3.35 \(\int \sqrt{1-d x} \sqrt{1+d x} (e+f x)^3 \left (A+B x+C x^2\right ) \, dx\)

Optimal. Leaf size=415 \[ -\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 \left (7 d^2 f (2 A f+B e)-C \left (3 d^2 e^2-8 f^2\right )\right )}{70 d^4 f}+\frac{x \sqrt{1-d^2 x^2} \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^4}+\frac{\left (1-d^2 x^2\right )^{3/2} \left (3 d^2 f x \left (-98 A d^2 e f^2-14 B d^2 e^2 f-35 B f^3+6 C d^2 e^3-41 C e f^2\right )+8 \left (C \left (3 d^4 e^4-30 d^2 e^2 f^2-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )\right )}{840 d^6 f}+\frac{\sin ^{-1}(d x) \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^5}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^3 (3 C e-7 B f)}{42 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^4}{7 d^2 f} \]

[Out]

((2*C*d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)
*x*Sqrt[1 - d^2*x^2])/(16*d^4) - ((7*d^2*f*(B*e + 2*A*f) - C*(3*d^2*e^2 - 8*f^2)
)*(e + f*x)^2*(1 - d^2*x^2)^(3/2))/(70*d^4*f) + ((3*C*e - 7*B*f)*(e + f*x)^3*(1
- d^2*x^2)^(3/2))/(42*d^2*f) - (C*(e + f*x)^4*(1 - d^2*x^2)^(3/2))/(7*d^2*f) + (
(8*(C*(3*d^4*e^4 - 30*d^2*e^2*f^2 - 8*f^4) - 7*d^2*f*(2*A*f*(6*d^2*e^2 + f^2) +
B*(d^2*e^3 + 6*e*f^2))) + 3*d^2*f*(6*C*d^2*e^3 - 14*B*d^2*e^2*f - 41*C*e*f^2 - 9
8*A*d^2*e*f^2 - 35*B*f^3)*x)*(1 - d^2*x^2)^(3/2))/(840*d^6*f) + ((2*C*d^2*e^3 +
8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*ArcSin[d*x])/(1
6*d^5)

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Rubi [A]  time = 1.47305, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 \left (7 d^2 f (2 A f+B e)-C \left (3 d^2 e^2-8 f^2\right )\right )}{70 d^4 f}+\frac{x \sqrt{1-d^2 x^2} \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^4}+\frac{\left (1-d^2 x^2\right )^{3/2} \left (3 d^2 f x \left (-98 A d^2 e f^2-14 B d^2 e^2 f-35 B f^3+6 C d^2 e^3-41 C e f^2\right )+8 \left (C \left (3 d^4 e^4-30 d^2 e^2 f^2-8 f^4\right )-7 d^2 f \left (2 A f \left (6 d^2 e^2+f^2\right )+B \left (d^2 e^3+6 e f^2\right )\right )\right )\right )}{840 d^6 f}+\frac{\sin ^{-1}(d x) \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )}{16 d^5}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^3 (3 C e-7 B f)}{42 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^4}{7 d^2 f} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3*(A + B*x + C*x^2),x]

[Out]

((2*C*d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)
*x*Sqrt[1 - d^2*x^2])/(16*d^4) - ((7*d^2*f*(B*e + 2*A*f) - C*(3*d^2*e^2 - 8*f^2)
)*(e + f*x)^2*(1 - d^2*x^2)^(3/2))/(70*d^4*f) + ((3*C*e - 7*B*f)*(e + f*x)^3*(1
- d^2*x^2)^(3/2))/(42*d^2*f) - (C*(e + f*x)^4*(1 - d^2*x^2)^(3/2))/(7*d^2*f) + (
(8*(C*(3*d^4*e^4 - 30*d^2*e^2*f^2 - 8*f^4) - 7*d^2*f*(2*A*f*(6*d^2*e^2 + f^2) +
B*(d^2*e^3 + 6*e*f^2))) + 3*d^2*f*(6*C*d^2*e^3 - 14*B*d^2*e^2*f - 41*C*e*f^2 - 9
8*A*d^2*e*f^2 - 35*B*f^3)*x)*(1 - d^2*x^2)^(3/2))/(840*d^6*f) + ((2*C*d^2*e^3 +
8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*ArcSin[d*x])/(1
6*d^5)

