\(\int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx\) [34]

Optimal result

Integrand size = 17, antiderivative size = 241 \[ \int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx=-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^4}+\frac {\operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^3}-\frac {a^2 d^2 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^3}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 b^5}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^4} \] Output:

-1/2*a^2*d*cos(d*x+c)/b^4/(b*x+a)-2*a*d*cos(-c+a*d/b)*Ci(a*d/b+d*x)/b^4-Ci 
(a*d/b+d*x)*sin(-c+a*d/b)/b^3+1/2*a^2*d^2*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^5- 
1/2*a^2*sin(d*x+c)/b^3/(b*x+a)^2+2*a*sin(d*x+c)/b^3/(b*x+a)+cos(-c+a*d/b)* 
Si(a*d/b+d*x)/b^3-1/2*a^2*d^2*cos(-c+a*d/b)*Si(a*d/b+d*x)/b^5-2*a*d*sin(-c 
+a*d/b)*Si(a*d/b+d*x)/b^4
 

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.64 \[ \int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx=-\frac {-\operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-4 a b d \cos \left (c-\frac {a d}{b}\right )+\left (2 b^2-a^2 d^2\right ) \sin \left (c-\frac {a d}{b}\right )\right )+\frac {a b (a d (a+b x) \cos (c+d x)-b (3 a+4 b x) \sin (c+d x))}{(a+b x)^2}+\left (\left (-2 b^2+a^2 d^2\right ) \cos \left (c-\frac {a d}{b}\right )-4 a b d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{2 b^5} \] Input:

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

Output:

-1/2*(-(CosIntegral[d*(a/b + x)]*(-4*a*b*d*Cos[c - (a*d)/b] + (2*b^2 - a^2 
*d^2)*Sin[c - (a*d)/b])) + (a*b*(a*d*(a + b*x)*Cos[c + d*x] - b*(3*a + 4*b 
*x)*Sin[c + d*x]))/(a + b*x)^2 + ((-2*b^2 + a^2*d^2)*Cos[c - (a*d)/b] - 4* 
a*b*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^5
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7293, 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.

\(\displaystyle \int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {a^2 \sin (c+d x)}{b^2 (a+b x)^3}-\frac {2 a \sin (c+d x)}{b^2 (a+b x)^2}+\frac {\sin (c+d x)}{b^2 (a+b x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^2 d^2 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^5}-\frac {a^2 d \cos (c+d x)}{2 b^4 (a+b x)}-\frac {a^2 \sin (c+d x)}{2 b^3 (a+b x)^2}-\frac {2 a d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {2 a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^3}+\frac {2 a \sin (c+d x)}{b^3 (a+b x)}\)

Input:

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

Output:

-1/2*(a^2*d*Cos[c + d*x])/(b^4*(a + b*x)) - (2*a*d*Cos[c - (a*d)/b]*CosInt 
egral[(a*d)/b + d*x])/b^4 + (CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/ 
b^3 - (a^2*d^2*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/(2*b^5) - (a^2 
*Sin[c + d*x])/(2*b^3*(a + b*x)^2) + (2*a*Sin[c + d*x])/(b^3*(a + b*x)) + 
(Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^3 - (a^2*d^2*Cos[c - (a*d) 
/b]*SinIntegral[(a*d)/b + d*x])/(2*b^5) + (2*a*d*Sin[c - (a*d)/b]*SinInteg 
ral[(a*d)/b + d*x])/b^4
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.27 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.58

