3.797 \(\int \frac{\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx\)

Optimal. Leaf size=276 \[ \frac{x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac{6 a c}{d^2}+\frac{b^2}{d^2}+\frac{3 c^2}{d^4}\right )}{d^6 \sqrt{1-d^2 x^2}}-\frac{3 \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{8 d^7}+\frac{c x \sqrt{1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{8 d^6}+\frac{b \sqrt{1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4} \]

[Out]

(b*(3*a^2 + (3*c^2)/d^4 + b^2/d^2 + (6*a*c)/d^2)*d^4 + (c + a*d^2)*(c^2 + 3*b^2*
d^2 + 2*a*c*d^2 + a^2*d^4)*x)/(d^6*Sqrt[1 - d^2*x^2]) + (b*(5*c^2 + b^2*d^2 + 6*
a*c*d^2)*Sqrt[1 - d^2*x^2])/d^6 + (c*(7*c^2 + 12*b^2*d^2 + 12*a*c*d^2)*x*Sqrt[1
- d^2*x^2])/(8*d^6) + (b*c^2*x^2*Sqrt[1 - d^2*x^2])/d^4 + (c^3*x^3*Sqrt[1 - d^2*
x^2])/(4*d^4) - (3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*
d^4)*ArcSin[d*x])/(8*d^7)

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Rubi [A]  time = 1.02247, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b \left (3 a^2 d^4+6 a c d^2+b^2 d^2+3 c^2\right )}{d^6 \sqrt{1-d^2 x^2}}-\frac{3 \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{8 d^7}+\frac{c x \sqrt{1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{8 d^6}+\frac{b \sqrt{1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)),x]

[Out]

(b*(3*c^2 + b^2*d^2 + 6*a*c*d^2 + 3*a^2*d^4) + (c + a*d^2)*(c^2 + 3*b^2*d^2 + 2*
a*c*d^2 + a^2*d^4)*x)/(d^6*Sqrt[1 - d^2*x^2]) + (b*(5*c^2 + b^2*d^2 + 6*a*c*d^2)
*Sqrt[1 - d^2*x^2])/d^6 + (c*(7*c^2 + 12*b^2*d^2 + 12*a*c*d^2)*x*Sqrt[1 - d^2*x^
2])/(8*d^6) + (b*c^2*x^2*Sqrt[1 - d^2*x^2])/d^4 + (c^3*x^3*Sqrt[1 - d^2*x^2])/(4
*d^4) - (3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*Arc
Sin[d*x])/(8*d^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(-d*x+1)**(3/2)/(d*x+1)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.976828, size = 226, normalized size = 0.82 \[ -\frac{3 \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )+\frac{d \left (8 b \left (-3 a^2 d^4+6 a c d^2 \left (d^2 x^2-2\right )+c^2 \left (d^4 x^4+4 d^2 x^2-8\right )\right )+x \left (-8 a^3 d^6-24 a^2 c d^4+12 a c^2 d^2 \left (d^2 x^2-3\right )+c^3 \left (2 d^4 x^4+5 d^2 x^2-15\right )\right )+12 b^2 d^2 x \left (c \left (d^2 x^2-3\right )-2 a d^2\right )+8 b^3 d^2 \left (d^2 x^2-2\right )\right )}{\sqrt{1-d^2 x^2}}}{8 d^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)),x]

[Out]

-((d*(8*b^3*d^2*(-2 + d^2*x^2) + 12*b^2*d^2*x*(-2*a*d^2 + c*(-3 + d^2*x^2)) + 8*
b*(-3*a^2*d^4 + 6*a*c*d^2*(-2 + d^2*x^2) + c^2*(-8 + 4*d^2*x^2 + d^4*x^4)) + x*(
-24*a^2*c*d^4 - 8*a^3*d^6 + 12*a*c^2*d^2*(-3 + d^2*x^2) + c^3*(-15 + 5*d^2*x^2 +
2*d^4*x^4))))/Sqrt[1 - d^2*x^2] + 3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*
b^2*d^4 + 8*a^2*c*d^4)*ArcSin[d*x])/(8*d^7)

