\(\int (e+f x)^2 \sin (a+\frac {b}{\sqrt {c+d x}}) \, dx\) [197]

Optimal result

Integrand size = 22, antiderivative size = 611 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {b^5 f^2 \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{360 d^3}-\frac {b^3 f (d e-c f) \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^3}+\frac {b (d e-c f)^2 \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^3}-\frac {b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{180 d^3}+\frac {b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{3 d^3}+\frac {b f^2 (c+d x)^{5/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{15 d^3}+\frac {b^6 f^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{360 d^3}-\frac {b^4 f (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{6 d^3}+\frac {b^2 (d e-c f)^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{d^3}+\frac {b^4 f^2 (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{360 d^3}-\frac {b^2 f (d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^3}-\frac {b^2 f^2 (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{60 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{3 d^3}+\frac {b^6 f^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{360 d^3}-\frac {b^4 f (d e-c f) \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{6 d^3}+\frac {b^2 (d e-c f)^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d^3} \] Output:

1/360*b^5*f^2*(d*x+c)^(1/2)*cos(a+b/(d*x+c)^(1/2))/d^3-1/6*b^3*f*(-c*f+d*e 
)*(d*x+c)^(1/2)*cos(a+b/(d*x+c)^(1/2))/d^3+b*(-c*f+d*e)^2*(d*x+c)^(1/2)*co 
s(a+b/(d*x+c)^(1/2))/d^3-1/180*b^3*f^2*(d*x+c)^(3/2)*cos(a+b/(d*x+c)^(1/2) 
)/d^3+1/3*b*f*(-c*f+d*e)*(d*x+c)^(3/2)*cos(a+b/(d*x+c)^(1/2))/d^3+1/15*b*f 
^2*(d*x+c)^(5/2)*cos(a+b/(d*x+c)^(1/2))/d^3+1/360*b^6*f^2*Ci(b/(d*x+c)^(1/ 
2))*sin(a)/d^3-1/6*b^4*f*(-c*f+d*e)*Ci(b/(d*x+c)^(1/2))*sin(a)/d^3+b^2*(-c 
*f+d*e)^2*Ci(b/(d*x+c)^(1/2))*sin(a)/d^3+1/360*b^4*f^2*(d*x+c)*sin(a+b/(d* 
x+c)^(1/2))/d^3-1/6*b^2*f*(-c*f+d*e)*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^3+(- 
c*f+d*e)^2*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^3-1/60*b^2*f^2*(d*x+c)^2*sin(a 
+b/(d*x+c)^(1/2))/d^3+f*(-c*f+d*e)*(d*x+c)^2*sin(a+b/(d*x+c)^(1/2))/d^3+1/ 
3*f^2*(d*x+c)^3*sin(a+b/(d*x+c)^(1/2))/d^3+1/360*b^6*f^2*cos(a)*Si(b/(d*x+ 
c)^(1/2))/d^3-1/6*b^4*f*(-c*f+d*e)*cos(a)*Si(b/(d*x+c)^(1/2))/d^3+b^2*(-c* 
f+d*e)^2*cos(a)*Si(b/(d*x+c)^(1/2))/d^3
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.44 (sec) , antiderivative size = 557, normalized size of antiderivative = 0.91 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {i e^{-i a} \left (e^{-\frac {i b}{\sqrt {c+d x}}} \sqrt {c+d x} \left (-i b^5 f^2+b^4 f^2 \sqrt {c+d x}+2 i b^3 f (30 d e-29 c f+d f x)-6 b^2 f \sqrt {c+d x} (10 d e-9 c f+d f x)+120 \sqrt {c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )\right )-e^{i \left (2 a+\frac {b}{\sqrt {c+d x}}\right )} \sqrt {c+d x} \left (i b^5 f^2+b^4 f^2 \sqrt {c+d x}-2 i b^3 f (30 d e-29 c f+d f x)-6 b^2 f \sqrt {c+d x} (10 d e-9 c f+d f x)+120 \sqrt {c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )+24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )\right )+b^2 \left (360 d^2 e^2-60 \left (b^2+12 c\right ) d e f+\left (b^4+60 b^2 c+360 c^2\right ) f^2\right ) \operatorname {ExpIntegralEi}\left (-\frac {i b}{\sqrt {c+d x}}\right )-b^2 e^{2 i a} \left (360 d^2 e^2-60 \left (b^2+12 c\right ) d e f+\left (b^4+60 b^2 c+360 c^2\right ) f^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i b}{\sqrt {c+d x}}\right )\right )}{720 d^3} \] Input:

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

Output:

((I/720)*((Sqrt[c + d*x]*((-I)*b^5*f^2 + b^4*f^2*Sqrt[c + d*x] + (2*I)*b^3 
*f*(30*d*e - 29*c*f + d*f*x) - 6*b^2*f*Sqrt[c + d*x]*(10*d*e - 9*c*f + d*f 
*x) + 120*Sqrt[c + d*x]*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f* 
x + f^2*x^2)) - (24*I)*b*(11*c^2*f^2 - c*d*f*(25*e + 3*f*x) + d^2*(15*e^2 
+ 5*e*f*x + f^2*x^2))))/E^((I*b)/Sqrt[c + d*x]) - E^(I*(2*a + b/Sqrt[c + d 
*x]))*Sqrt[c + d*x]*(I*b^5*f^2 + b^4*f^2*Sqrt[c + d*x] - (2*I)*b^3*f*(30*d 
*e - 29*c*f + d*f*x) - 6*b^2*f*Sqrt[c + d*x]*(10*d*e - 9*c*f + d*f*x) + 12 
0*Sqrt[c + d*x]*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2* 
x^2)) + (24*I)*b*(11*c^2*f^2 - c*d*f*(25*e + 3*f*x) + d^2*(15*e^2 + 5*e*f* 
x + f^2*x^2))) + b^2*(360*d^2*e^2 - 60*(b^2 + 12*c)*d*e*f + (b^4 + 60*b^2* 
c + 360*c^2)*f^2)*ExpIntegralEi[((-I)*b)/Sqrt[c + d*x]] - b^2*E^((2*I)*a)* 
(360*d^2*e^2 - 60*(b^2 + 12*c)*d*e*f + (b^4 + 60*b^2*c + 360*c^2)*f^2)*Exp 
IntegralEi[(I*b)/Sqrt[c + d*x]]))/(d^3*E^(I*a))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.03, 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.

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

\(\Big \downarrow \) 3912

\(\displaystyle -\frac {2 \int \left (\frac {f^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) (c+d x)^{7/2}}{d^2}+\frac {2 f (d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) (c+d x)^{5/2}}{d^2}+\frac {(d e-c f)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) (c+d x)^{3/2}}{d^2}\right )d\frac {1}{\sqrt {c+d x}}}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {b^6 f^2 \sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}-\frac {b^6 f^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}-\frac {b^5 f^2 \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}+\frac {b^4 f \sin (a) (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^4 f \cos (a) (d e-c f) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}-\frac {b^4 f^2 (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}+\frac {b^3 f \sqrt {c+d x} (d e-c f) \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{360 d^2}-\frac {b^2 \sin (a) (d e-c f)^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {b^2 \cos (a) (d e-c f)^2 \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}+\frac {b^2 f (c+d x) (d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^2 f^2 (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{120 d^2}-\frac {f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {(c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {b f (c+d x)^{3/2} (d e-c f) \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^2}-\frac {b \sqrt {c+d x} (d e-c f)^2 \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^2}-\frac {b f^2 (c+d x)^{5/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{30 d^2}\right )}{d}\)

Input:

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

Output:

(-2*(-1/720*(b^5*f^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/d^2 + (b^3*f* 
(d*e - c*f)*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(12*d^2) - (b*(d*e - c 
*f)^2*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(2*d^2) + (b^3*f^2*(c + d*x) 
^(3/2)*Cos[a + b/Sqrt[c + d*x]])/(360*d^2) - (b*f*(d*e - c*f)*(c + d*x)^(3 
/2)*Cos[a + b/Sqrt[c + d*x]])/(6*d^2) - (b*f^2*(c + d*x)^(5/2)*Cos[a + b/S 
qrt[c + d*x]])/(30*d^2) - (b^6*f^2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(7 
20*d^2) + (b^4*f*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(12*d^2) 
 - (b^2*(d*e - c*f)^2*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(2*d^2) - (b^4* 
f^2*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/(720*d^2) + (b^2*f*(d*e - c*f)*(c 
+ d*x)*Sin[a + b/Sqrt[c + d*x]])/(12*d^2) - ((d*e - c*f)^2*(c + d*x)*Sin[a 
 + b/Sqrt[c + d*x]])/(2*d^2) + (b^2*f^2*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x 
]])/(120*d^2) - (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]])/(2*d^ 
2) - (f^2*(c + d*x)^3*Sin[a + b/Sqrt[c + d*x]])/(6*d^2) - (b^6*f^2*Cos[a]* 
SinIntegral[b/Sqrt[c + d*x]])/(720*d^2) + (b^4*f*(d*e - c*f)*Cos[a]*SinInt 
egral[b/Sqrt[c + d*x]])/(12*d^2) - (b^2*(d*e - c*f)^2*Cos[a]*SinIntegral[b 
/Sqrt[c + d*x]])/(2*d^2)))/d
 

