 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac{\left (6 (d e-c f)^2\right ) \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}\\ &=-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{\left (168 f^2\right ) \operatorname{Subst}\left (\int x^6 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{(120 f (d e-c f)) \operatorname{Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{\left (6 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}\\ &=\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{\left (1008 f^2\right ) \operatorname{Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{(360 f (d e-c f)) \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}\\ &=\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{\left (5040 f^2\right ) \operatorname{Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{(720 f (d e-c f)) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}\\ &=\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{\left (20160 f^2\right ) \operatorname{Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{(720 f (d e-c f)) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}\\ &=\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{\left (60480 f^2\right ) \operatorname{Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d^3}\\ &=\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{\left (120960 f^2\right ) \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^7 d^3}\\ &=\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{\left (120960 f^2\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^8 d^3}\\ &=-\frac{120960 f^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac{6 (d e-c f)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{60480 f^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{3 (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 f (d e-c f) (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{5040 f^2 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 f (d e-c f) (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{168 f^2 (c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{3 f^2 (c+d x)^{8/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{720 f (d e-c f) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 f^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{6 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{360 f (d e-c f) (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 f^2 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{30 f (d e-c f) (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 f^2 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 f^2 (c+d x)^{7/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}\\ \end{align*}

Mathematica [A]  time = 2.50915, size = 256, normalized size = 0.4 \[ \frac{6 b \sin \left (a+b \sqrt [3]{c+d x}\right ) \left (b^6 d \sqrt [3]{c+d x} (e+f x) (3 c f+d (e+4 f x))-12 b^4 f (c+d x)^{2/3} (9 c f+5 d e+14 d f x)+120 b^2 f (27 c f+d (e+28 f x))-20160 f^2 \sqrt [3]{c+d x}\right )-3 \cos \left (a+b \sqrt [3]{c+d x}\right ) \left (-2 b^6 \left (9 c^2 f^2+18 c d f (e+2 f x)+d^2 \left (e^2+20 e f x+28 f^2 x^2\right )\right )+b^8 d^2 (c+d x)^{2/3} (e+f x)^2+240 b^4 f \sqrt [3]{c+d x} (6 c f+d (e+7 f x))-20160 b^2 f^2 (c+d x)^{2/3}+40320 f^2\right )}{b^9 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(e + f*x)^2*Sin[a + b*(c + d*x)^(1/3)],x]

[Out]

(-3*(40320*f^2 - 20160*b^2*f^2*(c + d*x)^(2/3) + b^8*d^2*(c + d*x)^(2/3)*(e + f*x)^2 + 240*b^4*f*(c + d*x)^(1/
3)*(6*c*f + d*(e + 7*f*x)) - 2*b^6*(9*c^2*f^2 + 18*c*d*f*(e + 2*f*x) + d^2*(e^2 + 20*e*f*x + 28*f^2*x^2)))*Cos
[a + b*(c + d*x)^(1/3)] + 6*b*(-20160*f^2*(c + d*x)^(1/3) - 12*b^4*f*(c + d*x)^(2/3)*(5*d*e + 9*c*f + 14*d*f*x
) + b^6*d*(c + d*x)^(1/3)*(e + f*x)*(3*c*f + d*(e + 4*f*x)) + 120*b^2*f*(27*c*f + d*(e + 28*f*x)))*Sin[a + b*(
c + d*x)^(1/3)])/(b^9*d^3)

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Maple [B]  time = 0.013, size = 2704, normalized size = 4.3 \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)^(1/3)),x)

[Out]

