\(\int (e+f x)^2 \sin (a+\frac {b}{\sqrt [3]{c+d x}}) \, dx\) [217]

Optimal result

Integrand size = 22, antiderivative size = 855 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx =\text {Too large to display} \] Output:

1/120*b^5*f*(-c*f+d*e)*(d*x+c)^(1/3)*cos(a+b/(d*x+c)^(1/3))/d^3-1/120960*b 
^7*f^2*(d*x+c)^(2/3)*cos(a+b/(d*x+c)^(1/3))/d^3+1/2*b*(-c*f+d*e)^2*(d*x+c) 
^(2/3)*cos(a+b/(d*x+c)^(1/3))/d^3-1/60*b^3*f*(-c*f+d*e)*(d*x+c)*cos(a+b/(d 
*x+c)^(1/3))/d^3+1/20160*b^5*f^2*(d*x+c)^(4/3)*cos(a+b/(d*x+c)^(1/3))/d^3+ 
1/5*b*f*(-c*f+d*e)*(d*x+c)^(5/3)*cos(a+b/(d*x+c)^(1/3))/d^3-1/1008*b^3*f^2 
*(d*x+c)^2*cos(a+b/(d*x+c)^(1/3))/d^3+1/24*b*f^2*(d*x+c)^(8/3)*cos(a+b/(d* 
x+c)^(1/3))/d^3-1/120960*b^9*f^2*cos(a)*Ci(b/(d*x+c)^(1/3))/d^3+1/2*b^3*(- 
c*f+d*e)^2*cos(a)*Ci(b/(d*x+c)^(1/3))/d^3+1/120*b^6*f*(-c*f+d*e)*Ci(b/(d*x 
+c)^(1/3))*sin(a)/d^3+1/120960*b^8*f^2*(d*x+c)^(1/3)*sin(a+b/(d*x+c)^(1/3) 
)/d^3-1/2*b^2*(-c*f+d*e)^2*(d*x+c)^(1/3)*sin(a+b/(d*x+c)^(1/3))/d^3+1/120* 
b^4*f*(-c*f+d*e)*(d*x+c)^(2/3)*sin(a+b/(d*x+c)^(1/3))/d^3-1/60480*b^6*f^2* 
(d*x+c)*sin(a+b/(d*x+c)^(1/3))/d^3+(-c*f+d*e)^2*(d*x+c)*sin(a+b/(d*x+c)^(1 
/3))/d^3-1/20*b^2*f*(-c*f+d*e)*(d*x+c)^(4/3)*sin(a+b/(d*x+c)^(1/3))/d^3+1/ 
5040*b^4*f^2*(d*x+c)^(5/3)*sin(a+b/(d*x+c)^(1/3))/d^3+f*(-c*f+d*e)*(d*x+c) 
^2*sin(a+b/(d*x+c)^(1/3))/d^3-1/168*b^2*f^2*(d*x+c)^(7/3)*sin(a+b/(d*x+c)^ 
(1/3))/d^3+1/3*f^2*(d*x+c)^3*sin(a+b/(d*x+c)^(1/3))/d^3+1/120*b^6*f*(-c*f+ 
d*e)*cos(a)*Si(b/(d*x+c)^(1/3))/d^3+1/120960*b^9*f^2*sin(a)*Si(b/(d*x+c)^( 
1/3))/d^3-1/2*b^3*(-c*f+d*e)^2*sin(a)*Si(b/(d*x+c)^(1/3))/d^3
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.31 (sec) , antiderivative size = 929, normalized size of antiderivative = 1.09 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