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Maple [C]  time = 0.053, size = 755, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(-d*x+1)^(3/2)/(d*x+1)^(3/2),x)

[Out]

1/8*(-d*x+1)^(1/2)*(-15*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x^2*c^3*d^2-16*(-
d^2*x^2+1)^(1/2)*b^3*d^3*csgn(d)+24*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a^2*c
*d^4+24*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*a*b^2*d^4+36*arctan(csgn(d)*d*x/(
-d^2*x^2+1)^(1/2))*a*c^2*d^2+36*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*b^2*c*d^2
+15*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*c^3-24*arctan(csgn(d)*d*x/(-d^2*x^2+1
)^(1/2))*x^2*a*b^2*d^6-8*(-d^2*x^2+1)^(1/2)*csgn(d)*d^7*x*a^3-36*arctan(csgn(d)*
d*x/(-d^2*x^2+1)^(1/2))*x^2*a*c^2*d^4-36*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*
x^2*b^2*c*d^4+2*csgn(d)*x^5*c^3*d^5*(-d^2*x^2+1)^(1/2)-24*arctan(csgn(d)*d*x/(-d
^2*x^2+1)^(1/2))*x^2*a^2*c*d^6+8*csgn(d)*x^4*b*c^2*d^5*(-d^2*x^2+1)^(1/2)+12*csg
n(d)*x^3*a*c^2*d^5*(-d^2*x^2+1)^(1/2)+12*csgn(d)*x^3*b^2*c*d^5*(-d^2*x^2+1)^(1/2
)-24*x*(-d^2*x^2+1)^(1/2)*a^2*c*d^5*csgn(d)-24*x*(-d^2*x^2+1)^(1/2)*a*b^2*d^5*cs
gn(d)+48*csgn(d)*x^2*a*b*c*d^5*(-d^2*x^2+1)^(1/2)+32*csgn(d)*d^3*(-d^2*x^2+1)^(1
/2)*x^2*b*c^2-36*x*(-d^2*x^2+1)^(1/2)*a*c^2*d^3*csgn(d)-36*x*(-d^2*x^2+1)^(1/2)*
b^2*c*d^3*csgn(d)-96*(-d^2*x^2+1)^(1/2)*a*b*c*d^3*csgn(d)+8*csgn(d)*x^2*b^3*d^5*
(-d^2*x^2+1)^(1/2)+5*c^3*x^3*(-d^2*x^2+1)^(1/2)*d^3*csgn(d)-24*(-d^2*x^2+1)^(1/2
)*a^2*b*d^5*csgn(d)-15*x*(-d^2*x^2+1)^(1/2)*c^3*csgn(d)*d-64*(-d^2*x^2+1)^(1/2)*
b*c^2*csgn(d)*d)*csgn(d)/(d*x-1)/(-d^2*x^2+1)^(1/2)/d^7/(d*x+1)^(1/2)

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Maxima [A]  time = 0.775117, size = 549, normalized size = 1.99 \[ -\frac{c^{3} x^{5}}{4 \, \sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{b c^{2} x^{4}}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{a^{3} x}{\sqrt{-d^{2} x^{2} + 1}} - \frac{5 \, c^{3} x^{3}}{8 \, \sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{3 \,{\left (b^{2} c + a c^{2}\right )} x^{3}}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{3 \, a^{2} b}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{4 \, b c^{2} x^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{{\left (b^{3} + 6 \, a b c\right )} x^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{3 \,{\left (a b^{2} + a^{2} c\right )} x}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{3 \,{\left (a b^{2} + a^{2} c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}} d^{2}} + \frac{15 \, c^{3} x}{8 \, \sqrt{-d^{2} x^{2} + 1} d^{6}} + \frac{9 \,{\left (b^{2} c + a c^{2}\right )} x}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{15 \, c^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{6}} - \frac{9 \,{\left (b^{2} c + a c^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{4}} + \frac{8 \, b c^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{6}} + \frac{2 \,{\left (b^{3} + 6 \, a b c\right )}}{\sqrt{-d^{2} x^{2} + 1} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/((d*x + 1)^(3/2)*(-d*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