3/d^3/b^3*(c^2*f^2*(-(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+2*cos(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(1/3
))*sin(a+b*(d*x+c)^(1/3)))+d^2*e^2*(-(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+2*cos(a+b*(d*x+c)^(1/3))+2*(
a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+1/b^6*f^2*(-(a+b*(d*x+c)^(1/3))^8*cos(a+b*(d*x+c)^(1/3))+8*(a+b*(d*
x+c)^(1/3))^7*sin(a+b*(d*x+c)^(1/3))+56*(a+b*(d*x+c)^(1/3))^6*cos(a+b*(d*x+c)^(1/3))-336*(a+b*(d*x+c)^(1/3))^5
*sin(a+b*(d*x+c)^(1/3))-1680*(a+b*(d*x+c)^(1/3))^4*cos(a+b*(d*x+c)^(1/3))+6720*(a+b*(d*x+c)^(1/3))^3*sin(a+b*(
d*x+c)^(1/3))+20160*(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))-40320*cos(a+b*(d*x+c)^(1/3))-40320*(a+b*(d*x+
c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))-a^2*c^2*f^2*cos(a+b*(d*x+c)^(1/3))-2*a*c^2*f^2*(sin(a+b*(d*x+c)^(1/3))-(a+b*
(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-2*a*d^2*e^2*(sin(a+b*(d*x+c)^(1/3))-(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)
^(1/3)))-56/b^6*a^3*f^2*(-(a+b*(d*x+c)^(1/3))^5*cos(a+b*(d*x+c)^(1/3))+5*(a+b*(d*x+c)^(1/3))^4*sin(a+b*(d*x+c)
^(1/3))+20*(a+b*(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))+120*si
n(a+b*(d*x+c)^(1/3))-120*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))+70/b^6*a^4*f^2*(-(a+b*(d*x+c)^(1/3))^4*co
s(a+b*(d*x+c)^(1/3))+4*(a+b*(d*x+c)^(1/3))^3*sin(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(
1/3))-24*cos(a+b*(d*x+c)^(1/3))-24*(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))-56/b^6*a^5*f^2*(-(a+b*(d*x+c)^(
1/3))^3*cos(a+b*(d*x+c)^(1/3))+3*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))-6*sin(a+b*(d*x+c)^(1/3))+6*(a+b*
(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-1/b^6*a^8*f^2*cos(a+b*(d*x+c)^(1/3))-a^2*d^2*e^2*cos(a+b*(d*x+c)^(1/3))
+28/b^6*a^6*f^2*(-(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+2*cos(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(1/3))*
sin(a+b*(d*x+c)^(1/3)))-8/b^6*a^7*f^2*(sin(a+b*(d*x+c)^(1/3))-(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-2/b^
3*c*f^2*(-(a+b*(d*x+c)^(1/3))^5*cos(a+b*(d*x+c)^(1/3))+5*(a+b*(d*x+c)^(1/3))^4*sin(a+b*(d*x+c)^(1/3))+20*(a+b*
(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))+120*sin(a+b*(d*x+c)^(1
/3))-120*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-8/b^6*a*f^2*(-(a+b*(d*x+c)^(1/3))^7*cos(a+b*(d*x+c)^(1/3)
)+7*(a+b*(d*x+c)^(1/3))^6*sin(a+b*(d*x+c)^(1/3))+42*(a+b*(d*x+c)^(1/3))^5*cos(a+b*(d*x+c)^(1/3))-210*(a+b*(d*x
+c)^(1/3))^4*sin(a+b*(d*x+c)^(1/3))-840*(a+b*(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))+2520*(a+b*(d*x+c)^(1/3))^
2*sin(a+b*(d*x+c)^(1/3))-5040*sin(a+b*(d*x+c)^(1/3))+5040*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))+28/b^6*a
^2*f^2*(-(a+b*(d*x+c)^(1/3))^6*cos(a+b*(d*x+c)^(1/3))+6*(a+b*(d*x+c)^(1/3))^5*sin(a+b*(d*x+c)^(1/3))+30*(a+b*(
d*x+c)^(1/3))^4*cos(a+b*(d*x+c)^(1/3))-120*(a+b*(d*x+c)^(1/3))^3*sin(a+b*(d*x+c)^(1/3))-360*(a+b*(d*x+c)^(1/3)
)^2*cos(a+b*(d*x+c)^(1/3))+720*cos(a+b*(d*x+c)^(1/3))+720*(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+2*a^2*c*
d*e*f*cos(a+b*(d*x+c)^(1/3))+10/b^3*a*c*f^2*(-(a+b*(d*x+c)^(1/3))^4*cos(a+b*(d*x+c)^(1/3))+4*(a+b*(d*x+c)^(1/3
))^3*sin(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))-24*cos(a+b*(d*x+c)^(1/3))-24*(a+b*
(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))-20/b^3*a^2*c*f^2*(-(a+b*(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))+3*(a+b*
(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))-6*sin(a+b*(d*x+c)^(1/3))+6*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))
+20/b^3*a^3*c*f^2*(-(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+2*cos(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(1/3)
)*sin(a+b*(d*x+c)^(1/3)))-10/b^3*a^4*c*f^2*(sin(a+b*(d*x+c)^(1/3))-(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))
-2*c*d*e*f*(-(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+2*cos(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(1/3))*sin(a
+b*(d*x+c)^(1/3)))+2/b^3*d*e*f*(-(a+b*(d*x+c)^(1/3))^5*cos(a+b*(d*x+c)^(1/3))+5*(a+b*(d*x+c)^(1/3))^4*sin(a+b*
(d*x+c)^(1/3))+20*(a+b*(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))
+120*sin(a+b*(d*x+c)^(1/3))-120*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-2/b^3*a^5*c*f^2*cos(a+b*(d*x+c)^(1
/3))-20/b^3*a^3*d*e*f*(-(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+2*cos(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(
1/3))*sin(a+b*(d*x+c)^(1/3)))+10/b^3*a^4*d*e*f*(sin(a+b*(d*x+c)^(1/3))-(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/
3)))+20/b^3*a^2*d*e*f*(-(a+b*(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))+3*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(
1/3))-6*sin(a+b*(d*x+c)^(1/3))+6*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))+4*a*c*d*e*f*(sin(a+b*(d*x+c)^(1/3
))-(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))+2/b^3*a^5*d*e*f*cos(a+b*(d*x+c)^(1/3))-10/b^3*a*d*e*f*(-(a+b*(d
*x+c)^(1/3))^4*cos(a+b*(d*x+c)^(1/3))+4*(a+b*(d*x+c)^(1/3))^3*sin(a+b*(d*x+c)^(1/3))+12*(a+b*(d*x+c)^(1/3))^2*
cos(a+b*(d*x+c)^(1/3))-24*cos(a+b*(d*x+c)^(1/3))-24*(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3))))

