Integrand size = 19, antiderivative size = 285 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=-\frac {e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}+\frac {e^2 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}+\frac {(2 c d-b e)^2 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{4 c^{5/2}}+\frac {(2 c d-b e)^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{4 c^{5/2}}-\frac {e^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{2 c^{3/2}} \]
-1/4*e*(-b*e+2*c*d)*cos(c*x^2+b*x+a)/c^2-1/2*e*(e*x+d)*cos(c*x^2+b*x+a)/c+ 1/4*e^2*cos(a-1/4*b^2/c)*FresnelC(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))* 2^(1/2)*Pi^(1/2)/c^(3/2)+1/8*(-b*e+2*c*d)^2*cos(a-1/4*b^2/c)*FresnelS(1/2* (2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/c^(5/2)+1/8*(-b*e+2*c *d)^2*FresnelC(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*sin(a-1/4*b^2/c)*2^ (1/2)*Pi^(1/2)/c^(5/2)-1/4*e^2*FresnelS(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^( 1/2))*sin(a-1/4*b^2/c)*2^(1/2)*Pi^(1/2)/c^(3/2)
Time = 0.85 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.65 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {2 \sqrt {c} e (b e-2 c (2 d+e x)) \cos (a+x (b+c x))+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left ((-2 c d+b e)^2 \cos \left (a-\frac {b^2}{4 c}\right )-2 c e^2 \sin \left (a-\frac {b^2}{4 c}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left (2 c e^2 \cos \left (a-\frac {b^2}{4 c}\right )+(-2 c d+b e)^2 \sin \left (a-\frac {b^2}{4 c}\right )\right )}{8 c^{5/2}} \]
(2*Sqrt[c]*e*(b*e - 2*c*(2*d + e*x))*Cos[a + x*(b + c*x)] + Sqrt[2*Pi]*Fre snelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*((-2*c*d + b*e)^2*Cos[a - b^2/(4*c )] - 2*c*e^2*Sin[a - b^2/(4*c)]) + Sqrt[2*Pi]*FresnelC[(b + 2*c*x)/(Sqrt[c ]*Sqrt[2*Pi])]*(2*c*e^2*Cos[a - b^2/(4*c)] + (-2*c*d + b*e)^2*Sin[a - b^2/ (4*c)]))/(8*c^(5/2))
Time = 0.74 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3944, 3929, 3832, 3833, 3942, 3928, 3832, 3833}
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.
| \(\displaystyle \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 3944 |
| \(\displaystyle \frac {(2 c d-b e) \int (d+e x) \sin \left (c x^2+b x+a\right )dx}{2 c}+\frac {e^2 \int \cos \left (c x^2+b x+a\right )dx}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3929 |
| \(\displaystyle \frac {e^2 \left (\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\sin \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\right )}{2 c}+\frac {(2 c d-b e) \int (d+e x) \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {e^2 \left (\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}+\frac {(2 c d-b e) \int (d+e x) \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {(2 c d-b e) \int (d+e x) \sin \left (c x^2+b x+a\right )dx}{2 c}+\frac {e^2 \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3942 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \int \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {e^2 \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3928 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\cos \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\right )}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {e^2 \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {e^2 \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {(2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e \cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {e^2 \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}\) |
-1/2*(e*(d + e*x)*Cos[a + b*x + c*x^2])/c + (e^2*((Sqrt[Pi/2]*Cos[a - b^2/ (4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/Sqrt[c] - (Sqrt[Pi/2]*F resnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]))/(2 *c) + ((2*c*d - b*e)*(-1/2*(e*Cos[a + b*x + c*x^2])/c + ((2*c*d - b*e)*((S qrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/S qrt[c] + (Sqrt[Pi/2]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^ 2/(4*c)])/Sqrt[c]))/(2*c)))/(2*c)
