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

Optimal. Leaf size=443 \[ \frac{c \left (-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac{d^2 x \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac{c \left (-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac{d^2 x \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac{d^2 \left (b-x \left (a d^2+c\right )\right )}{\sqrt{1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 3.21421, antiderivative size = 443, 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 \[ \frac{c \left (-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac{d^2 x \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac{c \left (-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2\right ) \tanh ^{-1}\left (\frac{d^2 x \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac{d^2 \left (b-x \left (a d^2+c\right )\right )}{\sqrt{1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.60469, size = 658, normalized size = 1.49 \[ \frac{c \sqrt{2-2 d^2 x^2} \left (-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right )^{3/2} \log \left (\sqrt{b^2-4 a c}-b-2 c x\right )-c \sqrt{2-2 d^2 x^2} \left (b d^2 \left (\sqrt{b^2-4 a c}-b\right )+2 a c d^2+2 c^2\right )^{3/2} \log \left (\sqrt{b^2-4 a c}+b+2 c x\right )-c \sqrt{2-2 d^2 x^2} \left (-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2\right )^{3/2} \log \left (-\sqrt{1-d^2 x^2} \sqrt{2 b d^2 \left (\sqrt{b^2-4 a c}-b\right )+4 a c d^2+4 c^2}+d^2 x \sqrt{b^2-4 a c}-b d^2 x-2 c\right )+c \sqrt{2-2 d^2 x^2} \left (b d^2 \left (\sqrt{b^2-4 a c}-b\right )+2 a c d^2+2 c^2\right )^{3/2} \log \left (\sqrt{1-d^2 x^2} \sqrt{-2 b d^2 \left (\sqrt{b^2-4 a c}+b\right )+4 a c d^2+4 c^2}+d^2 x \sqrt{b^2-4 a c}+b d^2 x+2 c\right )-2 d^2 \sqrt{b^2-4 a c} \sqrt{b d^2 \left (\sqrt{b^2-4 a c}-b\right )+2 a c d^2+2 c^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b-x \left (a d^2+c\right )\right )}{2 \sqrt{1-d^2 x^2} \sqrt{b^2-4 a c} \sqrt{b d^2 \left (\sqrt{b^2-4 a c}-b\right )+2 a c d^2+2 c^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (a^2 d^4+2 a c d^2-b^2 d^2+c^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b^2 - 4*a*c]*d^2*Sqrt[2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c])*d^
2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*(b - (c + a*d^2)*x) +
c*(2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2)^(3/2)*Sqrt[2 - 2*d^2*x^2]
*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x] - c*(2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 -
4*a*c])*d^2)^(3/2)*Sqrt[2 - 2*d^2*x^2]*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x] - c*(
2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2)^(3/2)*Sqrt[2 - 2*d^2*x^2]*Log
[-2*c - b*d^2*x + Sqrt[b^2 - 4*a*c]*d^2*x - Sqrt[4*c^2 + 4*a*c*d^2 + 2*b*(-b + S
qrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2]] + c*(2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[
b^2 - 4*a*c])*d^2)^(3/2)*Sqrt[2 - 2*d^2*x^2]*Log[2*c + b*d^2*x + Sqrt[b^2 - 4*a*
c]*d^2*x + Sqrt[4*c^2 + 4*a*c*d^2 - 2*b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^
2*x^2]])/(2*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 + b*(-b + Sqrt[b^2 - 4*a*c]
)*d^2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*(c^2 - b^2*d^2 +
2*a*c*d^2 + a^2*d^4)*Sqrt[1 - d^2*x^2])

_______________________________________________________________________________________

Maple [C]  time = 0.13, size = 11142, normalized size = 25.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out