________________________________________________________________________________________

Maxima [B]  time = 1.42847, size = 2904, normalized size = 4.59 \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)^(1/3)),x, algorithm="maxima")

[Out]

-3*(a^2*e^2*cos((d*x + c)^(1/3)*b + a) - 2*a^2*c*e*f*cos((d*x + c)^(1/3)*b + a)/d + a^2*c^2*f^2*cos((d*x + c)^
(1/3)*b + a)/d^2 - 2*(((d*x + c)^(1/3)*b + a)*cos((d*x + c)^(1/3)*b + a) - sin((d*x + c)^(1/3)*b + a))*a*e^2 +
4*(((d*x + c)^(1/3)*b + a)*cos((d*x + c)^(1/3)*b + a) - sin((d*x + c)^(1/3)*b + a))*a*c*e*f/d - 2*(((d*x + c)
^(1/3)*b + a)*cos((d*x + c)^(1/3)*b + a) - sin((d*x + c)^(1/3)*b + a))*a*c^2*f^2/d^2 - 2*a^5*e*f*cos((d*x + c)
^(1/3)*b + a)/(b^3*d) + 2*a^5*c*f^2*cos((d*x + c)^(1/3)*b + a)/(b^3*d^2) + ((((d*x + c)^(1/3)*b + a)^2 - 2)*co
s((d*x + c)^(1/3)*b + a) - 2*((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a))*e^2 + 10*(((d*x + c)^(1/3)*b
+ a)*cos((d*x + c)^(1/3)*b + a) - sin((d*x + c)^(1/3)*b + a))*a^4*e*f/(b^3*d) - 2*((((d*x + c)^(1/3)*b + a)^2
- 2)*cos((d*x + c)^(1/3)*b + a) - 2*((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a))*c*e*f/d - 10*(((d*x +
c)^(1/3)*b + a)*cos((d*x + c)^(1/3)*b + a) - sin((d*x + c)^(1/3)*b + a))*a^4*c*f^2/(b^3*d^2) + ((((d*x + c)^(1
/3)*b + a)^2 - 2)*cos((d*x + c)^(1/3)*b + a) - 2*((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a))*c^2*f^2/d
^2 + a^8*f^2*cos((d*x + c)^(1/3)*b + a)/(b^6*d^2) - 20*((((d*x + c)^(1/3)*b + a)^2 - 2)*cos((d*x + c)^(1/3)*b
+ a) - 2*((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a))*a^3*e*f/(b^3*d) - 8*(((d*x + c)^(1/3)*b + a)*cos(
(d*x + c)^(1/3)*b + a) - sin((d*x + c)^(1/3)*b + a))*a^7*f^2/(b^6*d^2) + 20*((((d*x + c)^(1/3)*b + a)^2 - 2)*c
os((d*x + c)^(1/3)*b + a) - 2*((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a))*a^3*c*f^2/(b^3*d^2) + 20*(((
(d*x + c)^(1/3)*b + a)^3 - 6*(d*x + c)^(1/3)*b - 6*a)*cos((d*x + c)^(1/3)*b + a) - 3*(((d*x + c)^(1/3)*b + a)^
2 - 2)*sin((d*x + c)^(1/3)*b + a))*a^2*e*f/(b^3*d) + 28*((((d*x + c)^(1/3)*b + a)^2 - 2)*cos((d*x + c)^(1/3)*b
+ a) - 2*((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a))*a^6*f^2/(b^6*d^2) - 20*((((d*x + c)^(1/3)*b + a)
^3 - 6*(d*x + c)^(1/3)*b - 6*a)*cos((d*x + c)^(1/3)*b + a) - 3*(((d*x + c)^(1/3)*b + a)^2 - 2)*sin((d*x + c)^(
1/3)*b + a))*a^2*c*f^2/(b^3*d^2) - 10*((((d*x + c)^(1/3)*b + a)^4 - 12*((d*x + c)^(1/3)*b + a)^2 + 24)*cos((d*