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Rubi in Sympy [A]  time = 160.039, size = 415, normalized size = 1. \[ - \frac{C \left (e + f x\right )^{4} \left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}}}{7 d^{2} f} - \frac{\left (e + f x\right )^{3} \left (7 B f - 3 C e\right ) \left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}}}{42 d^{2} f} + \frac{x \sqrt{- d^{2} x^{2} + 1} \left (8 A d^{4} e^{3} + 6 A d^{2} e f^{2} + 6 B d^{2} e^{2} f + B f^{3} + 2 C d^{2} e^{3} + 3 C e f^{2}\right )}{16 d^{4}} - \frac{\left (e + f x\right )^{2} \left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}} \left (d^{2} e \left (7 B f - 3 C e\right ) + f^{2} \left (14 A d^{2} + 8 C\right )\right )}{70 d^{4} f} + \frac{\left (8 A d^{4} e^{3} + 6 A d^{2} e f^{2} + 6 B d^{2} e^{2} f + B f^{3} + 2 C d^{2} e^{3} + 3 C e f^{2}\right ) \operatorname{asin}{\left (d x \right )}}{16 d^{5}} - \frac{\left (- d^{2} x^{2} + 1\right )^{\frac{3}{2}} \left (2016 A d^{4} e^{2} f^{2} + 336 A d^{2} f^{4} + 168 B d^{4} e^{3} f + 1008 B d^{2} e f^{3} - 72 C d^{4} e^{4} + 720 C d^{2} e^{2} f^{2} + 192 C f^{4} + 9 d^{2} f x \left (98 A d^{2} e f^{2} + 14 B d^{2} e^{2} f + 35 B f^{3} - 6 C d^{2} e^{3} + 41 C e f^{2}\right )\right )}{2520 d^{6} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)**3*(C*x**2+B*x+A)*(-d*x+1)**(1/2)*(d*x+1)**(1/2),x)

[Out]

-C*(e + f*x)**4*(-d**2*x**2 + 1)**(3/2)/(7*d**2*f) - (e + f*x)**3*(7*B*f - 3*C*e
)*(-d**2*x**2 + 1)**(3/2)/(42*d**2*f) + x*sqrt(-d**2*x**2 + 1)*(8*A*d**4*e**3 +
6*A*d**2*e*f**2 + 6*B*d**2*e**2*f + B*f**3 + 2*C*d**2*e**3 + 3*C*e*f**2)/(16*d**
4) - (e + f*x)**2*(-d**2*x**2 + 1)**(3/2)*(d**2*e*(7*B*f - 3*C*e) + f**2*(14*A*d
**2 + 8*C))/(70*d**4*f) + (8*A*d**4*e**3 + 6*A*d**2*e*f**2 + 6*B*d**2*e**2*f + B
*f**3 + 2*C*d**2*e**3 + 3*C*e*f**2)*asin(d*x)/(16*d**5) - (-d**2*x**2 + 1)**(3/2
)*(2016*A*d**4*e**2*f**2 + 336*A*d**2*f**4 + 168*B*d**4*e**3*f + 1008*B*d**2*e*f
**3 - 72*C*d**4*e**4 + 720*C*d**2*e**2*f**2 + 192*C*f**4 + 9*d**2*f*x*(98*A*d**2
*e*f**2 + 14*B*d**2*e**2*f + 35*B*f**3 - 6*C*d**2*e**3 + 41*C*e*f**2))/(2520*d**
6*f)

