\(\int \frac {(e+f x)^2 (A+B x+C x^2)}{\sqrt {1-d x} \sqrt {1+d x}} \, dx\) [9]

Optimal result

Integrand size = 37, antiderivative size = 228 \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {(C e-4 B f) (e+f x)^2 \sqrt {1-d^2 x^2}}{12 d^2 f}-\frac {C (e+f x)^3 \sqrt {1-d^2 x^2}}{4 d^2 f}+\frac {\left (4 \left (C \left (d^2 e^3-8 e f^2\right )-4 f \left (3 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-f \left (3 \left (3 C+4 A d^2\right ) f^2-2 d^2 e (C e-4 B f)\right ) x\right ) \sqrt {1-d^2 x^2}}{24 d^4 f}+\frac {\left (C \left (4 d^2 e^2+3 f^2\right )+4 d^2 \left (2 B e f+A \left (2 d^2 e^2+f^2\right )\right )\right ) \arcsin (d x)}{8 d^5} \]

[Out]

1/8*(C*(4*d^2*e^2+3*f^2)+4*d^2*(2*B*e*f+A*(2*d^2*e^2+f^2)))*arcsin(d*x)/d^5+1/12*(-4*B*f+C*e)*(f*x+e)^2*(-d^2*
x^2+1)^(1/2)/d^2/f-1/4*C*(f*x+e)^3*(-d^2*x^2+1)^(1/2)/d^2/f+1/24*(4*C*(d^2*e^3-8*e*f^2)-16*f*(3*A*d^2*e*f+B*(d
^2*e^2+f^2))-f*(3*(4*A*d^2+3*C)*f^2-2*d^2*e*(-4*B*f+C*e))*x)*(-d^2*x^2+1)^(1/2)/d^4/f

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1623, 1668, 847, 794, 222} \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {\arcsin (d x) \left (4 d^2 \left (A \left (2 d^2 e^2+f^2\right )+2 B e f\right )+C \left (4 d^2 e^2+3 f^2\right )\right )}{8 d^5}+\frac {\sqrt {1-d^2 x^2} \left (4 \left (C \left (d^2 e^3-8 e f^2\right )-4 f \left (3 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-f x \left (3 f^2 \left (4 A d^2+3 C\right )-2 d^2 e (C e-4 B f)\right )\right )}{24 d^4 f}+\frac {\sqrt {1-d^2 x^2} (e+f x)^2 (C e-4 B f)}{12 d^2 f}-\frac {C \sqrt {1-d^2 x^2} (e+f x)^3}{4 d^2 f} \]

[In]

Int[((e + f*x)^2*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

((C*e - 4*B*f)*(e + f*x)^2*Sqrt[1 - d^2*x^2])/(12*d^2*f) - (C*(e + f*x)^3*Sqrt[1 - d^2*x^2])/(4*d^2*f) + ((4*(
C*(d^2*e^3 - 8*e*f^2) - 4*f*(3*A*d^2*e*f + B*(d^2*e^2 + f^2))) - f*(3*(3*C + 4*A*d^2)*f^2 - 2*d^2*e*(C*e - 4*B
*f))*x)*Sqrt[1 - d^2*x^2])/(24*d^4*f) + ((C*(4*d^2*e^2 + 3*f^2) + 4*d^2*(2*B*e*f + A*(2*d^2*e^2 + f^2)))*ArcSi
n[d*x])/(8*d^5)

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^
m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 1623