x + c)^(1/3)*b + a) - 4*(((d*x + c)^(1/3)*b + a)^3 - 6*(d*x + c)^(1/3)*b - 6*a)*sin((d*x + c)^(1/3)*b + a))*a*
e*f/(b^3*d) - 56*((((d*x + c)^(1/3)*b + a)^3 - 6*(d*x + c)^(1/3)*b - 6*a)*cos((d*x + c)^(1/3)*b + a) - 3*(((d*
x + c)^(1/3)*b + a)^2 - 2)*sin((d*x + c)^(1/3)*b + a))*a^5*f^2/(b^6*d^2) + 10*((((d*x + c)^(1/3)*b + a)^4 - 12
*((d*x + c)^(1/3)*b + a)^2 + 24)*cos((d*x + c)^(1/3)*b + a) - 4*(((d*x + c)^(1/3)*b + a)^3 - 6*(d*x + c)^(1/3)
*b - 6*a)*sin((d*x + c)^(1/3)*b + a))*a*c*f^2/(b^3*d^2) + 2*((((d*x + c)^(1/3)*b + a)^5 - 20*((d*x + c)^(1/3)*
b + a)^3 + 120*(d*x + c)^(1/3)*b + 120*a)*cos((d*x + c)^(1/3)*b + a) - 5*(((d*x + c)^(1/3)*b + a)^4 - 12*((d*x
+ c)^(1/3)*b + a)^2 + 24)*sin((d*x + c)^(1/3)*b + a))*e*f/(b^3*d) + 70*((((d*x + c)^(1/3)*b + a)^4 - 12*((d*x
+ c)^(1/3)*b + a)^2 + 24)*cos((d*x + c)^(1/3)*b + a) - 4*(((d*x + c)^(1/3)*b + a)^3 - 6*(d*x + c)^(1/3)*b - 6
*a)*sin((d*x + c)^(1/3)*b + a))*a^4*f^2/(b^6*d^2) - 2*((((d*x + c)^(1/3)*b + a)^5 - 20*((d*x + c)^(1/3)*b + a)
^3 + 120*(d*x + c)^(1/3)*b + 120*a)*cos((d*x + c)^(1/3)*b + a) - 5*(((d*x + c)^(1/3)*b + a)^4 - 12*((d*x + c)^
(1/3)*b + a)^2 + 24)*sin((d*x + c)^(1/3)*b + a))*c*f^2/(b^3*d^2) - 56*((((d*x + c)^(1/3)*b + a)^5 - 20*((d*x +
c)^(1/3)*b + a)^3 + 120*(d*x + c)^(1/3)*b + 120*a)*cos((d*x + c)^(1/3)*b + a) - 5*(((d*x + c)^(1/3)*b + a)^4
- 12*((d*x + c)^(1/3)*b + a)^2 + 24)*sin((d*x + c)^(1/3)*b + a))*a^3*f^2/(b^6*d^2) + 28*((((d*x + c)^(1/3)*b +
a)^6 - 30*((d*x + c)^(1/3)*b + a)^4 + 360*((d*x + c)^(1/3)*b + a)^2 - 720)*cos((d*x + c)^(1/3)*b + a) - 6*(((
d*x + c)^(1/3)*b + a)^5 - 20*((d*x + c)^(1/3)*b + a)^3 + 120*(d*x + c)^(1/3)*b + 120*a)*sin((d*x + c)^(1/3)*b
+ a))*a^2*f^2/(b^6*d^2) - 8*((((d*x + c)^(1/3)*b + a)^7 - 42*((d*x + c)^(1/3)*b + a)^5 + 840*((d*x + c)^(1/3)*
b + a)^3 - 5040*(d*x + c)^(1/3)*b - 5040*a)*cos((d*x + c)^(1/3)*b + a) - 7*(((d*x + c)^(1/3)*b + a)^6 - 30*((d
*x + c)^(1/3)*b + a)^4 + 360*((d*x + c)^(1/3)*b + a)^2 - 720)*sin((d*x + c)^(1/3)*b + a))*a*f^2/(b^6*d^2) + ((
((d*x + c)^(1/3)*b + a)^8 - 56*((d*x + c)^(1/3)*b + a)^6 + 1680*((d*x + c)^(1/3)*b + a)^4 - 20160*((d*x + c)^(
1/3)*b + a)^2 + 40320)*cos((d*x + c)^(1/3)*b + a) - 8*(((d*x + c)^(1/3)*b + a)^7 - 42*((d*x + c)^(1/3)*b + a)^
5 + 840*((d*x + c)^(1/3)*b + a)^3 - 5040*(d*x + c)^(1/3)*b - 5040*a)*sin((d*x + c)^(1/3)*b + a))*f^2/(b^6*d^2)
)/(b^3*d)

