3.227 \(\int (c e+d e x)^{4/3} \sin (a+b \sqrt [3]{c+d x}) \, dx\)

Optimal. Leaf size=289 \[ \frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac{2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}+\frac{90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac{2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]

[Out]

(2160*e*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^7*d*(c + d*x)^(1/3)) - (1080*e*(c + d*x)^(1/3)*(e*(
c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d) + (90*e*(c + d*x)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^
(1/3)])/(b^3*d) - (3*e*(c + d*x)^(5/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d) + (2160*e*(e*(c +
d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d) - (360*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d
*x)^(1/3)])/(b^4*d) + (18*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d)

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Rubi [A]  time = 0.268518, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3431, 15, 3296, 2638} \[ \frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac{2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}+\frac{90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac{2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)],x]

[Out]

(2160*e*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^7*d*(c + d*x)^(1/3)) - (1080*e*(c + d*x)^(1/3)*(e*(
c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d) + (90*e*(c + d*x)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^
(1/3)])/(b^3*d) - (3*e*(c + d*x)^(5/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d) + (2160*e*(e*(c +
d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d) - (360*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d
*x)^(1/3)])/(b^4*d) + (18*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d)

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 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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]

Rubi steps

\begin{align*} \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{3 \operatorname{Subst}\left (\int x^2 \left (e x^3\right )^{4/3} \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{\left (3 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x^6 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{\left (18 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d \sqrt [3]{c+d x}}\\ &=-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{\left (90 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{c+d x}}\\ &=\frac{90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{\left (360 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d \sqrt [3]{c+d x}}\\ &=\frac{90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac{360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac{\left (1080 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}\\ &=-\frac{1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac{90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac{360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac{\left (2160 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d \sqrt [3]{c+d x}}\\ &=-\frac{1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac{90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac{360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{\left (2160 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d \sqrt [3]{c+d x}}\\ &=\frac{2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac{1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac{90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac{360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac{18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}\\ \end{align*}

Mathematica [A]  time = 0.547398, size = 226, normalized size = 0.78 \[ \frac{3 (e (c+d x))^{4/3} \left (\sin \left (b \sqrt [3]{c+d x}\right ) \left (\sin (a) \left (b^6 (c+d x)^2-30 b^4 (c+d x)^{4/3}+360 b^2 (c+d x)^{2/3}-720\right )+6 b \cos (a) \left (b^4 (c+d x)^{5/3}-20 b^2 (c+d x)+120 \sqrt [3]{c+d x}\right )\right )-\cos \left (b \sqrt [3]{c+d x}\right ) \left (\cos (a) \left (b^6 (c+d x)^2-30 b^4 (c+d x)^{4/3}+360 b^2 (c+d x)^{2/3}-720\right )-6 b \sin (a) \left (b^4 (c+d x)^{5/3}-20 b^2 (c+d x)+120 \sqrt [3]{c+d x}\right )\right )\right )}{b^7 d (c+d x)^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)],x]

[Out]

(3*(e*(c + d*x))^(4/3)*(-(Cos[b*(c + d*x)^(1/3)]*((-720 + 360*b^2*(c + d*x)^(2/3) - 30*b^4*(c + d*x)^(4/3) + b
^6*(c + d*x)^2)*Cos[a] - 6*b*(120*(c + d*x)^(1/3) - 20*b^2*(c + d*x) + b^4*(c + d*x)^(5/3))*Sin[a])) + (6*b*(1
20*(c + d*x)^(1/3) - 20*b^2*(c + d*x) + b^4*(c + d*x)^(5/3))*Cos[a] + (-720 + 360*b^2*(c + d*x)^(2/3) - 30*b^4
*(c + d*x)^(4/3) + b^6*(c + d*x)^2)*Sin[a])*Sin[b*(c + d*x)^(1/3)]))/(b^7*d*(c + d*x)^(4/3))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{4}{3}}}\sin \left ( a+b\sqrt [3]{dx+c} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x)

[Out]

int((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 7.63023, size = 568, normalized size = 1.97 \begin{align*} \frac{3 \,{\left ({\left (30 \, b^{4} d^{2} e x^{2} + 60 \, b^{4} c d e x + 30 \, b^{4} c^{2} e -{\left (b^{6} d^{2} e x^{2} + 2 \, b^{6} c d e x +{\left (b^{6} c^{2} - 720\right )} e\right )}{\left (d x + c\right )}^{\frac{2}{3}} - 360 \,{\left (b^{2} d e x + b^{2} c e\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )}{\left (d e x + c e\right )}^{\frac{1}{3}} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) + 6 \,{\left (120 \, b d e x + 120 \, b c e - 20 \,{\left (b^{3} d e x + b^{3} c e\right )}{\left (d x + c\right )}^{\frac{2}{3}} +{\left (b^{5} d^{2} e x^{2} + 2 \, b^{5} c d e x + b^{5} c^{2} e\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )}{\left (d e x + c e\right )}^{\frac{1}{3}} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{7} d^{2} x + b^{7} c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x, algorithm="fricas")