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 = 4.40 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(-\frac {2 b^{2} \left (-2 b^{2} c \,f^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )-2 c d e f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+2 b^{2} d e f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )+c^{2} f^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+d^{2} e^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+b^{4} f^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{3}}{6 b^{6}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {5}{2}}}{30 b^{5}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{120 b^{4}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{360 b^{3}}-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{720 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{720 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{720}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{720}\right )\right )}{d^{3}}\)	\(696\)
default	\(-\frac {2 b^{2} \left (-2 b^{2} c \,f^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )-2 c d e f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+2 b^{2} d e f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )+c^{2} f^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+d^{2} e^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+b^{4} f^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{3}}{6 b^{6}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {5}{2}}}{30 b^{5}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{120 b^{4}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{360 b^{3}}-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{720 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{720 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{720}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{720}\right )\right )}{d^{3}}\)	\(696\)
parts	\(\text {Expression too large to display}\)	\(1697\)

Input:

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

Output:

-2/d^3*b^2*(-2*b^2*c*f^2*(-1/4*sin(a+b/(d*x+c)^(1/2))/b^4*(d*x+c)^2-1/12*c 
os(a+b/(d*x+c)^(1/2))/b^3*(d*x+c)^(3/2)+1/24*sin(a+b/(d*x+c)^(1/2))/b^2*(d 
*x+c)+1/24*cos(a+b/(d*x+c)^(1/2))/b*(d*x+c)^(1/2)+1/24*Si(b/(d*x+c)^(1/2)) 
*cos(a)+1/24*Ci(b/(d*x+c)^(1/2))*sin(a))-2*c*d*e*f*(-1/2*sin(a+b/(d*x+c)^( 
1/2))/b^2*(d*x+c)-1/2*cos(a+b/(d*x+c)^(1/2))/b*(d*x+c)^(1/2)-1/2*Si(b/(d*x 
+c)^(1/2))*cos(a)-1/2*Ci(b/(d*x+c)^(1/2))*sin(a))+2*b^2*d*e*f*(-1/4*sin(a+ 
b/(d*x+c)^(1/2))/b^4*(d*x+c)^2-1/12*cos(a+b/(d*x+c)^(1/2))/b^3*(d*x+c)^(3/ 
2)+1/24*sin(a+b/(d*x+c)^(1/2))/b^2*(d*x+c)+1/24*cos(a+b/(d*x+c)^(1/2))/b*( 
d*x+c)^(1/2)+1/24*Si(b/(d*x+c)^(1/2))*cos(a)+1/24*Ci(b/(d*x+c)^(1/2))*sin( 
a))+c^2*f^2*(-1/2*sin(a+b/(d*x+c)^(1/2))/b^2*(d*x+c)-1/2*cos(a+b/(d*x+c)^( 
1/2))/b*(d*x+c)^(1/2)-1/2*Si(b/(d*x+c)^(1/2))*cos(a)-1/2*Ci(b/(d*x+c)^(1/2 
))*sin(a))+d^2*e^2*(-1/2*sin(a+b/(d*x+c)^(1/2))/b^2*(d*x+c)-1/2*cos(a+b/(d 
*x+c)^(1/2))/b*(d*x+c)^(1/2)-1/2*Si(b/(d*x+c)^(1/2))*cos(a)-1/2*Ci(b/(d*x+ 
c)^(1/2))*sin(a))+b^4*f^2*(-1/6*sin(a+b/(d*x+c)^(1/2))/b^6*(d*x+c)^3-1/30* 
cos(a+b/(d*x+c)^(1/2))/b^5*(d*x+c)^(5/2)+1/120*sin(a+b/(d*x+c)^(1/2))/b^4* 
(d*x+c)^2+1/360*cos(a+b/(d*x+c)^(1/2))/b^3*(d*x+c)^(3/2)-1/720*sin(a+b/(d* 
x+c)^(1/2))/b^2*(d*x+c)-1/720*cos(a+b/(d*x+c)^(1/2))/b*(d*x+c)^(1/2)-1/720 
*Si(b/(d*x+c)^(1/2))*cos(a)-1/720*Ci(b/(d*x+c)^(1/2))*sin(a)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.64 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {{\left (360 \, b^{2} d^{2} e^{2} - 60 \, {\left (b^{4} + 12 \, b^{2} c\right )} d e f + {\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \operatorname {Ci}\left (\frac {b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) + {\left (360 \, b^{2} d^{2} e^{2} - 60 \, {\left (b^{4} + 12 \, b^{2} c\right )} d e f + {\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{\sqrt {d x + c}}\right ) + {\left (24 \, b d^{2} f^{2} x^{2} + 360 \, b d^{2} e^{2} - 60 \, {\left (b^{3} + 10 \, b c\right )} d e f + {\left (b^{5} + 58 \, b^{3} c + 264 \, b c^{2}\right )} f^{2} + 2 \, {\left (60 \, b d^{2} e f - {\left (b^{3} + 36 \, b c\right )} d f^{2}\right )} x\right )} \sqrt {d x + c} \cos \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right ) + {\left (120 \, d^{3} f^{2} x^{3} + 360 \, c d^{2} e^{2} - 60 \, {\left (b^{2} c + 6 \, c^{2}\right )} d e f + {\left (b^{4} c + 54 \, b^{2} c^{2} + 120 \, c^{3}\right )} f^{2} - 6 \, {\left (b^{2} d^{2} f^{2} - 60 \, d^{3} e f\right )} x^{2} - {\left (60 \, b^{2} d^{2} e f - 360 \, d^{3} e^{2} - {\left (b^{4} + 48 \, b^{2} c\right )} d f^{2}\right )} x\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{360 \, d^{3}} \] Input:

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

Output:

1/360*((360*b^2*d^2*e^2 - 60*(b^4 + 12*b^2*c)*d*e*f + (b^6 + 60*b^4*c + 36 
0*b^2*c^2)*f^2)*cos_integral(b/sqrt(d*x + c))*sin(a) + (360*b^2*d^2*e^2 - 
60*(b^4 + 12*b^2*c)*d*e*f + (b^6 + 60*b^4*c + 360*b^2*c^2)*f^2)*cos(a)*sin 
_integral(b/sqrt(d*x + c)) + (24*b*d^2*f^2*x^2 + 360*b*d^2*e^2 - 60*(b^3 + 
 10*b*c)*d*e*f + (b^5 + 58*b^3*c + 264*b*c^2)*f^2 + 2*(60*b*d^2*e*f - (b^3 
 + 36*b*c)*d*f^2)*x)*sqrt(d*x + c)*cos((a*d*x + a*c + sqrt(d*x + c)*b)/(d* 
x + c)) + (120*d^3*f^2*x^3 + 360*c*d^2*e^2 - 60*(b^2*c + 6*c^2)*d*e*f + (b 
^4*c + 54*b^2*c^2 + 120*c^3)*f^2 - 6*(b^2*d^2*f^2 - 60*d^3*e*f)*x^2 - (60* 
b^2*d^2*e*f - 360*d^3*e^2 - (b^4 + 48*b^2*c)*d*f^2)*x)*sin((a*d*x + a*c + 
sqrt(d*x + c)*b)/(d*x + c)))/d^3
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.44 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/720*(360*(((-I*Ei(I*b/sqrt(d*x + c)) + I*Ei(-I*b/sqrt(d*x + c)))*cos(a) 
+ (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x + c)))*sin(a))*b^2 + 2*sqrt(d* 
x + c)*b*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + 2*(d*x + c)*sin((sqrt( 
d*x + c)*a + b)/sqrt(d*x + c)))*e^2 - 720*(((-I*Ei(I*b/sqrt(d*x + c)) + I* 
Ei(-I*b/sqrt(d*x + c)))*cos(a) + (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x 
 + c)))*sin(a))*b^2 + 2*sqrt(d*x + c)*b*cos((sqrt(d*x + c)*a + b)/sqrt(d*x 
 + c)) + 2*(d*x + c)*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*c*e*f/d + 3 
60*(((-I*Ei(I*b/sqrt(d*x + c)) + I*Ei(-I*b/sqrt(d*x + c)))*cos(a) + (Ei(I* 
b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x + c)))*sin(a))*b^2 + 2*sqrt(d*x + c)*b 
*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + 2*(d*x + c)*sin((sqrt(d*x + c) 
*a + b)/sqrt(d*x + c)))*c^2*f^2/d^2 + 60*(((I*Ei(I*b/sqrt(d*x + c)) - I*Ei 
(-I*b/sqrt(d*x + c)))*cos(a) - (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt(d*x + 
 c)))*sin(a))*b^4 - 2*(sqrt(d*x + c)*b^3 - 2*(d*x + c)^(3/2)*b)*cos((sqrt( 
d*x + c)*a + b)/sqrt(d*x + c)) - 2*((d*x + c)*b^2 - 6*(d*x + c)^2)*sin((sq 
rt(d*x + c)*a + b)/sqrt(d*x + c)))*e*f/d - 60*(((I*Ei(I*b/sqrt(d*x + c)) - 
 I*Ei(-I*b/sqrt(d*x + c)))*cos(a) - (Ei(I*b/sqrt(d*x + c)) + Ei(-I*b/sqrt( 
d*x + c)))*sin(a))*b^4 - 2*(sqrt(d*x + c)*b^3 - 2*(d*x + c)^(3/2)*b)*cos(( 
sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 2*((d*x + c)*b^2 - 6*(d*x + c)^2)*si 
n((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*c*f^2/d^2 + (((-I*Ei(I*b/sqrt(d*x 
+ c)) + I*Ei(-I*b/sqrt(d*x + c)))*cos(a) + (Ei(I*b/sqrt(d*x + c)) + Ei(...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6587 vs. \(2 (537) = 1074\).