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Fricas [A]  time = 1.76319, size = 776, normalized size = 1.23 \begin{align*} \frac{3 \,{\left ({\left (56 \, b^{6} d^{2} f^{2} x^{2} + 2 \, b^{6} d^{2} e^{2} + 36 \, b^{6} c d e f + 18 \,{\left (b^{6} c^{2} - 2240\right )} f^{2} + 8 \,{\left (5 \, b^{6} d^{2} e f + 9 \, b^{6} c d f^{2}\right )} x -{\left (b^{8} d^{2} f^{2} x^{2} + 2 \, b^{8} d^{2} e f x + b^{8} d^{2} e^{2} - 20160 \, b^{2} f^{2}\right )}{\left (d x + c\right )}^{\frac{2}{3}} - 240 \,{\left (7 \, b^{4} d f^{2} x + b^{4} d e f + 6 \, b^{4} c f^{2}\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) + 2 \,{\left (3360 \, b^{3} d f^{2} x + 120 \, b^{3} d e f + 3240 \, b^{3} c f^{2} - 12 \,{\left (14 \, b^{5} d f^{2} x + 5 \, b^{5} d e f + 9 \, b^{5} c f^{2}\right )}{\left (d x + c\right )}^{\frac{2}{3}} +{\left (4 \, b^{7} d^{2} f^{2} x^{2} + b^{7} d^{2} e^{2} + 3 \, b^{7} c d e f - 20160 \, b f^{2} +{\left (5 \, b^{7} d^{2} e f + 3 \, b^{7} c d f^{2}\right )} x\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{9} 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)^(1/3)),x, algorithm="fricas")