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Mathematica [A]  time = 0.770045, size = 355, normalized size = 0.86 \[ \frac{105 d \sin ^{-1}(d x) \left (8 A d^4 e^3+6 A d^2 e f^2+6 B d^2 e^2 f+B f^3+2 C d^2 e^3+3 C e f^2\right )+\sqrt{1-d^2 x^2} \left (14 A d^2 \left (6 d^4 x \left (10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right )-d^2 f \left (120 e^2+45 e f x+8 f^2 x^2\right )-16 f^3\right )+7 B \left (4 d^6 x^2 \left (20 e^3+45 e^2 f x+36 e f^2 x^2+10 f^3 x^3\right )-2 d^4 \left (40 e^3+45 e^2 f x+24 e f^2 x^2+5 f^3 x^3\right )-3 d^2 f^2 (32 e+5 f x)\right )-C \left (-12 d^6 x^3 \left (35 e^3+84 e^2 f x+70 e f^2 x^2+20 f^3 x^3\right )+6 d^4 x \left (35 e^3+56 e^2 f x+35 e f^2 x^2+8 f^3 x^3\right )+d^2 f \left (672 e^2+315 e f x+64 f^2 x^2\right )+128 f^3\right )\right )}{1680 d^6} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)^3*(A + B*x + C*x^2),x]

[Out]

(Sqrt[1 - d^2*x^2]*(14*A*d^2*(-16*f^3 - d^2*f*(120*e^2 + 45*e*f*x + 8*f^2*x^2) +
6*d^4*x*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3)) + 7*B*(-3*d^2*f^2*(32
*e + 5*f*x) - 2*d^4*(40*e^3 + 45*e^2*f*x + 24*e*f^2*x^2 + 5*f^3*x^3) + 4*d^6*x^2
*(20*e^3 + 45*e^2*f*x + 36*e*f^2*x^2 + 10*f^3*x^3)) - C*(128*f^3 + d^2*f*(672*e^
2 + 315*e*f*x + 64*f^2*x^2) + 6*d^4*x*(35*e^3 + 56*e^2*f*x + 35*e*f^2*x^2 + 8*f^
3*x^3) - 12*d^6*x^3*(35*e^3 + 84*e^2*f*x + 70*e*f^2*x^2 + 20*f^3*x^3))) + 105*d*
(2*C*d^2*e^3 + 8*A*d^4*e^3 + 6*B*d^2*e^2*f + 3*C*e*f^2 + 6*A*d^2*e*f^2 + B*f^3)*
ArcSin[d*x])/(1680*d^6)

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Maple [C]  time = 0.042, size = 959, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^3*(C*x^2+B*x+A)*(-d*x+1)^(1/2)*(d*x+1)^(1/2),x)

[Out]