3.1.29.3.1 Defintions of rubi rules used
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Cos[(b^2 - 4* a*c)/(4*c)] Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] - Simp[Sin[(b^2 - 4*a*c) /(4*c)] Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && Ne Q[b^2 - 4*a*c, 0]
Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Cos[(b^2 - 4* a*c)/(4*c)] Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] + Simp[Sin[(b^2 - 4*a*c) /(4*c)] Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && Ne Q[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(-e)*(Cos[a + b*x + c*x^2]/(2*c)), x] + Simp[(2*c*d - b*e)/(2*c) Int[Sin[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym bol] :> Simp[(-e)*(d + e*x)^(m - 1)*(Cos[a + b*x + c*x^2]/(2*c)), x] + (-Si mp[(b*e - 2*c*d)/(2*c) Int[(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2], x], x] + Simp[e^2*((m - 1)/(2*c)) Int[(d + e*x)^(m - 2)*Cos[a + b*x + c*x^2], x ], x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
Time = 1.51 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(-\frac {e^{2} x \cos \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {e^{2} b \left (-\frac {\cos \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}\right )}{2 c}+\frac {e^{2} \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}-\frac {d e \cos \left (c \,x^{2}+b x +a \right )}{c}-\frac {d e b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 c^{\frac {3}{2}}}+\frac {\sqrt {2}\, \sqrt {\pi }\, d^{2} \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 \sqrt {c}}\) | \(399\) |
| risch | \(\frac {i \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, d^{2} {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {-i c}}+\frac {i e^{2} \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, b^{2} {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{16 \sqrt {-i c}\, c^{2}}-\frac {e^{2} \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{8 \sqrt {-i c}\, c}-\frac {i d e \,\operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {-i c}\, c}+\frac {i \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, d^{2} {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {i c}}+\frac {i e^{2} \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, b^{2} {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{16 \sqrt {i c}\, c^{2}}+\frac {e^{2} \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{8 \sqrt {i c}\, c}-\frac {i d e \,\operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right ) \sqrt {\pi }\, b \,{\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}}}{4 \sqrt {i c}\, c}+2 \left (\frac {i e^{2} \left (\frac {i x}{2 c}-\frac {i b}{4 c^{2}}\right )}{2}-\frac {d e}{2 c}\right ) \cos \left (c \,x^{2}+b x +a \right )\) | \(486\) |
| parts | \(\text {Expression too large to display}\) | \(875\) |
-1/2*e^2/c*x*cos(c*x^2+b*x+a)-1/2*e^2*b/c*(-1/2*cos(c*x^2+b*x+a)/c-1/4*b/c ^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-a*c)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^ (1/2)*(c*x+1/2*b))-sin((1/4*b^2-a*c)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)* (c*x+1/2*b))))+1/4*e^2/c^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-a*c)/c)*Fres nelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))+sin((1/4*b^2-a*c)/c)*FresnelS(2 ^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))-d*e/c*cos(c*x^2+b*x+a)-1/2*d*e*b/c^( 3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-a*c)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^(1 /2)*(c*x+1/2*b))-sin((1/4*b^2-a*c)/c)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c *x+1/2*b)))+1/2*2^(1/2)*Pi^(1/2)/c^(1/2)*d^2*(cos((1/4*b^2-a*c)/c)*Fresnel S(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))-sin((1/4*b^2-a*c)/c)*FresnelC(2^(1 /2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))