[Out]

3*((56*b^6*d^2*f^2*x^2 + 2*b^6*d^2*e^2 + 36*b^6*c*d*e*f + 18*(b^6*c^2 - 2240)*f^2 + 8*(5*b^6*d^2*e*f + 9*b^6*c
*d*f^2)*x - (b^8*d^2*f^2*x^2 + 2*b^8*d^2*e*f*x + b^8*d^2*e^2 - 20160*b^2*f^2)*(d*x + c)^(2/3) - 240*(7*b^4*d*f
^2*x + b^4*d*e*f + 6*b^4*c*f^2)*(d*x + c)^(1/3))*cos((d*x + c)^(1/3)*b + a) + 2*(3360*b^3*d*f^2*x + 120*b^3*d*
e*f + 3240*b^3*c*f^2 - 12*(14*b^5*d*f^2*x + 5*b^5*d*e*f + 9*b^5*c*f^2)*(d*x + c)^(2/3) + (4*b^7*d^2*f^2*x^2 +
b^7*d^2*e^2 + 3*b^7*c*d*e*f - 20160*b*f^2 + (5*b^7*d^2*e*f + 3*b^7*c*d*f^2)*x)*(d*x + c)^(1/3))*sin((d*x + c)^
(1/3)*b + a))/(b^9*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 \sqrt [3]{c + d x} \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)**(1/3)),x)

[Out]

Integral((e + f*x)**2*sin(a + b*(c + d*x)**(1/3)), x)

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Giac [B]  time = 1.62132, size = 2103, normalized size = 3.32 \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)^(1/3)),x, algorithm="giac")

[Out]