-1/4*c^3*x^5/(sqrt(-d^2*x^2 + 1)*d^2) - b*c^2*x^4/(sqrt(-d^2*x^2 + 1)*d^2) + a^3
*x/sqrt(-d^2*x^2 + 1) - 5/8*c^3*x^3/(sqrt(-d^2*x^2 + 1)*d^4) - 3/2*(b^2*c + a*c^
2)*x^3/(sqrt(-d^2*x^2 + 1)*d^2) + 3*a^2*b/(sqrt(-d^2*x^2 + 1)*d^2) - 4*b*c^2*x^2
/(sqrt(-d^2*x^2 + 1)*d^4) - (b^3 + 6*a*b*c)*x^2/(sqrt(-d^2*x^2 + 1)*d^2) + 3*(a*
b^2 + a^2*c)*x/(sqrt(-d^2*x^2 + 1)*d^2) - 3*(a*b^2 + a^2*c)*arcsin(d^2*x/sqrt(d^
2))/(sqrt(d^2)*d^2) + 15/8*c^3*x/(sqrt(-d^2*x^2 + 1)*d^6) + 9/2*(b^2*c + a*c^2)*
x/(sqrt(-d^2*x^2 + 1)*d^4) - 15/8*c^3*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^6) -
9/2*(b^2*c + a*c^2)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4) + 8*b*c^2/(sqrt(-d^2
*x^2 + 1)*d^6) + 2*(b^3 + 6*a*b*c)/(sqrt(-d^2*x^2 + 1)*d^4)

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Fricas [A]  time = 0.30135, size = 1362, normalized size = 4.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/((d*x + 1)^(3/2)*(-d*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

-1/8*(10*c^3*d^9*x^9 + 40*b*c^2*d^9*x^8 - 192*a^2*b*d^7*x^2 - 15*(c^3*d^7 - 4*(b
^2*c + a*c^2)*d^9)*x^7 - 8*(3*a^2*b*d^11 + 8*b*c^2*d^7 - 3*(b^3 + 6*a*b*c)*d^9)*
x^6 - (40*a^3*d^11 + 120*(a*b^2 + a^2*c)*d^9 + 143*c^3*d^5 + 420*(b^2*c + a*c^2)
*d^7)*x^5 + 32*(6*a^2*b*d^9 - (b^3 + 6*a*b*c)*d^7)*x^4 + 4*(40*a^3*d^9 + 120*(a*
b^2 + a^2*c)*d^7 + 95*c^3*d^3 + 228*(b^2*c + a*c^2)*d^5)*x^3 - (2*c^3*d^9*x^9 +
8*b*c^2*d^9*x^8 - 192*a^2*b*d^7*x^2 - (19*c^3*d^7 - 12*(b^2*c + a*c^2)*d^9)*x^7
- 8*(8*b*c^2*d^7 - (b^3 + 6*a*b*c)*d^9)*x^6 - (8*a^3*d^11 + 24*(a*b^2 + a^2*c)*d
^9 + 43*c^3*d^5 + 180*(b^2*c + a*c^2)*d^7)*x^5 + 32*(3*a^2*b*d^9 - (b^3 + 6*a*b*
c)*d^7)*x^4 + 4*(24*a^3*d^9 + 72*(a*b^2 + a^2*c)*d^7 + 65*c^3*d^3 + 156*(b^2*c +
a*c^2)*d^5)*x^3 - 16*(8*a^3*d^7 + 24*(a*b^2 + a^2*c)*d^5 + 15*c^3*d + 36*(b^2*c
+ a*c^2)*d^3)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 16*(8*a^3*d^7 + 24*(a*b^2 + a^2
*c)*d^5 + 15*c^3*d + 36*(b^2*c + a*c^2)*d^3)*x - 6*((8*(a*b^2 + a^2*c)*d^10 + 5*
c^3*d^6 + 12*(b^2*c + a*c^2)*d^8)*x^6 - 128*(a*b^2 + a^2*c)*d^4 - 13*(8*(a*b^2 +
a^2*c)*d^8 + 5*c^3*d^4 + 12*(b^2*c + a*c^2)*d^6)*x^4 - 80*c^3 - 192*(b^2*c + a*
c^2)*d^2 + 28*(8*(a*b^2 + a^2*c)*d^6 + 5*c^3*d^2 + 12*(b^2*c + a*c^2)*d^4)*x^2 +
(128*(a*b^2 + a^2*c)*d^4 + 5*(8*(a*b^2 + a^2*c)*d^8 + 5*c^3*d^4 + 12*(b^2*c + a
*c^2)*d^6)*x^4 + 80*c^3 + 192*(b^2*c + a*c^2)*d^2 - 20*(8*(a*b^2 + a^2*c)*d^6 +
5*c^3*d^2 + 12*(b^2*c + a*c^2)*d^4)*x^2)*sqrt(d*x + 1)*sqrt(-d*x + 1))*arctan((s
qrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^13*x^6 - 13*d^11*x^4 + 28*d^9*x^2 -
16*d^7 + (5*d^11*x^4 - 20*d^9*x^2 + 16*d^7)*sqrt(d*x + 1)*sqrt(-d*x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{3}}{\left (- d x + 1\right )^{\frac{3}{2}} \left (d x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**3/(-d*x+1)**(3/2)/(d*x+1)**(3/2),x)

