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

Optimal. Leaf size=89 \[ \frac{3 \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b^2 d \sqrt [3]{c+d x}}-\frac{3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]

[Out]

(-3*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d) + (3*(e*(c + d*x))^(1/3)*Sin[a + b
*(c + d*x)^(2/3)])/(2*b^2*d*(c + d*x)^(1/3))

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Rubi [A]  time = 0.0861527, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3435, 3381, 3379, 3296, 2637} \[ \frac{3 \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b^2 d \sqrt [3]{c+d x}}-\frac{3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d) + (3*(e*(c + d*x))^(1/3)*Sin[a + b
*(c + d*x)^(2/3)])/(2*b^2*d*(c + d*x)^(1/3))

Rule 3435

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

Rule 3381

Int[((e_)*(x_))^(m_)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[(e^IntPart[m]*(e*x)
^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] &&
IntegerQ[Simplify[(m + 1)/n]]

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 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 \sqrt [3]{c e+d e x} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt [3]{e x} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{\sqrt [3]{e (c+d x)} \operatorname{Subst}\left (\int \sqrt [3]{x} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d \sqrt [3]{c+d x}}\\ &=\frac{\left (3 \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{2 d \sqrt [3]{c+d x}}\\ &=-\frac{3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{\left (3 \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{2 b d \sqrt [3]{c+d x}}\\ &=-\frac{3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{3 \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b^2 d \sqrt [3]{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0604944, size = 72, normalized size = 0.81 \[ -\frac{3 \sqrt [3]{e (c+d x)} \left (b (c+d x)^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )-\sin \left (a+b (c+d x)^{2/3}\right )\right )}{2 b^2 d \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(e*(c + d*x))^(1/3)*(b*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(2/3)] - Sin[a + b*(c + d*x)^(2/3)]))/(2*b^2*d*
(c + d*x)^(1/3))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{dex+ce}\sin \left ( a+b \left ( dx+c \right ) ^{{\frac{2}{3}}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*e*x+c*e)^(1/3)*sin(a+b*(d*x+c)^(2/3)),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)^(1/3)*sin(a+b*(d*x+c)^(2/3)),x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 7.91939, size = 232, normalized size = 2.61 \begin{align*} -\frac{3 \,{\left ({\left (b d x + b c\right )}{\left (d e x + c e\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{1}{3}} \cos \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right ) -{\left (d e x + c e\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \sin \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )\right )}}{2 \,{\left (b^{2} d^{2} x + b^{2} c d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(1/3)*sin(a+b*(d*x+c)^(2/3)),x, algorithm="fricas")

[Out]

-3/2*((b*d*x + b*c)*(d*e*x + c*e)^(1/3)*(d*x + c)^(1/3)*cos((d*x + c)^(2/3)*b + a) - (d*e*x + c*e)^(1/3)*(d*x
+ c)^(2/3)*sin((d*x + c)^(2/3)*b + a))/(b^2*d^2*x + b^2*c*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{e \left (c + d x\right )} \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((d*e*x+c*e)**(1/3)*sin(a+b*(d*x+c)**(2/3)),x)

[Out]

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

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Giac [A]  time = 1.20059, size = 171, normalized size = 1.92 \begin{align*} -\frac{3 \,{\left ({\left (d x e + c e\right )}^{\frac{2}{3}} b \cos \left ({\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} + a\right ) e^{\left (-\frac{2}{3}\right )} +{\left (d x e + c e\right )}^{\frac{2}{3}} b \cos \left (-{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} - a\right ) e^{\left (-\frac{2}{3}\right )} - \sin \left ({\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} + a\right ) + \sin \left (-{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} - a\right )\right )} e^{\frac{1}{3}}}{4 \, b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(1/3)*sin(a+b*(d*x+c)^(2/3)),x, algorithm="giac")

[Out]

-3/4*((d*x*e + c*e)^(2/3)*b*cos((d*x*e + c*e)^(2/3)*b*e^(-2/3) + a)*e^(-2/3) + (d*x*e + c*e)^(2/3)*b*cos(-(d*x
*e + c*e)^(2/3)*b*e^(-2/3) - a)*e^(-2/3) - sin((d*x*e + c*e)^(2/3)*b*e^(-2/3) + a) + sin(-(d*x*e + c*e)^(2/3)*
b*e^(-2/3) - a))*e^(1/3)/(b^2*d)