-3*(f^2*((((d*x + c)^(1/3)*b + a)^2*b^6*c^2 - 2*((d*x + c)^(1/3)*b + a)*a*b^6*c^2 + a^2*b^6*c^2 - 2*((d*x + c)
^(1/3)*b + a)^5*b^3*c + 10*((d*x + c)^(1/3)*b + a)^4*a*b^3*c - 20*((d*x + c)^(1/3)*b + a)^3*a^2*b^3*c + 20*((d
*x + c)^(1/3)*b + a)^2*a^3*b^3*c - 10*((d*x + c)^(1/3)*b + a)*a^4*b^3*c + 2*a^5*b^3*c + ((d*x + c)^(1/3)*b + a
)^8 - 8*((d*x + c)^(1/3)*b + a)^7*a + 28*((d*x + c)^(1/3)*b + a)^6*a^2 - 56*((d*x + c)^(1/3)*b + a)^5*a^3 + 70
*((d*x + c)^(1/3)*b + a)^4*a^4 - 56*((d*x + c)^(1/3)*b + a)^3*a^5 + 28*((d*x + c)^(1/3)*b + a)^2*a^6 - 8*((d*x
+ c)^(1/3)*b + a)*a^7 + a^8 - 2*b^6*c^2 + 40*((d*x + c)^(1/3)*b + a)^3*b^3*c - 120*((d*x + c)^(1/3)*b + a)^2*
a*b^3*c + 120*((d*x + c)^(1/3)*b + a)*a^2*b^3*c - 40*a^3*b^3*c - 56*((d*x + c)^(1/3)*b + a)^6 + 336*((d*x + c)
^(1/3)*b + a)^5*a - 840*((d*x + c)^(1/3)*b + a)^4*a^2 + 1120*((d*x + c)^(1/3)*b + a)^3*a^3 - 840*((d*x + c)^(1
/3)*b + a)^2*a^4 + 336*((d*x + c)^(1/3)*b + a)*a^5 - 56*a^6 - 240*((d*x + c)^(1/3)*b + a)*b^3*c + 240*a*b^3*c
+ 1680*((d*x + c)^(1/3)*b + a)^4 - 6720*((d*x + c)^(1/3)*b + a)^3*a + 10080*((d*x + c)^(1/3)*b + a)^2*a^2 - 67
20*((d*x + c)^(1/3)*b + a)*a^3 + 1680*a^4 - 20160*((d*x + c)^(1/3)*b + a)^2 + 40320*((d*x + c)^(1/3)*b + a)*a
- 20160*a^2 + 40320)*cos((d*x + c)^(1/3)*b + a)/(b^8*d^2) - 2*(((d*x + c)^(1/3)*b + a)*b^6*c^2 - a*b^6*c^2 - 5
*((d*x + c)^(1/3)*b + a)^4*b^3*c + 20*((d*x + c)^(1/3)*b + a)^3*a*b^3*c - 30*((d*x + c)^(1/3)*b + a)^2*a^2*b^3
*c + 20*((d*x + c)^(1/3)*b + a)*a^3*b^3*c - 5*a^4*b^3*c + 4*((d*x + c)^(1/3)*b + a)^7 - 28*((d*x + c)^(1/3)*b
+ a)^6*a + 84*((d*x + c)^(1/3)*b + a)^5*a^2 - 140*((d*x + c)^(1/3)*b + a)^4*a^3 + 140*((d*x + c)^(1/3)*b + a)^
3*a^4 - 84*((d*x + c)^(1/3)*b + a)^2*a^5 + 28*((d*x + c)^(1/3)*b + a)*a^6 - 4*a^7 + 60*((d*x + c)^(1/3)*b + a)
^2*b^3*c - 120*((d*x + c)^(1/3)*b + a)*a*b^3*c + 60*a^2*b^3*c - 168*((d*x + c)^(1/3)*b + a)^5 + 840*((d*x + c)
^(1/3)*b + a)^4*a - 1680*((d*x + c)^(1/3)*b + a)^3*a^2 + 1680*((d*x + c)^(1/3)*b + a)^2*a^3 - 840*((d*x + c)^(
1/3)*b + a)*a^4 + 168*a^5 - 120*b^3*c + 3360*((d*x + c)^(1/3)*b + a)^3 - 10080*((d*x + c)^(1/3)*b + a)^2*a + 1
0080*((d*x + c)^(1/3)*b + a)*a^2 - 3360*a^3 - 20160*(d*x + c)^(1/3)*b)*sin((d*x + c)^(1/3)*b + a)/(b^8*d^2)) -
(2*(d*x + c)^(1/3)*sin((d*x + c)^(1/3)*b + a)/b - (((d*x + c)^(1/3)*b + a)^2 - 2*((d*x + c)^(1/3)*b + a)*a +
a^2 - 2)*cos((d*x + c)^(1/3)*b + a)/b^2)*e^2 - 2*f*((((d*x + c)^(1/3)*b + a)^2*b^3*c - 2*((d*x + c)^(1/3)*b +
a)*a*b^3*c + a^2*b^3*c - ((d*x + c)^(1/3)*b + a)^5 + 5*((d*x + c)^(1/3)*b + a)^4*a - 10*((d*x + c)^(1/3)*b + a
)^3*a^2 + 10*((d*x + c)^(1/3)*b + a)^2*a^3 - 5*((d*x + c)^(1/3)*b + a)*a^4 + a^5 - 2*b^3*c + 20*((d*x + c)^(1/
3)*b + a)^3 - 60*((d*x + c)^(1/3)*b + a)^2*a + 60*((d*x + c)^(1/3)*b + a)*a^2 - 20*a^3 - 120*(d*x + c)^(1/3)*b
)*cos((d*x + c)^(1/3)*b + a)/b^5 - (2*((d*x + c)^(1/3)*b + a)*b^3*c - 2*a*b^3*c - 5*((d*x + c)^(1/3)*b + a)^4
+ 20*((d*x + c)^(1/3)*b + a)^3*a - 30*((d*x + c)^(1/3)*b + a)^2*a^2 + 20*((d*x + c)^(1/3)*b + a)*a^3 - 5*a^4 +
60*((d*x + c)^(1/3)*b + a)^2 - 120*((d*x + c)^(1/3)*b + a)*a + 60*a^2 - 120)*sin((d*x + c)^(1/3)*b + a)/b^5)*
e/d)/(b*d)