[Out]

3*((30*b^4*d^2*e*x^2 + 60*b^4*c*d*e*x + 30*b^4*c^2*e - (b^6*d^2*e*x^2 + 2*b^6*c*d*e*x + (b^6*c^2 - 720)*e)*(d*
x + c)^(2/3) - 360*(b^2*d*e*x + b^2*c*e)*(d*x + c)^(1/3))*(d*e*x + c*e)^(1/3)*cos((d*x + c)^(1/3)*b + a) + 6*(
120*b*d*e*x + 120*b*c*e - 20*(b^3*d*e*x + b^3*c*e)*(d*x + c)^(2/3) + (b^5*d^2*e*x^2 + 2*b^5*c*d*e*x + b^5*c^2*
e)*(d*x + c)^(1/3))*(d*e*x + c*e)^(1/3)*sin((d*x + c)^(1/3)*b + a))/(b^7*d^2*x + b^7*c*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(4/3)*sin(a+b*(d*x+c)**(1/3)),x)

[Out]

Timed out

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Giac [A]  time = 1.25369, size = 594, normalized size = 2.06 \begin{align*} -\frac{3 \,{\left (c{\left (\frac{{\left ({\left (d x e + c e\right )} b^{3} e^{3} - 6 \,{\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{11}{3}}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac{8}{3}\right )}}{b^{4}} - \frac{3 \,{\left ({\left (d x e + c e\right )}^{\frac{2}{3}} b^{2} e^{\frac{10}{3}} - 2 \, e^{4}\right )} e^{\left (-\frac{8}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{4}}\right )} -{\left ({\left (\frac{{\left ({\left (d x e + c e\right )} b^{3} c e^{3} - 6 \,{\left (d x e + c e\right )}^{\frac{1}{3}} b c e^{\frac{11}{3}}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac{8}{3}\right )}}{b^{4}} - \frac{3 \,{\left ({\left (d x e + c e\right )}^{\frac{2}{3}} b^{2} c e^{\frac{10}{3}} - 2 \, c e^{4}\right )} e^{\left (-\frac{8}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{4}}\right )} e - \frac{{\left ({\left (d x e + c e\right )}^{2} b^{6} e^{5} - 30 \,{\left (d x e + c e\right )}^{\frac{4}{3}} b^{4} e^{\frac{17}{3}} + 360 \,{\left (d x e + c e\right )}^{\frac{2}{3}} b^{2} e^{\frac{19}{3}} - 720 \, e^{7}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac{14}{3}\right )}}{b^{7}} + \frac{6 \,{\left ({\left (d x e + c e\right )}^{\frac{5}{3}} b^{5} e^{\frac{16}{3}} - 20 \,{\left (d x e + c e\right )} b^{3} e^{6} + 120 \,{\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{20}{3}}\right )} e^{\left (-\frac{14}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{7}}\right )} e^{\left (-1\right )}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x, algorithm="giac")

[Out]

-3*(c*(((d*x*e + c*e)*b^3*e^3 - 6*(d*x*e + c*e)^(1/3)*b*e^(11/3))*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^
(-1))*e^(-8/3)/b^4 - 3*((d*x*e + c*e)^(2/3)*b^2*e^(10/3) - 2*e^4)*e^(-8/3)*sin(((d*x*e + c*e)^(1/3)*b*e^(2/3)
+ a*e)*e^(-1))/b^4) - ((((d*x*e + c*e)*b^3*c*e^3 - 6*(d*x*e + c*e)^(1/3)*b*c*e^(11/3))*cos(((d*x*e + c*e)^(1/3
)*b*e^(2/3) + a*e)*e^(-1))*e^(-8/3)/b^4 - 3*((d*x*e + c*e)^(2/3)*b^2*c*e^(10/3) - 2*c*e^4)*e^(-8/3)*sin(((d*x*
e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))/b^4)*e - ((d*x*e + c*e)^2*b^6*e^5 - 30*(d*x*e + c*e)^(4/3)*b^4*e^(17/3
) + 360*(d*x*e + c*e)^(2/3)*b^2*e^(19/3) - 720*e^7)*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))*e^(-14/3
)/b^7 + 6*((d*x*e + c*e)^(5/3)*b^5*e^(16/3) - 20*(d*x*e + c*e)*b^3*e^6 + 120*(d*x*e + c*e)^(1/3)*b*e^(20/3))*e
^(-14/3)*sin(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))/b^7)*e^(-1))/d