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

Optimal. Leaf size=138 \[ \frac{x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac{c}{d^2}\right )}{d^4 \sqrt{1-d^2 x^2}}-\frac{\sin ^{-1}(d x) \left (4 a c d^2+2 b^2 d^2+3 c^2\right )}{2 d^5}+\frac{2 b c \sqrt{1-d^2 x^2}}{d^4}+\frac{c^2 x \sqrt{1-d^2 x^2}}{2 d^4} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.32729, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b \left (a d^2+c\right )}{d^4 \sqrt{1-d^2 x^2}}-\frac{\sin ^{-1}(d x) \left (4 a c d^2+2 b^2 d^2+3 c^2\right )}{2 d^5}+\frac{2 b c \sqrt{1-d^2 x^2}}{d^4}+\frac{c^2 x \sqrt{1-d^2 x^2}}{2 d^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 97.2099, size = 155, normalized size = 1.12 \[ \frac{2 b c x^{2} \sqrt{- d^{2} x^{2} + 1}}{d^{2}} + \frac{c^{2} x^{3} \sqrt{- d^{2} x^{2} + 1}}{d^{2}} + \frac{x \left (a + b x + c x^{2}\right )^{2}}{\sqrt{- d^{2} x^{2} + 1}} + \frac{\left (96 b \left (a d^{2} + 2 c\right ) + x \left (96 a c d^{2} + 48 b^{2} d^{2} + 72 c^{2}\right )\right ) \sqrt{- d^{2} x^{2} + 1}}{48 d^{4}} - \frac{\left (4 a c d^{2} + 2 b^{2} d^{2} + 3 c^{2}\right ) \operatorname{asin}{\left (d x \right )}}{2 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*c*x**2*sqrt(-d**2*x**2 + 1)/d**2 + c**2*x**3*sqrt(-d**2*x**2 + 1)/d**2 + x*(
a + b*x + c*x**2)**2/sqrt(-d**2*x**2 + 1) + (96*b*(a*d**2 + 2*c) + x*(96*a*c*d**
2 + 48*b**2*d**2 + 72*c**2))*sqrt(-d**2*x**2 + 1)/(48*d**4) - (4*a*c*d**2 + 2*b*
*2*d**2 + 3*c**2)*asin(d*x)/(2*d**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.261702, size = 114, normalized size = 0.83 \[ \frac{\frac{d \left (x \left (2 a^2 d^4+4 a c d^2+c^2 \left (3-d^2 x^2\right )\right )+4 b \left (a d^2+c \left (2-d^2 x^2\right )\right )+2 b^2 d^2 x\right )}{\sqrt{1-d^2 x^2}}-\sin ^{-1}(d x) \left (4 a c d^2+2 b^2 d^2+3 c^2\right )}{2 d^5} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.036, size = 380, normalized size = 2.8 \[{\frac{{\it csgn} \left ( d \right ) }{ \left ( 2\,dx-2 \right ){d}^{5}}\sqrt{-dx+1} \left ({\it csgn} \left ( d \right ){x}^{3}{c}^{2}{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}-2\,\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ){d}^{5}x{a}^{2}+4\,{\it csgn} \left ( d \right ){x}^{2}bc{d}^{3}\sqrt{-{d}^{2}{x}^{2}+1}-4\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){x}^{2}ac{d}^{4}-2\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){x}^{2}{b}^{2}{d}^{4}-4\,acx\sqrt{-{d}^{2}{x}^{2}+1}{d}^{3}{\it csgn} \left ( d \right ) -2\,{b}^{2}x\sqrt{-{d}^{2}{x}^{2}+1}{d}^{3}{\it csgn} \left ( d \right ) -4\,\sqrt{-{d}^{2}{x}^{2}+1}ab{d}^{3}{\it csgn} \left ( d \right ) -3\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){x}^{2}{c}^{2}{d}^{2}-3\,{c}^{2}x\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) d-8\,\sqrt{-{d}^{2}{x}^{2}+1}bc{\it csgn} \left ( d \right ) d+4\,ac\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){d}^{2}+2\,{b}^{2}\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ){d}^{2}+3\,{c}^{2}\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}{\frac{1}{\sqrt{dx+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(-d*x+1)^(1/2)*(csgn(d)*x^3*c^2*d^3*(-d^2*x^2+1)^(1/2)-2*(-d^2*x^2+1)^(1/2)*