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[P
x*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d,
0] && EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d^2 x^2}} \, dx \\ & = -\frac {C (e+f x)^3 \sqrt {1-d^2 x^2}}{4 d^2 f}-\frac {\int \frac {(e+f x)^2 \left (-\left (\left (3 C+4 A d^2\right ) f^2\right )+d^2 f (C e-4 B f) x\right )}{\sqrt {1-d^2 x^2}} \, dx}{4 d^2 f^2} \\ & = \frac {(C e-4 B f) (e+f x)^2 \sqrt {1-d^2 x^2}}{12 d^2 f}-\frac {C (e+f x)^3 \sqrt {1-d^2 x^2}}{4 d^2 f}+\frac {\int \frac {(e+f x) \left (d^2 f^2 \left (7 C e+12 A d^2 e+8 B f\right )+d^2 f \left (3 \left (3 C+4 A d^2\right ) f^2-2 d^2 e (C e-4 B f)\right ) x\right )}{\sqrt {1-d^2 x^2}} \, dx}{12 d^4 f^2} \\ & = \frac {(C e-4 B f) (e+f x)^2 \sqrt {1-d^2 x^2}}{12 d^2 f}-\frac {C (e+f x)^3 \sqrt {1-d^2 x^2}}{4 d^2 f}+\frac {\left (4 \left (C \left (d^2 e^3-8 e f^2\right )-4 f \left (3 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-f \left (3 \left (3 C+4 A d^2\right ) f^2-2 d^2 e (C e-4 B f)\right ) x\right ) \sqrt {1-d^2 x^2}}{24 d^4 f}+\frac {\left (C \left (4 d^2 e^2+3 f^2\right )+4 d^2 \left (2 B e f+A \left (2 d^2 e^2+f^2\right )\right )\right ) \int \frac {1}{\sqrt {1-d^2 x^2}} \, dx}{8 d^4} \\ & = \frac {(C e-4 B f) (e+f x)^2 \sqrt {1-d^2 x^2}}{12 d^2 f}-\frac {C (e+f x)^3 \sqrt {1-d^2 x^2}}{4 d^2 f}+\frac {\left (4 \left (C \left (d^2 e^3-8 e f^2\right )-4 f \left (3 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-f \left (3 \left (3 C+4 A d^2\right ) f^2-2 d^2 e (C e-4 B f)\right ) x\right ) \sqrt {1-d^2 x^2}}{24 d^4 f}+\frac {\left (C \left (4 d^2 e^2+3 f^2\right )+4 d^2 \left (2 B e f+A \left (2 d^2 e^2+f^2\right )\right )\right ) \sin ^{-1}(d x)}{8 d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.78 \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\frac {-d \sqrt {1-d^2 x^2} \left (12 A d^2 f (4 e+f x)+C \left (12 d^2 e^2 x+16 e f \left (2+d^2 x^2\right )+3 f^2 x \left (3+2 d^2 x^2\right )\right )+8 B \left (2 f^2+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right )\right )+6 \left (C \left (4 d^2 e^2+3 f^2\right )+4 d^2 \left (2 B e f+A \left (2 d^2 e^2+f^2\right )\right )\right ) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{24 d^5} \]

[In]

Integrate[((e + f*x)^2*(A + B*x + C*x^2))/(Sqrt[1 - d*x]*Sqrt[1 + d*x]),x]

[Out]

(-(d*Sqrt[1 - d^2*x^2]*(12*A*d^2*f*(4*e + f*x) + C*(12*d^2*e^2*x + 16*e*f*(2 + d^2*x^2) + 3*f^2*x*(3 + 2*d^2*x
^2)) + 8*B*(2*f^2 + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)))) + 6*(C*(4*d^2*e^2 + 3*f^2) + 4*d^2*(2*B*e*f + A*(2*d^2*
e^2 + f^2)))*ArcTan[(d*x)/(-1 + Sqrt[1 - d^2*x^2])])/(24*d^5)

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {\left (6 C \,d^{2} f^{2} x^{3}+8 B \,d^{2} f^{2} x^{2}+16 C \,d^{2} e f \,x^{2}+12 A \,d^{2} f^{2} x +24 B \,d^{2} e f x +12 C \,d^{2} e^{2} x +48 A \,d^{2} e f +24 B \,d^{2} e^{2}+9 C \,f^{2} x +16 B \,f^{2}+32 C e f \right ) \sqrt {d x +1}\, \left (d x -1\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{24 d^{4} \sqrt {-\left (d x +1\right ) \left (d x -1\right )}\, \sqrt {-d x +1}}+\frac {\left (8 A \,d^{4} e^{2}+4 A \,d^{2} f^{2}+8 B \,d^{2} e f +4 C \,d^{2} e^{2}+3 C \,f^{2}\right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{8 d^{4} \sqrt {d^{2}}\, \sqrt {-d x +1}\, \sqrt {d x +1}}\)	\(256\)
default	\(-\frac {\sqrt {-d x +1}\, \sqrt {d x +1}\, \left (6 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} f^{2} x^{3}+8 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} f^{2} x^{2}+16 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e f \,x^{2}+12 A \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} f^{2} x +24 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e f x +12 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e^{2} x +48 A \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e f -24 A \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{4} e^{2}+24 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} e^{2}+9 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d \,f^{2} x -12 A \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} f^{2}+16 B \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d \,f^{2}-24 B \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} e f +32 C \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d e f -12 C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d^{2} e^{2}-9 C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) f^{2}\right ) \operatorname {csgn}\left (d \right )}{24 d^{5} \sqrt {-d^{2} x^{2}+1}}\)	\(423\)