Time = 3.41 (sec) , antiderivative size = 6587, normalized size of antiderivative = 10.78 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/360*(360*(a^2*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) 
*sin(a) - a^2*b^3*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + 
 c)) - 2*(sqrt(d*x + c)*a + b)*a*b^3*cos_integral(-a + (sqrt(d*x + c)*a + 
b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 2*(sqrt(d*x + c)*a + b)*a*b^3*cos 
(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + 
(sqrt(d*x + c)*a + b)^2*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d 
*x + c))*sin(a)/(d*x + c) - (sqrt(d*x + c)*a + b)^2*b^3*cos(a)*sin_integra 
l(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/(d*x + c) - a*b^3*cos((sqrt(d*x 
 + c)*a + b)/sqrt(d*x + c)) + (sqrt(d*x + c)*a + b)*b^3*cos((sqrt(d*x + c) 
*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + b^3*sin((sqrt(d*x + c)*a + b)/sqrt( 
d*x + c)))*e^2/((a^2 - 2*(sqrt(d*x + c)*a + b)*a/sqrt(d*x + c) + (sqrt(d*x 
 + c)*a + b)^2/(d*x + c))*b) - 60*(a^4*b^5*cos_integral(-a + (sqrt(d*x + c 
)*a + b)/sqrt(d*x + c))*sin(a) - a^4*b^5*cos(a)*sin_integral(a - (sqrt(d*x 
 + c)*a + b)/sqrt(d*x + c)) - 4*(sqrt(d*x + c)*a + b)*a^3*b^5*cos_integral 
(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 4*(sqrt( 
d*x + c)*a + b)*a^3*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt 
(d*x + c))/sqrt(d*x + c) + 6*(sqrt(d*x + c)*a + b)^2*a^2*b^5*cos_integral( 
-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c) + 12*a^4*b^3*c* 
cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - 6*(sqrt(d* 
x + c)*a + b)^2*a^2*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/s...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sin((sqrt(c + d*x)*a + b)/sqrt(c + d*x)),x)*e**2 + int(sin((sqrt(c + d 
*x)*a + b)/sqrt(c + d*x))*x**2,x)*f**2 + 2*int(sin((sqrt(c + d*x)*a + b)/s 
qrt(c + d*x))*x,x)*e*f