Integrand size = 22, antiderivative size = 611 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {b^5 f^2 \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{360 d^3}-\frac {b^3 f (d e-c f) \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^3}+\frac {b (d e-c f)^2 \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^3}-\frac {b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{180 d^3}+\frac {b f (d e-c f) (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{3 d^3}+\frac {b f^2 (c+d x)^{5/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{15 d^3}+\frac {b^6 f^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{360 d^3}-\frac {b^4 f (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{6 d^3}+\frac {b^2 (d e-c f)^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{d^3}+\frac {b^4 f^2 (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{360 d^3}-\frac {b^2 f (d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^3}+\frac {(d e-c f)^2 (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^3}-\frac {b^2 f^2 (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{60 d^3}+\frac {f (d e-c f) (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^3}+\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{3 d^3}+\frac {b^6 f^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{360 d^3}-\frac {b^4 f (d e-c f) \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{6 d^3}+\frac {b^2 (d e-c f)^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d^3} \]
-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*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+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^5*f^2*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/d^3-1/6 *b^3*f*(-c*f+d*e)*cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/d^3+b*(-c*f+d*e)^2* cos(a+b/(d*x+c)^(1/2))*(d*x+c)^(1/2)/d^3
Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 557, normalized size of antiderivative = 0.91 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {i e^{-i a} \left (e^{-\frac {i b}{\sqrt {c+d x}}} \sqrt {c+d x} \left (-i b^5 f^2+b^4 f^2 \sqrt {c+d x}+2 i b^3 f (30 d e-29 c f+d f x)-6 b^2 f \sqrt {c+d x} (10 d e-9 c f+d f x)+120 \sqrt {c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )-24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )\right )-e^{i \left (2 a+\frac {b}{\sqrt {c+d x}}\right )} \sqrt {c+d x} \left (i b^5 f^2+b^4 f^2 \sqrt {c+d x}-2 i b^3 f (30 d e-29 c f+d f x)-6 b^2 f \sqrt {c+d x} (10 d e-9 c f+d f x)+120 \sqrt {c+d x} \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )+24 i b \left (11 c^2 f^2-c d f (25 e+3 f x)+d^2 \left (15 e^2+5 e f x+f^2 x^2\right )\right )\right )+b^2 \left (360 d^2 e^2-60 \left (b^2+12 c\right ) d e f+\left (b^4+60 b^2 c+360 c^2\right ) f^2\right ) \operatorname {ExpIntegralEi}\left (-\frac {i b}{\sqrt {c+d x}}\right )-b^2 e^{2 i a} \left (360 d^2 e^2-60 \left (b^2+12 c\right ) d e f+\left (b^4+60 b^2 c+360 c^2\right ) f^2\right ) \operatorname {ExpIntegralEi}\left (\frac {i b}{\sqrt {c+d x}}\right )\right )}{720 d^3} \]
((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))
Time = 0.88 (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 |
\(\displaystyle -\frac {2 \left (-\frac {b^6 f^2 \sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}-\frac {b^6 f^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}-\frac {b^5 f^2 \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}+\frac {b^4 f \sin (a) (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^4 f \cos (a) (d e-c f) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}-\frac {b^4 f^2 (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{720 d^2}+\frac {b^3 f \sqrt {c+d x} (d e-c f) \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^3 f^2 (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{360 d^2}-\frac {b^2 \sin (a) (d e-c f)^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {b^2 \cos (a) (d e-c f)^2 \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}+\frac {b^2 f (c+d x) (d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^2 f^2 (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{120 d^2}-\frac {f (c+d x)^2 (d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {(c+d x) (d e-c f)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {b f (c+d x)^{3/2} (d e-c f) \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^2}-\frac {b \sqrt {c+d x} (d e-c f)^2 \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {f^2 (c+d x)^3 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^2}-\frac {b f^2 (c+d x)^{5/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{30 d^2}\right )}{d}\) |
(-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
3.2.97.3.1 Defintions of rubi rules used
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]
Time = 1.62 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(-\frac {2 b^{2} \left (-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} 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 )+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 )+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 )+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 )\right )}{d^{3}}\) | \(696\) |
default | \(-\frac {2 b^{2} \left (-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} 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 )+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 )+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 )+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 )\right )}{d^{3}}\) | \(696\) |
parts | \(\text {Expression too large to display}\) | \(1697\) |
-2/d^3*b^2*(-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*c*f^2*(-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))+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*C i(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))+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)+1/2 4*Si(b/(d*x+c)^(1/2))*cos(a)+1/24*Ci(b/(d*x+c)^(1/2))*sin(a)))
Time = 0.31 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.64 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {{\left (360 \, b^{2} d^{2} e^{2} - 60 \, {\left (b^{4} + 12 \, b^{2} c\right )} d e f + {\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \operatorname {Ci}\left (\frac {b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) + {\left (360 \, b^{2} d^{2} e^{2} - 60 \, {\left (b^{4} + 12 \, b^{2} c\right )} d e f + {\left (b^{6} + 60 \, b^{4} c + 360 \, b^{2} c^{2}\right )} f^{2}\right )} \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{\sqrt {d x + c}}\right ) + {\left (24 \, b d^{2} f^{2} x^{2} + 360 \, b d^{2} e^{2} - 60 \, {\left (b^{3} + 10 \, b c\right )} d e f + {\left (b^{5} + 58 \, b^{3} c + 264 \, b c^{2}\right )} f^{2} + 2 \, {\left (60 \, b d^{2} e f - {\left (b^{3} + 36 \, b c\right )} d f^{2}\right )} x\right )} \sqrt {d x + c} \cos \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right ) + {\left (120 \, d^{3} f^{2} x^{3} + 360 \, c d^{2} e^{2} - 60 \, {\left (b^{2} c + 6 \, c^{2}\right )} d e f + {\left (b^{4} c + 54 \, b^{2} c^{2} + 120 \, c^{3}\right )} f^{2} - 6 \, {\left (b^{2} d^{2} f^{2} - 60 \, d^{3} e f\right )} x^{2} - {\left (60 \, b^{2} d^{2} e f - 360 \, d^{3} e^{2} - {\left (b^{4} + 48 \, b^{2} c\right )} d f^{2}\right )} x\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{360 \, d^{3}} \]
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
\[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\int \left (e + f x\right )^{2} \sin {\left (a + \frac {b}{\sqrt {c + d x}} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 877, normalized size of antiderivative = 1.44 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\text {Too large to display} \]
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(...
Leaf count of result is larger than twice the leaf count of optimal. 6606 vs. \(2 (537) = 1074\).
Time = 0.64 (sec) , antiderivative size = 6606, normalized size of antiderivative = 10.81 \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\text {Too large to display} \]
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) + (a^6*b^7*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - a^6*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 6*(sqrt(d*x + c)*a + b)*a^5*b^7*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 6*(sqrt(d*x + c)*a + b)*a^5*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d* x + c))/sqrt(d*x + c) + 15*(sqrt(d*x + c)*a + b)^2*a^4*b^7*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c) + 60*a^6*b^5*c*co s_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - 15*(sqrt(d*x + c)*a + b)^2*a^4*b^7*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sq...
Timed out. \[ \int (e+f x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\int \sin \left (a+\frac {b}{\sqrt {c+d\,x}}\right )\,{\left (e+f\,x\right )}^2 \,d x \]