3.249 \(\int (c e+d e x)^{4/3} \sin (a+\frac{b}{(c+d x)^{2/3}}) \, dx\)

Optimal. Leaf size=299 \[ -\frac{8 \sqrt{2 \pi } b^{7/2} e \sin (a) \sqrt [3]{e (c+d x)} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac{8 \sqrt{2 \pi } b^{7/2} e \cos (a) \sqrt [3]{e (c+d x)} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac{4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}-\frac{8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}+\frac{6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d} \]

[Out]

(-8*b^3*e*(e*(c + d*x))^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/(35*d) + (6*b*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*
Cos[a + b/(c + d*x)^(2/3)])/(35*d) - (8*b^(7/2)*e*Sqrt[2*Pi]*(e*(c + d*x))^(1/3)*Cos[a]*FresnelS[(Sqrt[b]*Sqrt
[2/Pi])/(c + d*x)^(1/3)])/(35*d*(c + d*x)^(1/3)) - (8*b^(7/2)*e*Sqrt[2*Pi]*(e*(c + d*x))^(1/3)*FresnelC[(Sqrt[
b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(35*d*(c + d*x)^(1/3)) - (4*b^2*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*
Sin[a + b/(c + d*x)^(2/3)])/(35*d) + (3*e*(c + d*x)^2*(e*(c + d*x))^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(7*d)

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Rubi [A]  time = 0.287398, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3435, 3417, 3415, 3409, 3387, 3388, 3353, 3352, 3351} \[ -\frac{8 \sqrt{2 \pi } b^{7/2} e \sin (a) \sqrt [3]{e (c+d x)} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac{8 \sqrt{2 \pi } b^{7/2} e \cos (a) \sqrt [3]{e (c+d x)} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac{4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}-\frac{8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}+\frac{6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(4/3)*Sin[a + b/(c + d*x)^(2/3)],x]

[Out]

(-8*b^3*e*(e*(c + d*x))^(1/3)*Cos[a + b/(c + d*x)^(2/3)])/(35*d) + (6*b*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*
Cos[a + b/(c + d*x)^(2/3)])/(35*d) - (8*b^(7/2)*e*Sqrt[2*Pi]*(e*(c + d*x))^(1/3)*Cos[a]*FresnelS[(Sqrt[b]*Sqrt
[2/Pi])/(c + d*x)^(1/3)])/(35*d*(c + d*x)^(1/3)) - (8*b^(7/2)*e*Sqrt[2*Pi]*(e*(c + d*x))^(1/3)*FresnelC[(Sqrt[
b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a])/(35*d*(c + d*x)^(1/3)) - (4*b^2*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*
Sin[a + b/(c + d*x)^(2/3)])/(35*d) + (3*e*(c + d*x)^2*(e*(c + d*x))^(1/3)*Sin[a + b/(c + d*x)^(2/3)])/(7*d)

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 3417

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}, x] && Integ
erQ[p] && FractionQ[n]

Rule 3415

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Module[{k = Denominator[n]}, D
ist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}
, x] && IntegerQ[p] && FractionQ[n]

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]

Rubi steps

\begin{align*} \int (c e+d e x)^{4/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{4/3} \sin \left (a+\frac{b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x^{4/3} \sin \left (a+\frac{b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d \sqrt [3]{c+d x}}\\ &=\frac{\left (3 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int x^6 \sin \left (a+\frac{b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac{\left (3 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^8} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d \sqrt [3]{c+d x}}\\ &=\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}-\frac{\left (6 b e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (a+b x^2\right )}{x^6} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{7 d \sqrt [3]{c+d x}}\\ &=\frac{6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}+\frac{\left (12 b^2 e \sqrt [3]{e (c+d x)}\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 )}{35 d \sqrt [3]{c+d x}}\\ &=\frac{6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}-\frac{4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}+\frac{\left (8 b^3 e \sqrt [3]{e (c+d x)}\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 )}{35 d \sqrt [3]{c+d x}}\\ &=-\frac{8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}-\frac{4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}-\frac{\left (16 b^4 e \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}\\ &=-\frac{8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}-\frac{4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}-\frac{\left (16 b^4 e \sqrt [3]{e (c+d x)} \cos (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac{\left (16 b^4 e \sqrt [3]{e (c+d x)} \sin (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}\\ &=-\frac{8 b^3 e \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{6 b e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}-\frac{8 b^{7/2} e \sqrt{2 \pi } \sqrt [3]{e (c+d x)} \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{35 d \sqrt [3]{c+d x}}-\frac{8 b^{7/2} e \sqrt{2 \pi } \sqrt [3]{e (c+d x)} C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{35 d \sqrt [3]{c+d x}}-\frac{4 b^2 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{35 d}+\frac{3 e (c+d x)^2 \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.906377, size = 237, normalized size = 0.79 \[ \frac{(e (c+d x))^{4/3} \left (-\frac{8 \sqrt{2 \pi } b^{7/2} \left (\sin (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )+\cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )\right )}{(c+d x)^{4/3}}+\frac{\cos \left (\frac{b}{(c+d x)^{2/3}}\right ) \left (-4 b^2 \sin (a) (c+d x)^{2/3}-8 b^3 \cos (a)+6 b \cos (a) (c+d x)^{4/3}+15 \sin (a) (c+d x)^2\right )}{c+d x}+\frac{\sin \left (\frac{b}{(c+d x)^{2/3}}\right ) \left (-4 b^2 \cos (a) (c+d x)^{2/3}+8 b^3 \sin (a)-6 b \sin (a) (c+d x)^{4/3}+15 \cos (a) (c+d x)^2\right )}{c+d x}\right )}{35 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(4/3)*Sin[a + b/(c + d*x)^(2/3)],x]

[Out]

((e*(c + d*x))^(4/3)*((Cos[b/(c + d*x)^(2/3)]*(-8*b^3*Cos[a] + 6*b*(c + d*x)^(4/3)*Cos[a] - 4*b^2*(c + d*x)^(2
/3)*Sin[a] + 15*(c + d*x)^2*Sin[a]))/(c + d*x) - (8*b^(7/2)*Sqrt[2*Pi]*(Cos[a]*FresnelS[(Sqrt[b]*Sqrt[2/Pi])/(
c + d*x)^(1/3)] + FresnelC[(Sqrt[b]*Sqrt[2/Pi])/(c + d*x)^(1/3)]*Sin[a]))/(c + d*x)^(4/3) + ((-4*b^2*(c + d*x)
^(2/3)*Cos[a] + 15*(c + d*x)^2*Cos[a] + 8*b^3*Sin[a] - 6*b*(c + d*x)^(4/3)*Sin[a])*Sin[b/(c + d*x)^(2/3)])/(c
+ d*x)))/(35*d)

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{4}{3}}}\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)^(4/3)*sin(a+b/(d*x+c)^(2/3)),x)

[Out]

int((d*e*x+c*e)^(4/3)*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)^(4/3)*sin(a+b/(d*x+c)^(2/3)),x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d e x + c e\right )}^{\frac{4}{3}} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right ), x\right ) \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)^(2/3)),x, algorithm="fricas")

[Out]

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

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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)**(2/3)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{\frac{4}{3}} \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((d*e*x+c*e)^(4/3)*sin(a+b/(d*x+c)^(2/3)),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(4/3)*sin(a + b/(d*x + c)^(2/3)), x)