Integrand size = 27, antiderivative size = 267 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\frac {b^3 \cos \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac {b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}+\frac {b^4 \sqrt [3]{c+d x} \operatorname {CosIntegral}\left (b \sqrt [3]{c+d x}\right ) \sin (a)}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}+\frac {b^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac {b^4 \sqrt [3]{c+d x} \cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}} \] Output:
1/8*b^3*cos(a+b*(d*x+c)^(1/3))/d/e^2/(e*(d*x+c))^(1/3)-1/4*b*cos(a+b*(d*x+ c)^(1/3))/d/e^2/(d*x+c)^(2/3)/(e*(d*x+c))^(1/3)+1/8*b^4*(d*x+c)^(1/3)*Ci(b *(d*x+c)^(1/3))*sin(a)/d/e^2/(e*(d*x+c))^(1/3)-3/4*sin(a+b*(d*x+c)^(1/3))/ d/e^2/(d*x+c)/(e*(d*x+c))^(1/3)+1/8*b^2*sin(a+b*(d*x+c)^(1/3))/d/e^2/(d*x+ c)^(1/3)/(e*(d*x+c))^(1/3)+1/8*b^4*(d*x+c)^(1/3)*cos(a)*Si(b*(d*x+c)^(1/3) )/d/e^2/(e*(d*x+c))^(1/3)
Time = 0.53 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.69 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\frac {b^3 c \cos \left (a+b \sqrt [3]{c+d x}\right )+b^3 d x \cos \left (a+b \sqrt [3]{c+d x}\right )-2 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )+b^4 (c+d x)^{4/3} \operatorname {CosIntegral}\left (b \sqrt [3]{c+d x}\right ) \sin (a)-6 \sin \left (a+b \sqrt [3]{c+d x}\right )+b^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )+b^4 (c+d x)^{4/3} \cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )}{8 d e (e (c+d x))^{4/3}} \] Input:
Integrate[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3),x]
Output:
(b^3*c*Cos[a + b*(c + d*x)^(1/3)] + b^3*d*x*Cos[a + b*(c + d*x)^(1/3)] - 2 *b*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)] + b^4*(c + d*x)^(4/3)*CosInt egral[b*(c + d*x)^(1/3)]*Sin[a] - 6*Sin[a + b*(c + d*x)^(1/3)] + b^2*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)] + b^4*(c + d*x)^(4/3)*Cos[a]*SinInte gral[b*(c + d*x)^(1/3)])/(8*d*e*(e*(c + d*x))^(4/3))
Time = 0.85 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.69, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3912, 30, 3042, 3778, 3042, 3778, 25, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle \frac {3 \int \frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{(e (c+d x))^{7/3}}d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c+d x)^{5/3}}d\sqrt [3]{c+d x}}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c+d x)^{5/3}}d\sqrt [3]{c+d x}}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{(c+d x)^{4/3}}d\sqrt [3]{c+d x}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \int \frac {\sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )}{(c+d x)^{4/3}}d\sqrt [3]{c+d x}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (\frac {1}{3} b \int -\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{c+d x}d\sqrt [3]{c+d x}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{c+d x}d\sqrt [3]{c+d x}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{c+d x}d\sqrt [3]{c+d x}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{(c+d x)^{2/3}}d\sqrt [3]{c+d x}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \int \frac {\sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )}{(c+d x)^{2/3}}d\sqrt [3]{c+d x}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (b \int -\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \frac {\cos \left (b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}+\cos (a) \int \frac {\sin \left (b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \frac {\sin \left (\sqrt [3]{c+d x} b+\frac {\pi }{2}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}+\cos (a) \int \frac {\sin \left (b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \frac {\sin \left (\sqrt [3]{c+d x} b+\frac {\pi }{2}\right )}{\sqrt [3]{c+d x}}d\sqrt [3]{c+d x}+\cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {3 \sqrt [3]{c+d x} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \operatorname {CosIntegral}\left (b \sqrt [3]{c+d x}\right )+\cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{\sqrt [3]{c+d x}}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{2 (c+d x)^{2/3}}\right )-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (c+d x)}\right )-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{4 (c+d x)^{4/3}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\) |
Input:
Int[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3),x]
Output:
(3*(c + d*x)^(1/3)*(-1/4*Sin[a + b*(c + d*x)^(1/3)]/(c + d*x)^(4/3) + (b*( -1/3*Cos[a + b*(c + d*x)^(1/3)]/(c + d*x) - (b*(-1/2*Sin[a + b*(c + d*x)^( 1/3)]/(c + d*x)^(2/3) + (b*(-(Cos[a + b*(c + d*x)^(1/3)]/(c + d*x)^(1/3)) - b*(CosIntegral[b*(c + d*x)^(1/3)]*Sin[a] + Cos[a]*SinIntegral[b*(c + d*x )^(1/3)])))/2))/3))/4))/(d*e^2*(e*(c + d*x))^(1/3))
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f) Subst[Int[ExpandIntegra nd[(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]
\[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{\left (d e x +c e \right )^{\frac {7}{3}}}d x\]
Input:
int(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x)
Output:
int(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x)
\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac {7}{3}}} \,d x } \] Input:
integrate(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x, algorithm="fricas")
Output:
integral((d*e*x + c*e)^(2/3)*sin((d*x + c)^(1/3)*b + a)/(d^3*e^3*x^3 + 3*c *d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)
Timed out. \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\text {Timed out} \] Input:
integrate(sin(a+b*(d*x+c)**(1/3))/(d*e*x+c*e)**(7/3),x)
Output:
Timed out
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.48 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\frac {3 \, {\left ({\left (-i \, \Gamma \left (-4, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (-4, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (-4, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, \Gamma \left (-4, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-4, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-4, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-4, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (-4, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} b^{4}}{4 \, d e^{\frac {7}{3}}} \] Input:
integrate(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x, algorithm="maxima")
Output:
3/4*((-I*gamma(-4, I*b*conjugate((d*x + c)^(1/3))) + I*gamma(-4, -I*b*conj ugate((d*x + c)^(1/3))) - I*gamma(-4, I*(d*x + c)^(1/3)*b) + I*gamma(-4, - I*(d*x + c)^(1/3)*b))*cos(a) - (gamma(-4, I*b*conjugate((d*x + c)^(1/3))) + gamma(-4, -I*b*conjugate((d*x + c)^(1/3))) + gamma(-4, I*(d*x + c)^(1/3) *b) + gamma(-4, -I*(d*x + c)^(1/3)*b))*sin(a))*b^4/(d*e^(7/3))
\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac {7}{3}}} \,d x } \] Input:
integrate(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x, algorithm="giac")
Output:
integrate(sin((d*x + c)^(1/3)*b + a)/(d*e*x + c*e)^(7/3), x)
Timed out. \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{7/3}} \,d x \] Input:
int(sin(a + b*(c + d*x)^(1/3))/(c*e + d*e*x)^(7/3),x)
Output:
int(sin(a + b*(c + d*x)^(1/3))/(c*e + d*e*x)^(7/3), x)
\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx=\frac {\int \frac {\sin \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right )}{\left (d x +c \right )^{\frac {1}{3}} c^{2}+2 \left (d x +c \right )^{\frac {1}{3}} c d x +\left (d x +c \right )^{\frac {1}{3}} d^{2} x^{2}}d x}{e^{\frac {7}{3}}} \] Input:
int(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),x)
Output:
int(sin((c + d*x)**(1/3)*b + a)/((c + d*x)**(1/3)*c**2 + 2*(c + d*x)**(1/3 )*c*d*x + (c + d*x)**(1/3)*d**2*x**2),x)/(e**(1/3)*e**2)