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

Optimal. Leaf size=571 \[ -\frac{c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt{b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt{b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac{c \left (-4 c d^2 \left (b^2-2 a^2 d^2\right )-b d^2 \left (\sqrt{b^2-4 a c}+b\right ) \left (c-a d^2\right )-2 a b^2 d^4+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac{\sqrt{1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )} \]

[Out]

-(((b*(b^2*d^2 - c*(c + 3*a*d^2)) - c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)*Sqrt[1 -
d^2*x^2])/((b^2 - 4*a*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2))) - (c*(4*c
^3 + 12*a*c^2*d^2 - a*b*(b + Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*(5*b^2 - b*Sqrt[b^2
- 4*a*c] - 8*a^2*d^2))*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sq
rt[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]*(b^2 - 4*a*c)^(3/2)*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*(4*c^3 + 12*a*c^2*d^2 - 2*a*b^2*d^4 - b*(b + Sqr
t[b^2 - 4*a*c])*d^2*(c - a*d^2) - 4*c*d^2*(b^2 - 2*a^2*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]*(b^2 - 4*a*c)^(3/2)*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))

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Rubi [A]  time = 10.4805, antiderivative size = 571, 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 \[ -\frac{c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt{b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt{b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (b-\sqrt{b^2-4 a c}\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}+\frac{c \left (-4 c d^2 \left (b^2-2 a^2 d^2\right )-b d^2 \left (\sqrt{b^2-4 a c}+b\right ) \left (c-a d^2\right )-2 a b^2 d^4+12 a c^2 d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{d^2 x \left (\sqrt{b^2-4 a c}+b\right )+2 c}{\sqrt{2} \sqrt{1-d^2 x^2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{-b d^2 \left (\sqrt{b^2-4 a c}+b\right )+2 a c d^2+2 c^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac{\sqrt{1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(((b*(b^2*d^2 - c*(c + 3*a*d^2)) - c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)*Sqrt[1 -
d^2*x^2])/((b^2 - 4*a*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2))) - (c*(4*c
^3 + 12*a*c^2*d^2 - a*b*(b + Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*(5*b^2 - b*Sqrt[b^2
- 4*a*c] - 8*a^2*d^2))*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sq
rt[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]*(b^2 - 4*a*c)^(3/2)*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*(4*c^3 + 12*a*c^2*d^2 - 2*a*b^2*d^4 - b*(b + Sqr
t[b^2 - 4*a*c])*d^2*(c - a*d^2) - 4*c*d^2*(b^2 - 2*a^2*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]*(b^2 - 4*a*c)^(3/2)*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))

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

[Out]

Timed out

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Mathematica [A]  time = 4.03246, size = 800, normalized size = 1.4 \[ -\frac{-\frac{2 \sqrt{1-d^2 x^2} \left (d^2 b^3+c d^2 x b^2-c \left (3 a d^2+c\right ) b-2 c^2 \left (a d^2+c\right ) x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{c \left (-a b \left (b+\sqrt{b^2-4 a c}\right ) d^4+12 a c^2 d^2+c \left (-5 b^2+\sqrt{b^2-4 a c} b+8 a^2 d^2\right ) d^2+4 c^3\right ) \log \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{c^2+a d^2 c+\frac{1}{2} b \left (\sqrt{b^2-4 a c}-b\right ) d^2}}+\frac{c \left (a b \left (b-\sqrt{b^2-4 a c}\right ) d^4-12 a c^2 d^2+c \left (5 b^2+\sqrt{b^2-4 a c} b-8 a^2 d^2\right ) d^2-4 c^3\right ) \log \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{c^2+a d^2 c-\frac{1}{2} b \left (b+\sqrt{b^2-4 a c}\right ) d^2}}-\frac{c \left (-a b \left (b+\sqrt{b^2-4 a c}\right ) d^4+12 a c^2 d^2+c \left (-5 b^2+\sqrt{b^2-4 a c} b+8 a^2 d^2\right ) d^2+4 c^3\right ) \log \left (-b x d^2+\sqrt{b^2-4 a c} x d^2-2 c-\sqrt{4 c^2+4 a d^2 c+2 b \left (\sqrt{b^2-4 a c}-b\right ) d^2} \sqrt{1-d^2 x^2}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{c^2+a d^2 c+\frac{1}{2} b \left (\sqrt{b^2-4 a c}-b\right ) d^2}}+\frac{c \left (a b \left (\sqrt{b^2-4 a c}-b\right ) d^4+12 a c^2 d^2+c \left (-5 b^2-\sqrt{b^2-4 a c} b+8 a^2 d^2\right ) d^2+4 c^3\right ) \log \left (b x d^2+\sqrt{b^2-4 a c} x d^2+2 c+\sqrt{4 c^2+4 a d^2 c-2 b \left (b+\sqrt{b^2-4 a c}\right ) d^2} \sqrt{1-d^2 x^2}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{c^2+a d^2 c-\frac{1}{2} b \left (b+\sqrt{b^2-4 a c}\right ) d^2}}}{2 \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((-2*(b^3*d^2 - b*c*(c + 3*a*d^2) + b^2*c*d^2*x - 2*c^2*(c + a*d^2)*x)*Sqrt[1 -
d^2*x^2])/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (c*(4*c^3 + 12*a*c^2*d^2 - a*b*(b
+ Sqrt[b^2 - 4*a*c])*d^4 + c*d^2*(-5*b^2 + b*Sqrt[b^2 - 4*a*c] + 8*a^2*d^2))*Lo
g[-b + Sqrt[b^2 - 4*a*c] - 2*c*x])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 + (b*
(-b + Sqrt[b^2 - 4*a*c])*d^2)/2]) + (c*(-4*c^3 - 12*a*c^2*d^2 + a*b*(b - Sqrt[b^
2 - 4*a*c])*d^4 + c*d^2*(5*b^2 + b*Sqrt[b^2 - 4*a*c] - 8*a^2*d^2))*Log[b + Sqrt[
b^2 - 4*a*c] + 2*c*x])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 - (b*(b + Sqrt[b^
2 - 4*a*c])*d^2)/2]) - (c*(4*c^3 + 12*a*c^2*d^2 - a*b*(b + Sqrt[b^2 - 4*a*c])*d^
4 + c*d^2*(-5*b^2 + b*Sqrt[b^2 - 4*a*c] + 8*a^2*d^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]])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 + (b*(-b + Sqrt[b^2
- 4*a*c])*d^2)/2]) + (c*(4*c^3 + 12*a*c^2*d^2 + a*b*(-b + Sqrt[b^2 - 4*a*c])*d^4
+ c*d^2*(-5*b^2 - b*Sqrt[b^2 - 4*a*c] + 8*a^2*d^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]*Sqr
t[1 - d^2*x^2]])/((b^2 - 4*a*c)^(3/2)*Sqrt[c^2 + a*c*d^2 - (b*(b + Sqrt[b^2 - 4*
a*c])*d^2)/2]))/(2*(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4))

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Maple [C]  time = 0.77, size = 41837, normalized size = 73.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^2*sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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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)^2*sqrt(d*x + 1)*sqrt(-d*x + 1)),x, algorithm="giac")

[Out]

Timed out