Time = 0.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.81 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} {\left (2 \, \pi c e^{2} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \pi {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} {\left (2 \, \pi c e^{2} \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - \pi {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - 2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \cos \left (c x^{2} + b x + a\right )}{8 \, c^{3}} \]
1/8*(sqrt(2)*(2*pi*c*e^2*cos(-1/4*(b^2 - 4*a*c)/c) + pi*(4*c^2*d^2 - 4*b*c *d*e + b^2*e^2)*sin(-1/4*(b^2 - 4*a*c)/c))*sqrt(c/pi)*fresnel_cos(1/2*sqrt (2)*(2*c*x + b)*sqrt(c/pi)/c) - sqrt(2)*(2*pi*c*e^2*sin(-1/4*(b^2 - 4*a*c) /c) - pi*(4*c^2*d^2 - 4*b*c*d*e + b^2*e^2)*cos(-1/4*(b^2 - 4*a*c)/c))*sqrt (c/pi)*fresnel_sin(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c) - 2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cos(c*x^2 + b*x + a))/c^3
\[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right )^{2} \sin {\left (a + b x + c x^{2} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 2269, normalized size of antiderivative = 7.96 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]
-1/8*sqrt(2)*sqrt(pi)*((-(I + 1)*cos(-1/4*(b^2 - 4*a*c)/c) + (I - 1)*sin(- 1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(I*c)) + (-(I - 1)*cos(- 1/4*(b^2 - 4*a*c)/c) + (I + 1)*sin(-1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(-I*c)))*d^2/sqrt(c) + 1/8*((-(I + 1)*sqrt(2)*sqrt(pi)*(erf(1/ 2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + (I - 1)*sqrt(2)*sqrt(p i)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*cos(-1/4 *(b^2 - 4*a*c)/c) + ((I - 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - (I + 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4* I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^2*sin(-1/4*(b^2 - 4*a*c)/c) - 2 *(((I + 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2 )/c)) - 1) - (I - 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b* c*x + I*b^2)/c)) - 1))*b*c*cos(-1/4*(b^2 - 4*a*c)/c) + (-(I - 1)*sqrt(2)*s qrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + (I + 1) *sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1 ))*b*c*sin(-1/4*(b^2 - 4*a*c)/c))*x - 4*(c*(e^(1/4*(4*I*c^2*x^2 + 4*I*b*c* x + I*b^2)/c) + e^(-1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*cos(-1/4*(b^ 2 - 4*a*c)/c) + c*(I*e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - I*e^(-1 /4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*sin(-1/4*(b^2 - 4*a*c)/c))*sqrt(( 4*c^2*x^2 + 4*b*c*x + b^2)/c))*d*e/(c^2*sqrt((4*c^2*x^2 + 4*b*c*x + b^2)/c )) + 1/32*(8*((((I + 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4...
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.02 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=-\frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (-4 i \, c^{2} d^{2} + 4 i \, b c d e - i \, b^{2} e^{2} + 2 \, c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{16 \, c^{2}} - \frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (4 i \, c^{2} d^{2} - 4 i \, b c d e + i \, b^{2} e^{2} + 2 \, c e^{2}\right )} \operatorname {erf}\left (\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{16 \, c^{2}} \]
-1/16*(-I*sqrt(2)*sqrt(pi)*(-4*I*c^2*d^2 + 4*I*b*c*d*e - I*b^2*e^2 + 2*c*e ^2)*erf(-1/4*I*sqrt(2)*(2*x + b/c)*(I*c/abs(c) + 1)*sqrt(abs(c)))*e^(-1/4* (I*b^2 - 4*I*a*c)/c)/((I*c/abs(c) + 1)*sqrt(abs(c))) + 2*(c*e^2*(2*x + b/c ) + 4*c*d*e - 2*b*e^2)*e^(I*c*x^2 + I*b*x + I*a))/c^2 - 1/16*(I*sqrt(2)*sq rt(pi)*(4*I*c^2*d^2 - 4*I*b*c*d*e + I*b^2*e^2 + 2*c*e^2)*erf(1/4*I*sqrt(2) *(2*x + b/c)*(-I*c/abs(c) + 1)*sqrt(abs(c)))*e^(-1/4*(-I*b^2 + 4*I*a*c)/c) /((-I*c/abs(c) + 1)*sqrt(abs(c))) + 2*(c*e^2*(2*x + b/c) + 4*c*d*e - 2*b*e ^2)*e^(-I*c*x^2 - I*b*x - I*a))/c^2
Timed out. \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\int \sin \left (c\,x^2+b\,x+a\right )\,{\left (d+e\,x\right )}^2 \,d x \]