[Out]

Integral((a + b*x + c*x**2)**3/((-d*x + 1)**(3/2)*(d*x + 1)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.343008, size = 961, normalized size = 3.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/((d*x + 1)^(3/2)*(-d*x + 1)^(3/2)),x, algorithm="giac")

[Out]

-1/86016*(8*a*b^2*d^39 + 8*a^2*c*d^39 + 12*b^2*c*d^37 + 12*a*c^2*d^37 + 5*c^3*d^
35)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1)) - 1/516096*(4*a^3*d^41 + 12*a^2*b*d^40 + 1
2*a*b^2*d^39 + 12*a^2*c*d^39 + 20*b^3*d^38 + 120*a*b*c*d^38 - 12*b^2*c*d^37 - 12
*a*c^2*d^37 + 108*b*c^2*d^36 - 14*c^3*d^35 - (8*b^3*d^38 + 48*a*b*c*d^38 - 36*b^
2*c*d^37 - 36*a*c^2*d^37 + 80*b*c^2*d^36 - 35*c^3*d^35 + (12*b^2*c*d^37 + 12*a*c
^2*d^37 - 32*b*c^2*d^36 + 25*c^3*d^35 + 2*((d*x + 1)*c^3*d^35 + 4*b*c^2*d^36 - 5
*c^3*d^35)*(d*x + 1))*(d*x + 1))*(d*x + 1))*sqrt(d*x + 1)*sqrt(-d*x + 1)/(d*x -
1) + 1/4*(a^3*d^6*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 3*a^2*b*d^5*(sqrt(2
) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 3*a*b^2*d^4*(sqrt(2) - sqrt(-d*x + 1))/sqrt(
d*x + 1) + 3*a^2*c*d^4*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - b^3*d^3*(sqrt(
2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 6*a*b*c*d^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt
(d*x + 1) + 3*b^2*c*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 3*a*c^2*d^2*(
sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 3*b*c^2*d*(sqrt(2) - sqrt(-d*x + 1))/s
qrt(d*x + 1) + c^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1))/d^7 - 1/4*(a^3*d^6
- 3*a^2*b*d^5 + 3*a*b^2*d^4 + 3*a^2*c*d^4 - b^3*d^3 - 6*a*b*c*d^3 + 3*b^2*c*d^2
+ 3*a*c^2*d^2 - 3*b*c^2*d + c^3)*sqrt(d*x + 1)/(d^7*(sqrt(2) - sqrt(-d*x + 1)))