1/1680*(-d*x+1)^(1/2)*(d*x+1)^(1/2)*(-128*C*(-d^2*x^2+1)^(1/2)*csgn(d)*f^3+840*A
*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d^5*e^3+210*C*arctan(csgn(d)*d*x/(-d^2*x
^2+1)^(1/2))*d^3*e^3+105*B*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d*f^3-560*B*(-
d^2*x^2+1)^(1/2)*csgn(d)*d^4*e^3-630*A*(-d^2*x^2+1)^(1/2)*csgn(d)*x*d^4*e*f^2-63
0*B*(-d^2*x^2+1)^(1/2)*csgn(d)*x*d^4*e^2*f-315*C*(-d^2*x^2+1)^(1/2)*csgn(d)*x*d^
2*e*f^2+1260*A*csgn(d)*x^3*d^6*e*f^2*(-d^2*x^2+1)^(1/2)-336*C*(-d^2*x^2+1)^(1/2)
*csgn(d)*x^2*d^4*e^2*f+840*C*csgn(d)*x^5*d^6*e*f^2*(-d^2*x^2+1)^(1/2)+1008*B*csg
n(d)*x^4*d^6*e*f^2*(-d^2*x^2+1)^(1/2)+1008*C*csgn(d)*x^4*d^6*e^2*f*(-d^2*x^2+1)^
(1/2)+1260*B*csgn(d)*x^3*d^6*e^2*f*(-d^2*x^2+1)^(1/2)+1680*A*csgn(d)*x^2*d^6*e^2
*f*(-d^2*x^2+1)^(1/2)-210*C*(-d^2*x^2+1)^(1/2)*csgn(d)*x^3*d^4*e*f^2-336*B*(-d^2
*x^2+1)^(1/2)*csgn(d)*x^2*d^4*e*f^2-224*A*(-d^2*x^2+1)^(1/2)*csgn(d)*d^2*f^3+630
*A*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d^3*e*f^2+630*B*arctan(csgn(d)*d*x/(-d
^2*x^2+1)^(1/2))*d^3*e^2*f+315*C*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d*e*f^2+
840*A*(-d^2*x^2+1)^(1/2)*csgn(d)*x*d^6*e^3-210*C*(-d^2*x^2+1)^(1/2)*csgn(d)*x*d^
4*e^3-105*B*(-d^2*x^2+1)^(1/2)*csgn(d)*x*d^2*f^3+240*C*csgn(d)*x^6*d^6*f^3*(-d^2
*x^2+1)^(1/2)+280*B*csgn(d)*x^5*d^6*f^3*(-d^2*x^2+1)^(1/2)+336*A*csgn(d)*x^4*d^6
*f^3*(-d^2*x^2+1)^(1/2)+420*C*csgn(d)*x^3*d^6*e^3*(-d^2*x^2+1)^(1/2)+560*B*csgn(
d)*x^2*d^6*e^3*(-d^2*x^2+1)^(1/2)-48*C*(-d^2*x^2+1)^(1/2)*csgn(d)*x^4*d^4*f^3-70
*B*(-d^2*x^2+1)^(1/2)*csgn(d)*x^3*d^4*f^3-112*A*(-d^2*x^2+1)^(1/2)*csgn(d)*x^2*d
^4*f^3-1680*A*(-d^2*x^2+1)^(1/2)*csgn(d)*d^4*e^2*f-64*C*(-d^2*x^2+1)^(1/2)*csgn(
d)*x^2*d^2*f^3-672*B*(-d^2*x^2+1)^(1/2)*csgn(d)*d^2*e*f^2-672*C*(-d^2*x^2+1)^(1/
2)*csgn(d)*d^2*e^2*f)*csgn(d)/d^6/(-d^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.50095, size = 644, normalized size = 1.55 \[ -\frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{3} x^{4}}{7 \, d^{2}} + \frac{1}{2} \, \sqrt{-d^{2} x^{2} + 1} A e^{3} x + \frac{A e^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} B e^{3}}{3 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} A e^{2} f}{d^{2}} - \frac{4 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{3} x^{2}}{35 \, d^{4}} - \frac{{\left (3 \, C e f^{2} + B f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{3}}{6 \, d^{2}} - \frac{{\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}{5 \, d^{2}} - \frac{{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, d^{2}} + \frac{{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} \sqrt{-d^{2} x^{2} + 1} x}{8 \, d^{2}} - \frac{8 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{3}}{105 \, d^{6}} - \frac{{\left (3 \, C e f^{2} + B f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{8 \, d^{4}} + \frac{{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{2}} - \frac{2 \,{\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )}{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{15 \, d^{4}} + \frac{{\left (3 \, C e f^{2} + B f^{3}\right )} \sqrt{-d^{2} x^{2} + 1} x}{16 \, d^{4}} + \frac{{\left (3 \, C e f^{2} + B f^{3}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{16 \, \sqrt{d^{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)^3,x, algorithm="maxima")

[Out]

-1/7*(-d^2*x^2 + 1)^(3/2)*C*f^3*x^4/d^2 + 1/2*sqrt(-d^2*x^2 + 1)*A*e^3*x + 1/2*A
*e^3*arcsin(d^2*x/sqrt(d^2))/sqrt(d^2) - 1/3*(-d^2*x^2 + 1)^(3/2)*B*e^3/d^2 - (-
d^2*x^2 + 1)^(3/2)*A*e^2*f/d^2 - 4/35*(-d^2*x^2 + 1)^(3/2)*C*f^3*x^2/d^4 - 1/6*(
3*C*e*f^2 + B*f^3)*(-d^2*x^2 + 1)^(3/2)*x^3/d^2 - 1/5*(3*C*e^2*f + 3*B*e*f^2 + A
*f^3)*(-d^2*x^2 + 1)^(3/2)*x^2/d^2 - 1/4*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*(-d^2*x
^2 + 1)^(3/2)*x/d^2 + 1/8*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*sqrt(-d^2*x^2 + 1)*x/d
^2 - 8/105*(-d^2*x^2 + 1)^(3/2)*C*f^3/d^6 - 1/8*(3*C*e*f^2 + B*f^3)*(-d^2*x^2 +
1)^(3/2)*x/d^4 + 1/8*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*arcsin(d^2*x/sqrt(d^2))/(sq
rt(d^2)*d^2) - 2/15*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*(-d^2*x^2 + 1)^(3/2)/d^4 + 1
/16*(3*C*e*f^2 + B*f^3)*sqrt(-d^2*x^2 + 1)*x/d^4 + 1/16*(3*C*e*f^2 + B*f^3)*arcs
in(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4)