[In]

int((f*x+e)^2*(C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/24*(6*C*d^2*f^2*x^3+8*B*d^2*f^2*x^2+16*C*d^2*e*f*x^2+12*A*d^2*f^2*x+24*B*d^2*e*f*x+12*C*d^2*e^2*x+48*A*d^2*e
*f+24*B*d^2*e^2+9*C*f^2*x+16*B*f^2+32*C*e*f)*(d*x+1)^(1/2)*(d*x-1)/d^4/(-(d*x+1)*(d*x-1))^(1/2)*((-d*x+1)*(d*x
+1))^(1/2)/(-d*x+1)^(1/2)+1/8*(8*A*d^4*e^2+4*A*d^2*f^2+8*B*d^2*e*f+4*C*d^2*e^2+3*C*f^2)/d^4/(d^2)^(1/2)*arctan
((d^2)^(1/2)*x/(-d^2*x^2+1)^(1/2))*((-d*x+1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.84 \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {{\left (6 \, C d^{3} f^{2} x^{3} + 24 \, B d^{3} e^{2} + 16 \, B d f^{2} + 16 \, {\left (3 \, A d^{3} + 2 \, C d\right )} e f + 8 \, {\left (2 \, C d^{3} e f + B d^{3} f^{2}\right )} x^{2} + 3 \, {\left (4 \, C d^{3} e^{2} + 8 \, B d^{3} e f + {\left (4 \, A d^{3} + 3 \, C d\right )} f^{2}\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 6 \, {\left (8 \, B d^{2} e f + 4 \, {\left (2 \, A d^{4} + C d^{2}\right )} e^{2} + {\left (4 \, A d^{2} + 3 \, C\right )} f^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{24 \, d^{5}} \]

[In]

integrate((f*x+e)^2*(C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fricas")

[Out]

-1/24*((6*C*d^3*f^2*x^3 + 24*B*d^3*e^2 + 16*B*d*f^2 + 16*(3*A*d^3 + 2*C*d)*e*f + 8*(2*C*d^3*e*f + B*d^3*f^2)*x
^2 + 3*(4*C*d^3*e^2 + 8*B*d^3*e*f + (4*A*d^3 + 3*C*d)*f^2)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*(8*B*d^2*e*f +
4*(2*A*d^4 + C*d^2)*e^2 + (4*A*d^2 + 3*C)*f^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/d^5

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**2*(C*x**2+B*x+A)/(-d*x+1)**(1/2)/(d*x+1)**(1/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.01 \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {\sqrt {-d^{2} x^{2} + 1} C f^{2} x^{3}}{4 \, d^{2}} + \frac {A e^{2} \arcsin \left (d x\right )}{d} - \frac {\sqrt {-d^{2} x^{2} + 1} B e^{2}}{d^{2}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} A e f}{d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} {\left (2 \, C e f + B f^{2}\right )} x^{2}}{3 \, d^{2}} - \frac {\sqrt {-d^{2} x^{2} + 1} {\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{2 \, d^{2}} - \frac {3 \, \sqrt {-d^{2} x^{2} + 1} C f^{2} x}{8 \, d^{4}} + \frac {{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} \arcsin \left (d x\right )}{2 \, d^{3}} + \frac {3 \, C f^{2} \arcsin \left (d x\right )}{8 \, d^{5}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 1} {\left (2 \, C e f + B f^{2}\right )}}{3 \, d^{4}} \]