((-1/241920*I)*((Cos[a] + I*Sin[a])*((60480*I)*b^3*d^2*e^2*ExpIntegralEi[( 
I*b)/(c + d*x)^(1/3)] + 1008*b^6*d*e*f*ExpIntegralEi[(I*b)/(c + d*x)^(1/3) 
] - (120960*I)*b^3*c*d*e*f*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - I*b^9*f^ 
2*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - 1008*b^6*c*f^2*ExpIntegralEi[(I*b 
)/(c + d*x)^(1/3)] + (60480*I)*b^3*c^2*f^2*ExpIntegralEi[(I*b)/(c + d*x)^( 
1/3)] + (c + d*x)^(1/3)*(b^8*f^2 - I*b^7*f^2*(c + d*x)^(1/3) - 2*b^6*f^2*( 
c + d*x)^(2/3) + (24*I)*b^3*f*(c + d*x)^(2/3)*(-84*d*e + 79*c*f - 5*d*f*x) 
 + (6*I)*b^5*f*(168*d*e - 167*c*f + d*f*x) + 24*b^4*f*(c + d*x)^(1/3)*(42* 
d*e - 41*c*f + d*f*x) + 40320*(c + d*x)^(2/3)*(c^2*f^2 - c*d*f*(3*e + f*x) 
 + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)) + (1008*I)*b*(c + d*x)^(1/3)*(41*c^2*f 
^2 - 2*c*d*f*(48*e + 7*f*x) + d^2*(60*e^2 + 24*e*f*x + 5*f^2*x^2)) - 144*b 
^2*(383*c^2*f^2 - 2*c*d*f*(399*e + 16*f*x) + d^2*(420*e^2 + 42*e*f*x + 5*f 
^2*x^2)))*(Cos[b/(c + d*x)^(1/3)] + I*Sin[b/(c + d*x)^(1/3)])) - ((c + d*x 
)^(1/3)*(b^8*f^2 + I*b^7*f^2*(c + d*x)^(1/3) - 2*b^6*f^2*(c + d*x)^(2/3) - 
 (6*I)*b^5*f*(168*d*e - 167*c*f + d*f*x) + 24*b^4*f*(c + d*x)^(1/3)*(42*d* 
e - 41*c*f + d*f*x) + (24*I)*b^3*f*(c + d*x)^(2/3)*(84*d*e - 79*c*f + 5*d* 
f*x) + 40320*(c + d*x)^(2/3)*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3 
*e*f*x + f^2*x^2)) - (1008*I)*b*(c + d*x)^(1/3)*(41*c^2*f^2 - 2*c*d*f*(48* 
e + 7*f*x) + d^2*(60*e^2 + 24*e*f*x + 5*f^2*x^2)) - 144*b^2*(383*c^2*f^2 - 
 2*c*d*f*(399*e + 16*f*x) + d^2*(420*e^2 + 42*e*f*x + 5*f^2*x^2))) + I*...
 

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 866, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3912, 2009}

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.

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

\(\Big \downarrow \) 3912

\(\displaystyle -\frac {3 \int \left (\frac {f^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) (c+d x)^{10/3}}{d^2}+\frac {2 f (d e-c f) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) (c+d x)^{7/3}}{d^2}+\frac {(d e-c f)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) (c+d x)^{4/3}}{d^2}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \left (\frac {f^2 \cos (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) b^9}{362880 d^2}-\frac {f^2 \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) b^9}{362880 d^2}-\frac {f^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^8}{362880 d^2}+\frac {f^2 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^7}{362880 d^2}-\frac {f (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a) b^6}{360 d^2}+\frac {f^2 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^6}{181440 d^2}-\frac {f (d e-c f) \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) b^6}{360 d^2}-\frac {f^2 (c+d x)^{4/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^5}{60480 d^2}-\frac {f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^5}{360 d^2}-\frac {f^2 (c+d x)^{5/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^4}{15120 d^2}-\frac {f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^4}{360 d^2}+\frac {f^2 (c+d x)^2 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^3}{3024 d^2}+\frac {f (d e-c f) (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^3}{180 d^2}-\frac {(d e-c f)^2 \cos (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) b^3}{6 d^2}+\frac {(d e-c f)^2 \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) b^3}{6 d^2}+\frac {f^2 (c+d x)^{7/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^2}{504 d^2}+\frac {f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^2}{60 d^2}+\frac {(d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b^2}{6 d^2}-\frac {f^2 (c+d x)^{8/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b}{72 d^2}-\frac {f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b}{15 d^2}-\frac {(d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) b}{6 d^2}-\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{9 d^2}-\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 d^2}-\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 d^2}\right )}{d}\)

