\(\int (d+e x)^2 \sin (a+b x+c x^2) \, dx\) [29]

Optimal result

Integrand size = 19, antiderivative size = 285 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=-\frac {e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}+\frac {e^2 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}+\frac {(2 c d-b e)^2 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{4 c^{5/2}}+\frac {(2 c d-b e)^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{4 c^{5/2}}-\frac {e^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{2 c^{3/2}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3544, 3542, 3528, 3432, 3433, 3529} \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) (2 c d-b e)^2 \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{4 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) (2 c d-b e)^2 \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{4 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} e^2 \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} e^2 \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}-\frac {e (2 c d-b e) \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c} \]

[In]

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

[Out]

-1/4*(e*(2*c*d - b*e)*Cos[a + b*x + c*x^2])/c^2 - (e*(d + e*x)*Cos[a + b*x + c*x^2])/(2*c) + (e^2*Sqrt[Pi/2]*C
os[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(2*c^(3/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*Cos[a
- b^2/(4*c)]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])])/(4*c^(5/2)) + ((2*c*d - b*e)^2*Sqrt[Pi/2]*FresnelC[(b
+ 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(4*c^(5/2)) - (e^2*Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c
]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/(2*c^(3/2))

Rule 3432

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]

Rule 3433

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]

Rule 3528

Int[Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[Cos[(b^2 - 4*a*c)/(4*c)], Int[Sin[(b + 2*c*x)^2/
(4*c)], x], x] - Dist[Sin[(b^2 - 4*a*c)/(4*c)], Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] &&
NeQ[b^2 - 4*a*c, 0]

Rule 3529

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[Cos[(b^2 - 4*a*c)/(4*c)], Int[Cos[(b + 2*c*x)^2/
(4*c)], x], x] + Dist[Sin[(b^2 - 4*a*c)/(4*c)], Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] &&
NeQ[b^2 - 4*a*c, 0]

Rule 3542

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] + Dist[(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]

Rule 3544

Int[((d_.) + (e_.)*(x_))^(m_)*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(-e)*(d + e*x)^(m - 1)
*(Cos[a + b*x + c*x^2]/(2*c)), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2], x]
, x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {e (d+e x) \cos \left (a+b x+c x^2\right )}{2 c}+\frac {e^2 \int \cos \left (a+b x+c x^2\right ) \, dx}{2 c}-\frac {(-2 c d+b e) \int (d+e x) \sin \left (a+b x+c x^2\right ) \, dx}{2 c} \\ & = -\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 {(2 c d-b e)^2 \int \sin \left (a+b x+c x^2\right ) \, dx}{4 c^2}+\frac {\left (e^2 \cos \left (a-\frac {b^2}{4 c}\right )\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right ) \, dx}{2 c}-\frac {\left (e^2 \sin \left (a-\frac {b^2}{4 c}\right )\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right ) \, dx}{2 c} \\ & = -\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 {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}}+\frac {\left ((2 c d-b e)^2 \cos \left (a-\frac {b^2}{4 c}\right )\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right ) \, dx}{4 c^2}+\frac {\left ((2 c d-b e)^2 \sin \left (a-\frac {b^2}{4 c}\right )\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right ) \, dx}{4 c^2} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.65 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {2 \sqrt {c} e (b e-2 c (2 d+e x)) \cos (a+x (b+c x))+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left ((-2 c d+b e)^2 \cos \left (a-\frac {b^2}{4 c}\right )-2 c e^2 \sin \left (a-\frac {b^2}{4 c}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left (2 c e^2 \cos \left (a-\frac {b^2}{4 c}\right )+(-2 c d+b e)^2 \sin \left (a-\frac {b^2}{4 c}\right )\right )}{8 c^{5/2}} \]

[In]

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

[Out]

(2*Sqrt[c]*e*(b*e - 2*c*(2*d + e*x))*Cos[a + x*(b + c*x)] + Sqrt[2*Pi]*FresnelS[(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))

Maple [A] (verified)

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	\(\text {Expression too large to display}\)	\(875\)

[In]

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

[Out]

-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)*FresnelC(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)*F
resnelS(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)))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.81 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} {\left (2 \, \pi c e^{2} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \pi {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} {\left (2 \, \pi c e^{2} \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - \pi {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - 2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \cos \left (c x^{2} + b x + a\right )}{8 \, c^{3}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.24 (sec) , antiderivative size = 2269, normalized size of antiderivative = 7.96 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-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(pi)*(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)*s
qrt(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)*s
qrt(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*s
qrt(-(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*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*c^3 + 4*((I - 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4
*I*b*c*x + I*b^2)/c) - (I + 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*c^4)*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*c^3 + 4*((I + 1)*sqrt(2)*gamm
a(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - (I - 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I
*b^2)/c))*c^4)*sin(-1/4*(b^2 - 4*a*c)/c))*x^3 + 12*((((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^3*c^2 + 4*((I - 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - (I + 1)*sqrt(2)*gamma(3/2
, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*b*c^3)*cos(-1/4*(b^2 - 4*a*c)/c) + ((-(I - 1)*sqrt(2)*sqrt(pi)*(e
rf(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^3*c^2 + 4*((I + 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c
) - (I - 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*b*c^3)*sin(-1/4*(b^2 - 4*a*c)/c))*x^
2 + 8*(b*c^2*(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) + b*c^2*(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))*((4*c^2*x^2 + 4*b*c*x + b^2)/c)^(3/2) + 6*((((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^4*c + 4*((I - 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^
2)/c) - (I + 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*b^2*c^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)*sq
rt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b^4*c + 4*((I + 1)*sqrt(2)*gamma(3/2, 1/4*(4
*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - (I - 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*b^2
*c^2)*sin(-1/4*(b^2 - 4*a*c)/c))*x - ((-(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^5 - 4*((I
- 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - (I + 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^
2 + 4*I*b*c*x + I*b^2)/c))*b^3*c)*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^5 - 4*((I + 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - (I - 1)*sqrt(2)*gam
ma(3/2, -1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c))*b^3*c)*sin(-1/4*(b^2 - 4*a*c)/c))*e^2/(c^4*((4*c^2*x^2 + 4*
b*c*x + b^2)/c)^(3/2))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.02 \[ \int (d+e x)^2 \sin \left (a+b x+c x^2\right ) \, dx=-\frac {-\frac {i \, \sqrt {2} \sqrt {\pi } {\left (-4 i \, c^{2} d^{2} + 4 i \, b c d e - i \, b^{2} e^{2} + 2 \, c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{16 \, c^{2}} - \frac {\frac {i \, \sqrt {2} \sqrt {\pi } {\left (4 i \, c^{2} d^{2} - 4 i \, b c d e + i \, b^{2} e^{2} + 2 \, c e^{2}\right )} \operatorname {erf}\left (\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{16 \, c^{2}} \]

[In]

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

[Out]

-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)*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

Mupad [F(-1)]

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

[In]

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

[Out]

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