[In]

integrate((f*x+e)^2*(C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(-d^2*x^2 + 1)*C*f^2*x^3/d^2 + A*e^2*arcsin(d*x)/d - sqrt(-d^2*x^2 + 1)*B*e^2/d^2 - 2*sqrt(-d^2*x^2 +
1)*A*e*f/d^2 - 1/3*sqrt(-d^2*x^2 + 1)*(2*C*e*f + B*f^2)*x^2/d^2 - 1/2*sqrt(-d^2*x^2 + 1)*(C*e^2 + 2*B*e*f + A
*f^2)*x/d^2 - 3/8*sqrt(-d^2*x^2 + 1)*C*f^2*x/d^4 + 1/2*(C*e^2 + 2*B*e*f + A*f^2)*arcsin(d*x)/d^3 + 3/8*C*f^2*a
rcsin(d*x)/d^5 - 2/3*sqrt(-d^2*x^2 + 1)*(2*C*e*f + B*f^2)/d^4

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.02 \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=-\frac {{\left (24 \, B d^{3} e^{2} + 48 \, A d^{3} e f - 12 \, C d^{2} e^{2} - 24 \, B d^{2} e f - 12 \, A d^{2} f^{2} + 48 \, C d e f + 24 \, B d f^{2} - 15 \, C f^{2} + {\left (12 \, C d^{2} e^{2} + 24 \, B d^{2} e f + 12 \, A d^{2} f^{2} - 32 \, C d e f - 16 \, B d f^{2} + 27 \, C f^{2} + 2 \, {\left (8 \, C d e f + 3 \, {\left (d x + 1\right )} C f^{2} + 4 \, B d f^{2} - 9 \, C f^{2}\right )} {\left (d x + 1\right )}\right )} {\left (d x + 1\right )}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - 6 \, {\left (8 \, A d^{4} e^{2} + 4 \, C d^{2} e^{2} + 8 \, B d^{2} e f + 4 \, A d^{2} f^{2} + 3 \, C f^{2}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{24 \, d^{5}} \]

[In]

integrate((f*x+e)^2*(C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="giac")

[Out]

-1/24*((24*B*d^3*e^2 + 48*A*d^3*e*f - 12*C*d^2*e^2 - 24*B*d^2*e*f - 12*A*d^2*f^2 + 48*C*d*e*f + 24*B*d*f^2 - 1
5*C*f^2 + (12*C*d^2*e^2 + 24*B*d^2*e*f + 12*A*d^2*f^2 - 32*C*d*e*f - 16*B*d*f^2 + 27*C*f^2 + 2*(8*C*d*e*f + 3*
(d*x + 1)*C*f^2 + 4*B*d*f^2 - 9*C*f^2)*(d*x + 1))*(d*x + 1))*sqrt(d*x + 1)*sqrt(-d*x + 1) - 6*(8*A*d^4*e^2 + 4
*C*d^2*e^2 + 8*B*d^2*e*f + 4*A*d^2*f^2 + 3*C*f^2)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)))/d^5

Mupad [B] (verification not implemented)

Time = 35.09 (sec) , antiderivative size = 1732, normalized size of antiderivative = 7.60 \[ \int \frac {(e+f x)^2 \left (A+B x+C x^2\right )}{\sqrt {1-d x} \sqrt {1+d x}} \, dx=\text {Too large to display} \]

[In]

int(((e + f*x)^2*(A + B*x + C*x^2))/((1 - d*x)^(1/2)*(d*x + 1)^(1/2)),x)

[Out]