csgn(d)*d^5*x*a^2+4*csgn(d)*x^2*b*c*d^3*(-d^2*x^2+1)^(1/2)-4*arctan(csgn(d)*d*x/
(-d^2*x^2+1)^(1/2))*x^2*a*c*d^4-2*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*x^2*b^2
*d^4-4*a*c*x*(-d^2*x^2+1)^(1/2)*d^3*csgn(d)-2*b^2*x*(-d^2*x^2+1)^(1/2)*d^3*csgn(
d)-4*(-d^2*x^2+1)^(1/2)*a*b*d^3*csgn(d)-3*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))
*x^2*c^2*d^2-3*c^2*x*(-d^2*x^2+1)^(1/2)*csgn(d)*d-8*(-d^2*x^2+1)^(1/2)*b*c*csgn(
d)*d+4*a*c*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*d^2+2*b^2*arctan(csgn(d)*d*x/(
-d^2*x^2+1)^(1/2))*d^2+3*c^2*arctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2)))*csgn(d)/(d*
x-1)/(-d^2*x^2+1)^(1/2)/d^5/(d*x+1)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.773188, size = 270, normalized size = 1.96 \[ \frac{a^{2} x}{\sqrt{-d^{2} x^{2} + 1}} - \frac{c^{2} x^{3}}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{2 \, b c x^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{2 \, a b}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{{\left (b^{2} + 2 \, a c\right )} x}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{{\left (b^{2} + 2 \, a c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}} d^{2}} + \frac{3 \, c^{2} x}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{3 \, c^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{4}} + \frac{4 \, b c}{\sqrt{-d^{2} x^{2} + 1} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x/sqrt(-d^2*x^2 + 1) - 1/2*c^2*x^3/(sqrt(-d^2*x^2 + 1)*d^2) - 2*b*c*x^2/(sqr
t(-d^2*x^2 + 1)*d^2) + 2*a*b/(sqrt(-d^2*x^2 + 1)*d^2) + (b^2 + 2*a*c)*x/(sqrt(-d
^2*x^2 + 1)*d^2) - (b^2 + 2*a*c)*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^2) + 3/2*c
^2*x/(sqrt(-d^2*x^2 + 1)*d^4) - 3/2*c^2*arcsin(d^2*x/sqrt(d^2))/(sqrt(d^2)*d^4)
+ 4*b*c/(sqrt(-d^2*x^2 + 1)*d^4)

_______________________________________________________________________________________

Fricas [A]  time = 0.286159, size = 593, normalized size = 4.3 \[ -\frac{3 \, c^{2} d^{5} x^{5} + 8 \, a b d^{5} x^{2} - 4 \,{\left (a b d^{7} - b c d^{5}\right )} x^{4} -{\left (6 \, a^{2} d^{7} + 6 \,{\left (b^{2} + 2 \, a c\right )} d^{5} + 13 \, c^{2} d^{3}\right )} x^{3} -{\left (c^{2} d^{5} x^{5} + 4 \, b c d^{5} x^{4} + 8 \, a b d^{5} x^{2} -{\left (2 \, a^{2} d^{7} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{5} + 7 \, c^{2} d^{3}\right )} x^{3} + 4 \,{\left (2 \, a^{2} d^{5} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 4 \,{\left (2 \, a^{2} d^{5} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x - 2 \,{\left ({\left (2 \,{\left (b^{2} + 2 \, a c\right )} d^{6} + 3 \, c^{2} d^{4}\right )} x^{4} + 8 \,{\left (b^{2} + 2 \, a c\right )} d^{2} - 5 \,{\left (2 \,{\left (b^{2} + 2 \, a c\right )} d^{4} + 3 \, c^{2} d^{2}\right )} x^{2} -{\left (8 \,{\left (b^{2} + 2 \, a c\right )} d^{2} - 3 \,{\left (2 \,{\left (b^{2} + 2 \, a c\right )} d^{4} + 3 \, c^{2} d^{2}\right )} x^{2} + 12 \, c^{2}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 12 \, c^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{2 \,{\left (d^{9} x^{4} - 5 \, d^{7} x^{2} + 4 \, d^{5} +{\left (3 \, d^{7} x^{2} - 4 \, d^{5}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(3*c^2*d^5*x^5 + 8*a*b*d^5*x^2 - 4*(a*b*d^7 - b*c*d^5)*x^4 - (6*a^2*d^7 + 6