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Fricas [A]  time = 0.261212, size = 2512, normalized size = 6.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)^3,x, algorithm="fricas")

[Out]

1/1680*(240*C*d^13*f^3*x^14 + 280*(3*C*d^13*e*f^2 + B*d^13*f^3)*x^13 + 336*(3*C*
d^13*e^2*f + 3*B*d^13*e*f^2 + (A*d^13 - 18*C*d^11)*f^3)*x^12 + 70*(6*C*d^13*e^3
+ 18*B*d^13*e^2*f - 101*B*d^11*f^3 + 3*(6*A*d^13 - 101*C*d^11)*e*f^2)*x^11 + 112
*(5*B*d^13*e^3 - 228*B*d^11*e*f^2 + 3*(5*A*d^13 - 76*C*d^11)*e^2*f - (76*A*d^11
- 233*C*d^9)*f^3)*x^10 - 105*(306*B*d^11*e^2*f - 293*B*d^9*f^3 - 2*(4*A*d^13 - 5
1*C*d^11)*e^3 + 3*(102*A*d^11 - 293*C*d^9)*e*f^2)*x^9 - 560*(26*B*d^11*e^3 - 201
*B*d^9*e*f^2 + 3*(26*A*d^11 - 67*C*d^9)*e^2*f - (67*A*d^9 - 68*C*d^7)*f^3)*x^8 +
35*(4194*B*d^9*e^2*f - 1285*B*d^7*f^3 - 6*(100*A*d^11 - 233*C*d^9)*e^3 + 3*(139
8*A*d^9 - 1285*C*d^7)*e*f^2)*x^7 + 1120*(61*B*d^9*e^3 - 150*B*d^7*e*f^2 + 3*(61*
A*d^9 - 50*C*d^7)*e^2*f - 2*(25*A*d^7 - 8*C*d^5)*f^3)*x^6 - 280*(882*B*d^7*e^2*f
- 61*B*d^5*f^3 - 6*(52*A*d^9 - 49*C*d^7)*e^3 + 3*(294*A*d^7 - 61*C*d^5)*e*f^2)*
x^5 - 26880*(4*B*d^7*e^3 - 3*B*d^5*e*f^2 - A*d^5*f^3 + 3*(4*A*d^7 - C*d^5)*e^2*f
)*x^4 + 560*(306*B*d^5*e^2*f + 19*B*d^3*f^3 - 6*(36*A*d^7 - 17*C*d^5)*e^3 + 3*(1
02*A*d^5 + 19*C*d^3)*e*f^2)*x^3 + 53760*(B*d^5*e^3 + 3*A*d^5*e^2*f)*x^2 + 7*(240
*C*d^11*f^3*x^12 + 280*(3*C*d^11*e*f^2 + B*d^11*f^3)*x^11 + 48*(21*C*d^11*e^2*f
+ 21*B*d^11*e*f^2 + (7*A*d^11 - 41*C*d^9)*f^3)*x^10 + 210*(2*C*d^11*e^3 + 6*B*d^
11*e^2*f - 11*B*d^9*f^3 + 3*(2*A*d^11 - 11*C*d^9)*e*f^2)*x^9 + 80*(7*B*d^11*e^3
- 105*B*d^9*e*f^2 + 21*(A*d^11 - 5*C*d^9)*e^2*f - (35*A*d^9 - 52*C*d^7)*f^3)*x^8
- 105*(102*B*d^9*e^2*f - 47*B*d^7*f^3 - 2*(4*A*d^11 - 17*C*d^9)*e^3 + 3*(34*A*d
^9 - 47*C*d^7)*e*f^2)*x^7 - 160*(31*B*d^9*e^3 - 114*B*d^7*e*f^2 + 3*(31*A*d^9 -