- ((14*A*f^2*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 - (2*A*f^2*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/
2) - 1) - (14*A*f^2*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 + (2*A*f^2*((1 - d*x)^(1/2) - 1)^7)/((d*x
+ 1)^(1/2) - 1)^7 + (16*A*d*e*f*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (32*A*d*e*f*((1 - d*x)^(1/
2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (16*A*d*e*f*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6)/(d^3 + (4*d
^3*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (6*d^3*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4
+ (4*d^3*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (d^3*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1
)^8) - ((((1 - d*x)^(1/2) - 1)^4*(64*B*f^2 + 32*B*d^2*e^2))/((d*x + 1)^(1/2) - 1)^4 + (((1 - d*x)^(1/2) - 1)^8
*(64*B*f^2 + 32*B*d^2*e^2))/((d*x + 1)^(1/2) - 1)^8 - (((1 - d*x)^(1/2) - 1)^6*((128*B*f^2)/3 - 48*B*d^2*e^2))
/((d*x + 1)^(1/2) - 1)^6 + (8*B*d^2*e^2*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (8*B*d^2*e^2*((1 -
d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 + (20*B*d*e*f*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 +
(24*B*d*e*f*((1 - d*x)^(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 - (24*B*d*e*f*((1 - d*x)^(1/2) - 1)^7)/((d*x + 1)
^(1/2) - 1)^7 - (20*B*d*e*f*((1 - d*x)^(1/2) - 1)^9)/((d*x + 1)^(1/2) - 1)^9 + (4*B*d*e*f*((1 - d*x)^(1/2) - 1
)^11)/((d*x + 1)^(1/2) - 1)^11 - (4*B*d*e*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1))/(d^4 + (6*d^4*((1 -
d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (15*d^4*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (20*d^
4*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (15*d^4*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 1)^8
+ (6*d^4*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 + (d^4*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2)
- 1)^12) - ((((1 - d*x)^(1/2) - 1)^15*((3*C*f^2)/2 + 2*C*d^2*e^2))/((d*x + 1)^(1/2) - 1)^15 - (((1 - d*x)^(1/2
) - 1)^3*((23*C*f^2)/2 - 6*C*d^2*e^2))/((d*x + 1)^(1/2) - 1)^3 - (((1 - d*x)^(1/2) - 1)*((3*C*f^2)/2 + 2*C*d^2
*e^2))/((d*x + 1)^(1/2) - 1) + (((1 - d*x)^(1/2) - 1)^13*((23*C*f^2)/2 - 6*C*d^2*e^2))/((d*x + 1)^(1/2) - 1)^1
3 + (((1 - d*x)^(1/2) - 1)^5*((333*C*f^2)/2 + 30*C*d^2*e^2))/((d*x + 1)^(1/2) - 1)^5 - (((1 - d*x)^(1/2) - 1)^
11*((333*C*f^2)/2 + 30*C*d^2*e^2))/((d*x + 1)^(1/2) - 1)^11 - (((1 - d*x)^(1/2) - 1)^7*((671*C*f^2)/2 - 22*C*d
^2*e^2))/((d*x + 1)^(1/2) - 1)^7 + (((1 - d*x)^(1/2) - 1)^9*((671*C*f^2)/2 - 22*C*d^2*e^2))/((d*x + 1)^(1/2) -
1)^9 + (128*C*d*e*f*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (512*C*d*e*f*((1 - d*x)^(1/2) - 1)^6)/
(3*((d*x + 1)^(1/2) - 1)^6) + (256*C*d*e*f*((1 - d*x)^(1/2) - 1)^8)/(3*((d*x + 1)^(1/2) - 1)^8) + (512*C*d*e*f
*((1 - d*x)^(1/2) - 1)^10)/(3*((d*x + 1)^(1/2) - 1)^10) + (128*C*d*e*f*((1 - d*x)^(1/2) - 1)^12)/((d*x + 1)^(1
/2) - 1)^12)/(d^5 + (8*d^5*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (28*d^5*((1 - d*x)^(1/2) - 1)^4)
/((d*x + 1)^(1/2) - 1)^4 + (56*d^5*((1 - d*x)^(1/2) - 1)^6)/((d*x + 1)^(1/2) - 1)^6 + (70*d^5*((1 - d*x)^(1/2)
- 1)^8)/((d*x + 1)^(1/2) - 1)^8 + (56*d^5*((1 - d*x)^(1/2) - 1)^10)/((d*x + 1)^(1/2) - 1)^10 + (28*d^5*((1 -
d*x)^(1/2) - 1)^12)/((d*x + 1)^(1/2) - 1)^12 + (8*d^5*((1 - d*x)^(1/2) - 1)^14)/((d*x + 1)^(1/2) - 1)^14 + (d^
5*((1 - d*x)^(1/2) - 1)^16)/((d*x + 1)^(1/2) - 1)^16) - (C*atan((C*((1 - d*x)^(1/2) - 1)*(3*f^2 + 4*d^2*e^2))/
(((d*x + 1)^(1/2) - 1)*(3*C*f^2 + 4*C*d^2*e^2)))*(3*f^2 + 4*d^2*e^2))/(2*d^5) - (2*A*atan((A*(f^2 + 2*d^2*e^2)
*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) - 1)*(A*f^2 + 2*A*d^2*e^2)))*(f^2 + 2*d^2*e^2))/d^3 - (4*B*e*f*atan(
((1 - d*x)^(1/2) - 1)/((d*x + 1)^(1/2) - 1)))/d^3