method	result	size
risch	\(-\frac {i \left (2 i a^{2} b^{3} d^{4} x^{3}+6 i a^{3} b^{2} d^{4} x^{2}+6 i a^{4} b \,d^{4} x +2 i a^{5} d^{4}\right ) \cos \left (d x +c \right )}{4 b^{4} d \left (b x +a \right )^{2} \left (-x^{2} d^{2} b^{2}-2 a b \,d^{2} x -a^{2} d^{2}\right )}-\frac {\left (8 a \,b^{3} d^{3} x^{3}+22 a^{2} b^{2} d^{3} x^{2}+20 a^{3} b \,d^{3} x +6 a^{4} d^{3}\right ) \sin \left (d x +c \right )}{4 b^{3} d \left (b x +a \right )^{2} \left (-x^{2} d^{2} b^{2}-2 a b \,d^{2} x -a^{2} d^{2}\right )}-\frac {i \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a^{2} d^{2}}{4 b^{5}}+\frac {i \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a^{2} d^{2}}{4 b^{5}}+\frac {i \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{3}}+\frac {\cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a d}{b^{4}}-\frac {i \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{3}}+\frac {\cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a d}{b^{4}}-\frac {\sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a^{2} d^{2}}{4 b^{5}}-\frac {\sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a^{2} d^{2}}{4 b^{5}}+\frac {\sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{3}}-\frac {i \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right ) a d}{b^{4}}+\frac {\sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{3}}+\frac {i \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right ) a d}{b^{4}}\)	\(621\)
derivativedivides	\(\frac {d^{3} c^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )+\frac {2 d^{3} \left (a d -b c \right ) c \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )}{b}-\frac {2 d^{3} c \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{b}+\frac {d^{3} \left (a d -b c \right )^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )}{b^{2}}+\frac {d^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{b^{2}}-\frac {2 d^{3} \left (a d -b c \right ) \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{b^{2}}}{d^{3}}\)	\(779\)
default	\(\frac {d^{3} c^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )+\frac {2 d^{3} \left (a d -b c \right ) c \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )}{b}-\frac {2 d^{3} c \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{b}+\frac {d^{3} \left (a d -b c \right )^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (a d -b c +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}-\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}}{b}}{2 b}\right )}{b^{2}}+\frac {d^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{b^{2}}-\frac {2 d^{3} \left (a d -b c \right ) \left (-\frac {\sin \left (d x +c \right )}{\left (a d -b c +b \left (d x +c \right )\right ) b}+\frac {\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}+\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}}{b}\right )}{b^{2}}}{d^{3}}\)	\(779\)

Input:

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

Output:

-1/4*I/b^4/d*(6*I*a^3*b^2*d^4*x^2+6*I*a^4*b*d^4*x+2*I*a^2*b^3*d^4*x^3+2*I* 
a^5*d^4)/(b*x+a)^2/(-b^2*d^2*x^2-2*a*b*d^2*x-a^2*d^2)*cos(d*x+c)-1/4/b^3/d 
*(8*a*b^3*d^3*x^3+22*a^2*b^2*d^3*x^2+20*a^3*b*d^3*x+6*a^4*d^3)/(b*x+a)^2/( 
-b^2*d^2*x^2-2*a*b*d^2*x-a^2*d^2)*sin(d*x+c)-1/4*I/b^5*cos((a*d-b*c)/b)*Ei 
(1,-I*(b*x+a)*d/b)*a^2*d^2+1/4*I/b^5*cos((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)* 
a^2*d^2+1/2*I/b^3*cos((a*d-b*c)/b)*Ei(1,-I*(b*x+a)*d/b)+1/b^4*cos((a*d-b*c 
)/b)*Ei(1,-I*(b*x+a)*d/b)*a*d-1/2*I/b^3*cos((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/ 
b)+1/b^4*cos((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)*a*d-1/4/b^5*sin((a*d-b*c)/b) 
*Ei(1,-I*(b*x+a)*d/b)*a^2*d^2-1/4/b^5*sin((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b) 
*a^2*d^2+1/2/b^3*sin((a*d-b*c)/b)*Ei(1,-I*(b*x+a)*d/b)-I/b^4*sin((a*d-b*c) 
/b)*Ei(1,-I*(b*x+a)*d/b)*a*d+1/2/b^3*sin((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)+ 
I/b^4*sin((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)*a*d
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.35 \[ \int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx=-\frac {{\left (a^{2} b^{2} d x + a^{3} b d\right )} \cos \left (d x + c\right ) + {\left (4 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - {\left (4 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \sin \left (d x + c\right ) - {\left ({\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) - 4 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \] Input:

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

Output:

-1/2*((a^2*b^2*d*x + a^3*b*d)*cos(d*x + c) + (4*(a*b^3*d*x^2 + 2*a^2*b^2*d 
*x + a^3*b*d)*cos_integral((b*d*x + a*d)/b) + (a^4*d^2 - 2*a^2*b^2 + (a^2* 
b^2*d^2 - 2*b^4)*x^2 + 2*(a^3*b*d^2 - 2*a*b^3)*x)*sin_integral((b*d*x + a* 
d)/b))*cos(-(b*c - a*d)/b) - (4*a*b^3*x + 3*a^2*b^2)*sin(d*x + c) - ((a^4* 
d^2 - 2*a^2*b^2 + (a^2*b^2*d^2 - 2*b^4)*x^2 + 2*(a^3*b*d^2 - 2*a*b^3)*x)*c 
os_integral((b*d*x + a*d)/b) - 4*(a*b^3*d*x^2 + 2*a^2*b^2*d*x + a^3*b*d)*s 
in_integral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(b^7*x^2 + 2*a*b^6*x + 
a^2*b^5)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*((b*cos(c)^2 + b*sin(c)^2)*d*x^2*cos(d*x + c) + ((a*(I*exp_integral_e 
(4, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*cos( 
c)^2 + a*(I*exp_integral_e(4, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(4, - 
(I*b*d*x + I*a*d)/b))*sin(c)^2)*cos(-(b*c - a*d)/b) - (a*(exp_integral_e(4 
, (I*b*d*x + I*a*d)/b) + exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*cos(c)^2 
 + a*(exp_integral_e(4, (I*b*d*x + I*a*d)/b) + exp_integral_e(4, -(I*b*d*x 
 + I*a*d)/b))*sin(c)^2)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 + (b*cos(c)^2 
+ b*sin(c)^2)*x*sin(d*x + c) + ((a*(I*exp_integral_e(4, (I*b*d*x + I*a*d)/ 
b) - I*exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(I*exp_integr 
al_e(4, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(4, -(I*b*d*x + I*a*d)/b))* 
sin(c)^2)*cos(-(b*c - a*d)/b) - (a*(exp_integral_e(4, (I*b*d*x + I*a*d)/b) 
 + exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a*(exp_integral_e(4 
, (I*b*d*x + I*a*d)/b) + exp_integral_e(4, -(I*b*d*x + I*a*d)/b))*sin(c)^2 
)*sin(-(b*c - a*d)/b))*sin(d*x + c)^2 + ((b*d*x^2*cos(c) - b*x*sin(c))*cos 
(d*x + c)^2 + (b*d*x^2*cos(c) - b*x*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) 
 - 6*(((a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^3*x^3 + 3*(a^2*b^3*cos(c)^2 + a 
^2*b^3*sin(c)^2)*d^3*x^2 + 3*(a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^3*x + 
 (a^4*b*cos(c)^2 + a^4*b*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((a*b^4*cos(c)^2 
+ a*b^4*sin(c)^2)*d^3*x^3 + 3*(a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^3*x^ 
2 + 3*(a^3*b^2*cos(c)^2 + a^3*b^2*sin(c)^2)*d^3*x + (a^4*b*cos(c)^2 + a...
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.52 (sec) , antiderivative size = 15410, normalized size of antiderivative = 63.94 \[ \int \frac {x^2 \sin (c+d x)}{(a+b x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/4*(a^2*b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2* 
tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*b^2*d^2*x^2*imag_part(cos_integral(-d* 
x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^2*x 
^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b 
)^2 + 2*a^2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^ 
2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*b^2*d^2*x^2*real_part(cos_integral(- 
d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a^2*b^2*d^2*x 
^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2* 
a*d/b)^2 - 2*a^2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2 
*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a^3*b*d^2*x*imag_part(cos_integral 
(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*b*d^2* 
x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/ 
2*a*d/b)^2 + 4*a*b^3*d*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d* 
x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a*b^3*d*x^2*real_part(cos_integral( 
-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a^3*b*d^2* 
x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) 
^2 - a^2*b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*t 
an(1/2*c)^2 + a^2*b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/ 
2*d*x)^2*tan(1/2*c)^2 - 2*a^2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*ta 
n(1/2*d*x)^2*tan(1/2*c)^2 + 4*a^2*b^2*d^2*x^2*imag_part(cos_integral(d*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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