Input:

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

Output:

(-3*(-1/360*(b^5*f*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)]) 
/d^2 + (b^7*f^2*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(362880*d^2) - 
 (b*(d*e - c*f)^2*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(6*d^2) + (b 
^3*f*(d*e - c*f)*(c + d*x)*Cos[a + b/(c + d*x)^(1/3)])/(180*d^2) - (b^5*f^ 
2*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(1/3)])/(60480*d^2) - (b*f*(d*e - c* 
f)*(c + d*x)^(5/3)*Cos[a + b/(c + d*x)^(1/3)])/(15*d^2) + (b^3*f^2*(c + d* 
x)^2*Cos[a + b/(c + d*x)^(1/3)])/(3024*d^2) - (b*f^2*(c + d*x)^(8/3)*Cos[a 
 + b/(c + d*x)^(1/3)])/(72*d^2) + (b^9*f^2*Cos[a]*CosIntegral[b/(c + d*x)^ 
(1/3)])/(362880*d^2) - (b^3*(d*e - c*f)^2*Cos[a]*CosIntegral[b/(c + d*x)^( 
1/3)])/(6*d^2) - (b^6*f*(d*e - c*f)*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a]) 
/(360*d^2) - (b^8*f^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(362880* 
d^2) + (b^2*(d*e - c*f)^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(6*d 
^2) - (b^4*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(360* 
d^2) + (b^6*f^2*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/(181440*d^2) - ((d*e 
 - c*f)^2*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/(3*d^2) + (b^2*f*(d*e - c* 
f)*(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(60*d^2) - (b^4*f^2*(c + d* 
x)^(5/3)*Sin[a + b/(c + d*x)^(1/3)])/(15120*d^2) - (f*(d*e - c*f)*(c + d*x 
)^2*Sin[a + b/(c + d*x)^(1/3)])/(3*d^2) + (b^2*f^2*(c + d*x)^(7/3)*Sin[a + 
 b/(c + d*x)^(1/3)])/(504*d^2) - (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(1/3 
)])/(9*d^2) - (b^6*f*(d*e - c*f)*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3912

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]
 

Maple [A] (verified)

Time = 4.58 (sec) , antiderivative size = 936, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(936\)
default	\(\text {Expression too large to display}\)	\(936\)
parts	\(\text {Expression too large to display}\)	\(2775\)

Input:

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

Output:

-3/d^3*b^3*(2*b^3*d*e*f*(-1/6*sin(a+b/(d*x+c)^(1/3))/b^6*(d*x+c)^2-1/30*co 
s(a+b/(d*x+c)^(1/3))/b^5*(d*x+c)^(5/3)+1/120*sin(a+b/(d*x+c)^(1/3))/b^4*(d 
*x+c)^(4/3)+1/360*cos(a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/720*sin(a+b/(d*x+c) 
^(1/3))/b^2*(d*x+c)^(2/3)-1/720*cos(a+b/(d*x+c)^(1/3))/b*(d*x+c)^(1/3)-1/7 
20*Si(b/(d*x+c)^(1/3))*cos(a)-1/720*Ci(b/(d*x+c)^(1/3))*sin(a))-2*b^3*c*f^ 
2*(-1/6*sin(a+b/(d*x+c)^(1/3))/b^6*(d*x+c)^2-1/30*cos(a+b/(d*x+c)^(1/3))/b 
^5*(d*x+c)^(5/3)+1/120*sin(a+b/(d*x+c)^(1/3))/b^4*(d*x+c)^(4/3)+1/360*cos( 
a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/720*sin(a+b/(d*x+c)^(1/3))/b^2*(d*x+c)^(2 
/3)-1/720*cos(a+b/(d*x+c)^(1/3))/b*(d*x+c)^(1/3)-1/720*Si(b/(d*x+c)^(1/3)) 
*cos(a)-1/720*Ci(b/(d*x+c)^(1/3))*sin(a))-2*c*d*e*f*(-1/3*sin(a+b/(d*x+c)^ 
(1/3))/b^3*(d*x+c)-1/6*cos(a+b/(d*x+c)^(1/3))/b^2*(d*x+c)^(2/3)+1/6*sin(a+ 
b/(d*x+c)^(1/3))/b*(d*x+c)^(1/3)+1/6*Si(b/(d*x+c)^(1/3))*sin(a)-1/6*Ci(b/( 
d*x+c)^(1/3))*cos(a))+d^2*e^2*(-1/3*sin(a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/6 
*cos(a+b/(d*x+c)^(1/3))/b^2*(d*x+c)^(2/3)+1/6*sin(a+b/(d*x+c)^(1/3))/b*(d* 
x+c)^(1/3)+1/6*Si(b/(d*x+c)^(1/3))*sin(a)-1/6*Ci(b/(d*x+c)^(1/3))*cos(a))+ 
c^2*f^2*(-1/3*sin(a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/6*cos(a+b/(d*x+c)^(1/3) 
)/b^2*(d*x+c)^(2/3)+1/6*sin(a+b/(d*x+c)^(1/3))/b*(d*x+c)^(1/3)+1/6*Si(b/(d 
*x+c)^(1/3))*sin(a)-1/6*Ci(b/(d*x+c)^(1/3))*cos(a))+b^6*f^2*(-1/9*sin(a+b/ 
(d*x+c)^(1/3))/b^9*(d*x+c)^3-1/72*cos(a+b/(d*x+c)^(1/3))/b^8*(d*x+c)^(8/3) 
+1/504*sin(a+b/(d*x+c)^(1/3))/b^7*(d*x+c)^(7/3)+1/3024*cos(a+b/(d*x+c)^...
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.67 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=-\frac {{\left (120 \, b^{3} d^{2} f^{2} x^{2} + 2016 \, b^{3} c d e f - 1896 \, b^{3} c^{2} f^{2} + 48 \, {\left (42 \, b^{3} d^{2} e f - 37 \, b^{3} c d f^{2}\right )} x - {\left (5040 \, b d^{2} f^{2} x^{2} + 60480 \, b d^{2} e^{2} - 96768 \, b c d e f - {\left (b^{7} - 41328 \, b c^{2}\right )} f^{2} + 2016 \, {\left (12 \, b d^{2} e f - 7 \, b c d f^{2}\right )} x\right )} {\left (d x + c\right )}^{\frac {2}{3}} - 6 \, {\left (b^{5} d f^{2} x + 168 \, b^{5} d e f - 167 \, b^{5} c f^{2}\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) - {\left ({\left (60480 \, b^{3} d^{2} e^{2} - 120960 \, b^{3} c d e f - {\left (b^{9} - 60480 \, b^{3} c^{2}\right )} f^{2}\right )} \cos \left (a\right ) + 1008 \, {\left (b^{6} d e f - b^{6} c f^{2}\right )} \sin \left (a\right )\right )} \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - {\left (40320 \, d^{3} f^{2} x^{3} + 120960 \, d^{3} e f x^{2} + 120960 \, c d^{2} e^{2} - 120960 \, c^{2} d e f - 2 \, {\left (b^{6} c - 20160 \, c^{3}\right )} f^{2} - 2 \, {\left (b^{6} d f^{2} - 60480 \, d^{3} e^{2}\right )} x + 24 \, {\left (b^{4} d f^{2} x + 42 \, b^{4} d e f - 41 \, b^{4} c f^{2}\right )} {\left (d x + c\right )}^{\frac {2}{3}} - {\left (720 \, b^{2} d^{2} f^{2} x^{2} + 60480 \, b^{2} d^{2} e^{2} - 114912 \, b^{2} c d e f - {\left (b^{8} - 55152 \, b^{2} c^{2}\right )} f^{2} + 288 \, {\left (21 \, b^{2} d^{2} e f - 16 \, b^{2} c d f^{2}\right )} x\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) - {\left (1008 \, {\left (b^{6} d e f - b^{6} c f^{2}\right )} \cos \left (a\right ) - {\left (60480 \, b^{3} d^{2} e^{2} - 120960 \, b^{3} c d e f - {\left (b^{9} - 60480 \, b^{3} c^{2}\right )} f^{2}\right )} \sin \left (a\right )\right )} \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{120960 \, d^{3}} \] Input:

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

Output:

-1/120960*((120*b^3*d^2*f^2*x^2 + 2016*b^3*c*d*e*f - 1896*b^3*c^2*f^2 + 48 
*(42*b^3*d^2*e*f - 37*b^3*c*d*f^2)*x - (5040*b*d^2*f^2*x^2 + 60480*b*d^2*e 
^2 - 96768*b*c*d*e*f - (b^7 - 41328*b*c^2)*f^2 + 2016*(12*b*d^2*e*f - 7*b* 
c*d*f^2)*x)*(d*x + c)^(2/3) - 6*(b^5*d*f^2*x + 168*b^5*d*e*f - 167*b^5*c*f 
^2)*(d*x + c)^(1/3))*cos((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)) - (( 
60480*b^3*d^2*e^2 - 120960*b^3*c*d*e*f - (b^9 - 60480*b^3*c^2)*f^2)*cos(a) 
 + 1008*(b^6*d*e*f - b^6*c*f^2)*sin(a))*cos_integral(b/(d*x + c)^(1/3)) - 
(40320*d^3*f^2*x^3 + 120960*d^3*e*f*x^2 + 120960*c*d^2*e^2 - 120960*c^2*d* 
e*f - 2*(b^6*c - 20160*c^3)*f^2 - 2*(b^6*d*f^2 - 60480*d^3*e^2)*x + 24*(b^ 
4*d*f^2*x + 42*b^4*d*e*f - 41*b^4*c*f^2)*(d*x + c)^(2/3) - (720*b^2*d^2*f^ 
2*x^2 + 60480*b^2*d^2*e^2 - 114912*b^2*c*d*e*f - (b^8 - 55152*b^2*c^2)*f^2 
 + 288*(21*b^2*d^2*e*f - 16*b^2*c*d*f^2)*x)*(d*x + c)^(1/3))*sin((a*d*x + 
a*c + (d*x + c)^(2/3)*b)/(d*x + c)) - (1008*(b^6*d*e*f - b^6*c*f^2)*cos(a) 
 - (60480*b^3*d^2*e^2 - 120960*b^3*c*d*e*f - (b^9 - 60480*b^3*c^2)*f^2)*si 
n(a))*sin_integral(b/(d*x + c)^(1/3)))/d^3
 

Sympy [F]

\[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{\sqrt [3]{c + d x}} \right )}\, dx \] Input:

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 1003, normalized size of antiderivative = 1.17 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/241920*(60480*(((Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + c)^(1/3)))*cos 
(a) + (I*Ei(I*b/(d*x + c)^(1/3)) - I*Ei(-I*b/(d*x + c)^(1/3)))*sin(a))*b^3 
 + 2*(d*x + c)^(2/3)*b*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 2*(( 
d*x + c)^(1/3)*b^2 - 2*d*x - 2*c)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1 
/3)))*e^2 - 120960*(((Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + c)^(1/3)))* 
cos(a) + (I*Ei(I*b/(d*x + c)^(1/3)) - I*Ei(-I*b/(d*x + c)^(1/3)))*sin(a))* 
b^3 + 2*(d*x + c)^(2/3)*b*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 2 
*((d*x + c)^(1/3)*b^2 - 2*d*x - 2*c)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c) 
^(1/3)))*c*e*f/d + 60480*(((Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + c)^(1 
/3)))*cos(a) + (I*Ei(I*b/(d*x + c)^(1/3)) - I*Ei(-I*b/(d*x + c)^(1/3)))*si 
n(a))*b^3 + 2*(d*x + c)^(2/3)*b*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3 
)) - 2*((d*x + c)^(1/3)*b^2 - 2*d*x - 2*c)*sin(((d*x + c)^(1/3)*a + b)/(d* 
x + c)^(1/3)))*c^2*f^2/d^2 + 1008*(((-I*Ei(I*b/(d*x + c)^(1/3)) + I*Ei(-I* 
b/(d*x + c)^(1/3)))*cos(a) + (Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + c)^ 
(1/3)))*sin(a))*b^6 + 2*((d*x + c)^(1/3)*b^5 - 2*(d*x + c)*b^3 + 24*(d*x + 
 c)^(5/3)*b)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) + 2*((d*x + c)^( 
2/3)*b^4 - 6*(d*x + c)^(4/3)*b^2 + 120*(d*x + c)^2)*sin(((d*x + c)^(1/3)*a 
 + b)/(d*x + c)^(1/3)))*e*f/d - 1008*(((-I*Ei(I*b/(d*x + c)^(1/3)) + I*Ei( 
-I*b/(d*x + c)^(1/3)))*cos(a) + (Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + 
c)^(1/3)))*sin(a))*b^6 + 2*((d*x + c)^(1/3)*b^5 - 2*(d*x + c)*b^3 + 24*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11903 vs. \(2 (739) = 1478\).

Time = 0.67 (sec) , antiderivative size = 11903, normalized size of antiderivative = 13.92 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/120960*(60480*(a^3*b^4*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/ 
(d*x + c)^(1/3)) + a^3*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b) 
/(d*x + c)^(1/3)) - 3*((d*x + c)^(1/3)*a + b)*a^2*b^4*cos(a)*cos_integral( 
-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 3*((d*x + 
c)^(1/3)*a + b)*a^2*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d 
*x + c)^(1/3))/(d*x + c)^(1/3) + 3*((d*x + c)^(1/3)*a + b)^2*a*b^4*cos(a)* 
cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) 
 + 3*((d*x + c)^(1/3)*a + b)^2*a*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1 
/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) - ((d*x + c)^(1/3)*a + b)^3*b^ 
4*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + 
 c) - ((d*x + c)^(1/3)*a + b)^3*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1/ 
3)*a + b)/(d*x + c)^(1/3))/(d*x + c) + a^2*b^4*sin(((d*x + c)^(1/3)*a + b) 
/(d*x + c)^(1/3)) - 2*((d*x + c)^(1/3)*a + b)*a*b^4*sin(((d*x + c)^(1/3)*a 
 + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) + ((d*x + c)^(1/3)*a + b)^2*b^4*sin 
(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + a*b^4*cos(((d* 
x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - ((d*x + c)^(1/3)*a + b)*b^4*cos(((d 
*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 2*b^4*sin(((d*x + 
c)^(1/3)*a + b)/(d*x + c)^(1/3)))*e^2/((a^3 - 3*((d*x + c)^(1/3)*a + b)*a^ 
2/(d*x + c)^(1/3) + 3*((d*x + c)^(1/3)*a + b)^2*a/(d*x + c)^(2/3) - ((d*x 
+ c)^(1/3)*a + b)^3/(d*x + c))*b) + 1008*(a^6*b^7*cos_integral(-a + ((d...
 

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )\,{\left (e+f\,x\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\left (\int \sin \left (\frac {\left (d x +c \right )^{\frac {1}{3}} a +b}{\left (d x +c \right )^{\frac {1}{3}}}\right )d x \right ) e^{2}+\left (\int \sin \left (\frac {\left (d x +c \right )^{\frac {1}{3}} a +b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) x^{2}d x \right ) f^{2}+2 \left (\int \sin \left (\frac {\left (d x +c \right )^{\frac {1}{3}} a +b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) x d x \right ) e f \] Input:

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

Output:

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