38*C*d^7)*e^2*f - 2*(19*A*d^7 - 8*C*d^5)*f^3)*x^6 + 40*(630*B*d^7*e^2*f - 71*B*d
^5*f^3 - 42*(4*A*d^9 - 5*C*d^7)*e^3 + 3*(210*A*d^7 - 71*C*d^5)*e*f^2)*x^5 + 3840
*(3*B*d^7*e^3 - 3*B*d^5*e*f^2 - A*d^5*f^3 + 3*(3*A*d^7 - C*d^5)*e^2*f)*x^4 - 80*
(270*B*d^5*e^2*f + 13*B*d^3*f^3 - 6*(28*A*d^7 - 15*C*d^5)*e^3 + 3*(90*A*d^5 + 13
*C*d^3)*e*f^2)*x^3 - 7680*(B*d^5*e^3 + 3*A*d^5*e^2*f)*x^2 + 960*(6*B*d^3*e^2*f +
B*d*f^3 - 2*(4*A*d^5 - C*d^3)*e^3 + 3*(2*A*d^3 + C*d)*e*f^2)*x)*sqrt(d*x + 1)*s
qrt(-d*x + 1) - 6720*(6*B*d^3*e^2*f + B*d*f^3 - 2*(4*A*d^5 - C*d^3)*e^3 + 3*(2*A
*d^3 + C*d)*e*f^2)*x - 210*(7*(6*B*d^8*e^2*f + B*d^6*f^3 + 2*(4*A*d^10 + C*d^8)*
e^3 + 3*(2*A*d^8 + C*d^6)*e*f^2)*x^6 - 384*B*d^2*e^2*f - 56*(6*B*d^6*e^2*f + B*d
^4*f^3 + 2*(4*A*d^8 + C*d^6)*e^3 + 3*(2*A*d^6 + C*d^4)*e*f^2)*x^4 - 128*(4*A*d^4
+ C*d^2)*e^3 - 192*(2*A*d^2 + C)*e*f^2 - 64*B*f^3 + 112*(6*B*d^4*e^2*f + B*d^2*
f^3 + 2*(4*A*d^6 + C*d^4)*e^3 + 3*(2*A*d^4 + C*d^2)*e*f^2)*x^2 - ((6*B*d^8*e^2*f
+ B*d^6*f^3 + 2*(4*A*d^10 + C*d^8)*e^3 + 3*(2*A*d^8 + C*d^6)*e*f^2)*x^6 - 384*B
*d^2*e^2*f - 24*(6*B*d^6*e^2*f + B*d^4*f^3 + 2*(4*A*d^8 + C*d^6)*e^3 + 3*(2*A*d^
6 + C*d^4)*e*f^2)*x^4 - 128*(4*A*d^4 + C*d^2)*e^3 - 192*(2*A*d^2 + C)*e*f^2 - 64
*B*f^3 + 80*(6*B*d^4*e^2*f + B*d^2*f^3 + 2*(4*A*d^6 + C*d^4)*e^3 + 3*(2*A*d^4 +
C*d^2)*e*f^2)*x^2)*sqrt(d*x + 1)*sqrt(-d*x + 1))*arctan((sqrt(d*x + 1)*sqrt(-d*x
+ 1) - 1)/(d*x)))/(7*d^11*x^6 - 56*d^9*x^4 + 112*d^7*x^2 - 64*d^5 - (d^11*x^6 -
24*d^9*x^4 + 80*d^7*x^2 - 64*d^5)*sqrt(d*x + 1)*sqrt(-d*x + 1))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x+e)**3*(C*x**2+B*x+A)*(-d*x+1)**(1/2)*(d*x+1)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.317627, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*sqrt(d*x + 1)*sqrt(-d*x + 1)*(f*x + e)^3,x, algorithm="giac")

[Out]

Done