*(b^2 + 2*a*c)*d^5 + 13*c^2*d^3)*x^3 - (c^2*d^5*x^5 + 4*b*c*d^5*x^4 + 8*a*b*d^5*
x^2 - (2*a^2*d^7 + 2*(b^2 + 2*a*c)*d^5 + 7*c^2*d^3)*x^3 + 4*(2*a^2*d^5 + 2*(b^2
+ 2*a*c)*d^3 + 3*c^2*d)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 4*(2*a^2*d^5 + 2*(b^2
+ 2*a*c)*d^3 + 3*c^2*d)*x - 2*((2*(b^2 + 2*a*c)*d^6 + 3*c^2*d^4)*x^4 + 8*(b^2 +
2*a*c)*d^2 - 5*(2*(b^2 + 2*a*c)*d^4 + 3*c^2*d^2)*x^2 - (8*(b^2 + 2*a*c)*d^2 - 3*
(2*(b^2 + 2*a*c)*d^4 + 3*c^2*d^2)*x^2 + 12*c^2)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 1
2*c^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^9*x^4 - 5*d^7*x^2 +
4*d^5 + (3*d^7*x^2 - 4*d^5)*sqrt(d*x + 1)*sqrt(-d*x + 1))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.3037, size = 509, normalized size = 3.69 \[ -\frac{1}{384} \,{\left (2 \, b^{2} d^{17} + 4 \, a c d^{17} + 3 \, c^{2} d^{15}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right ) - \frac{{\left (a^{2} d^{19} + 2 \, a b d^{18} + b^{2} d^{17} + 2 \, a c d^{17} + 10 \, b c d^{16} - c^{2} d^{15} -{\left ({\left (d x + 1\right )} c^{2} d^{15} + 4 \, b c d^{16} - 3 \, c^{2} d^{15}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{-d x + 1}}{768 \,{\left (d x - 1\right )}} + \frac{\frac{a^{2} d^{4}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}} - \frac{2 \, a b d^{3}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}} + \frac{b^{2} d^{2}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}} + \frac{2 \, a c d^{2}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}} - \frac{2 \, b c d{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}} + \frac{c^{2}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}}{\sqrt{d x + 1}}}{4 \, d^{5}} - \frac{{\left (a^{2} d^{4} - 2 \, a b d^{3} + b^{2} d^{2} + 2 \, a c d^{2} - 2 \, b c d + c^{2}\right )} \sqrt{d x + 1}}{4 \, d^{5}{\left (\sqrt{2} - \sqrt{-d x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(2*b^2*d^17 + 4*a*c*d^17 + 3*c^2*d^15)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))
- 1/768*(a^2*d^19 + 2*a*b*d^18 + b^2*d^17 + 2*a*c*d^17 + 10*b*c*d^16 - c^2*d^15
- ((d*x + 1)*c^2*d^15 + 4*b*c*d^16 - 3*c^2*d^15)*(d*x + 1))*sqrt(d*x + 1)*sqrt(-
d*x + 1)/(d*x - 1) + 1/4*(a^2*d^4*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 2*a
*b*d^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + b^2*d^2*(sqrt(2) - sqrt(-d*x +
1))/sqrt(d*x + 1) + 2*a*c*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 2*b*c*
d*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + c^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt
(d*x + 1))/d^5 - 1/4*(a^2*d^4 - 2*a*b*d^3 + b^2*d^2 + 2*a*c*d^2 - 2*b*c*d + c^2)
*sqrt(d*x + 1)/(d^5*(sqrt(2) - sqrt(-d*x + 1)))