3.1164 \(\int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=98 \[ -\frac{1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}-\frac{12 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{12 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}} \]

[Out]

(-12*c)/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)) - 1/((b^2 - 4*a*c)*d^2*(b + 2*c*x)*(a
+ b*x + c*x^2)) + (12*c*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(
5/2)*d^2)

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Rubi [A]  time = 0.143872, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}-\frac{12 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{12 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-12*c)/((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)) - 1/((b^2 - 4*a*c)*d^2*(b + 2*c*x)*(a
+ b*x + c*x^2)) + (12*c*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(
5/2)*d^2)

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Rubi in Sympy [A]  time = 37.1828, size = 92, normalized size = 0.94 \[ \frac{12 c \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{2} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{12 c}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2}} - \frac{1}{d^{2} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*c*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(d**2*(-4*a*c + b**2)**(5/2)) - 12*c
/(d**2*(b + 2*c*x)*(-4*a*c + b**2)**2) - 1/(d**2*(b + 2*c*x)*(-4*a*c + b**2)*(a
+ b*x + c*x**2))

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Mathematica [A]  time = 0.239095, size = 84, normalized size = 0.86 \[ -\frac{\frac{12 c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}+\frac{8 c}{b+2 c x}}{d^2 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(((8*c)/(b + 2*c*x) + (b + 2*c*x)/(a + x*(b + c*x)) + (12*c*ArcTan[(b + 2*c*x)/
Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/((b^2 - 4*a*c)^2*d^2))

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Maple [A]  time = 0.016, size = 127, normalized size = 1.3 \[ -8\,{\frac{c}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \right ) }}-2\,{\frac{cx}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{b}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-12\,{\frac{c}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/d^2*c/(4*a*c-b^2)^2/(2*c*x+b)-2/d^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)*c*x-1/d^2/(4*
a*c-b^2)^2/(c*x^2+b*x+a)*b-12/d^2/(4*a*c-b^2)^(5/2)*c*arctan((2*c*x+b)/(4*a*c-b^
2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228622, size = 1, normalized size = 0.01 \[ \left [\frac{6 \,{\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c +{\left (b^{2} c + 2 \, a c^{2}\right )} x\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{{\left (2 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x^{3} + 3 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{2} x^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d^{2} x +{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{12 \,{\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c +{\left (b^{2} c + 2 \, a c^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}}{{\left (2 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x^{3} + 3 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{2} x^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d^{2} x +{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(6*(2*c^3*x^3 + 3*b*c^2*x^2 + a*b*c + (b^2*c + 2*a*c^2)*x)*log((b^3 - 4*a*b*c +
2*(b^2*c - 4*a*c^2)*x + (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/
(c*x^2 + b*x + a)) - (12*c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)*sqrt(b^2 - 4*a*c))/((
2*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*x^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2
*b*c^3)*d^2*x^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d^2*x + (a*b^5 - 8*a^2*b^3*c +
16*a^3*b*c^2)*d^2)*sqrt(b^2 - 4*a*c)), -(12*(2*c^3*x^3 + 3*b*c^2*x^2 + a*b*c + (
b^2*c + 2*a*c^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (12*
c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)*sqrt(-b^2 + 4*a*c))/((2*(b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4)*d^2*x^3 + 3*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^2*x^2 + (b^6 -
6*a*b^4*c + 32*a^3*c^3)*d^2*x + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d^2)*sqrt(
-b^2 + 4*a*c))]

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Sympy [A]  time = 8.82031, size = 457, normalized size = 4.66 \[ \frac{6 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{- 384 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} - \frac{6 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{384 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 6 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} - \frac{8 a c + b^{2} + 12 b c x + 12 c^{2} x^{2}}{16 a^{3} b c^{2} d^{2} - 8 a^{2} b^{3} c d^{2} + a b^{5} d^{2} + x^{3} \left (32 a^{2} c^{4} d^{2} - 16 a b^{2} c^{3} d^{2} + 2 b^{4} c^{2} d^{2}\right ) + x^{2} \left (48 a^{2} b c^{3} d^{2} - 24 a b^{3} c^{2} d^{2} + 3 b^{5} c d^{2}\right ) + x \left (32 a^{3} c^{3} d^{2} - 6 a b^{4} c d^{2} + b^{6} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (-384*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5
) + 288*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5) - 72*a*b**4*c**2*sqrt(-1/(4*a*
c - b**2)**5) + 6*b**6*c*sqrt(-1/(4*a*c - b**2)**5) + 6*b*c)/(12*c**2))/d**2 - 6
*c*sqrt(-1/(4*a*c - b**2)**5)*log(x + (384*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5)
- 288*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5) + 72*a*b**4*c**2*sqrt(-1/(4*a*c
- b**2)**5) - 6*b**6*c*sqrt(-1/(4*a*c - b**2)**5) + 6*b*c)/(12*c**2))/d**2 - (8*
a*c + b**2 + 12*b*c*x + 12*c**2*x**2)/(16*a**3*b*c**2*d**2 - 8*a**2*b**3*c*d**2
+ a*b**5*d**2 + x**3*(32*a**2*c**4*d**2 - 16*a*b**2*c**3*d**2 + 2*b**4*c**2*d**2
) + x**2*(48*a**2*b*c**3*d**2 - 24*a*b**3*c**2*d**2 + 3*b**5*c*d**2) + x*(32*a**
3*c**3*d**2 - 6*a*b**4*c*d**2 + b**6*d**2))

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GIAC/XCAS [A]  time = 0.217138, size = 297, normalized size = 3.03 \[ -\frac{8 \, c^{5} d^{7}}{{\left (b^{4} c^{4} d^{8} - 8 \, a b^{2} c^{5} d^{8} + 16 \, a^{2} c^{6} d^{8}\right )}{\left (2 \, c d x + b d\right )}} - \frac{12 \, c \arctan \left (\frac{\frac{b^{2} d}{2 \, c d x + b d} - \frac{4 \, a c d}{2 \, c d x + b d}}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c} d^{2}} + \frac{4 \, c}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}{\left (\frac{b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac{4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*c^5*d^7/((b^4*c^4*d^8 - 8*a*b^2*c^5*d^8 + 16*a^2*c^6*d^8)*(2*c*d*x + b*d)) -
12*c*arctan((b^2*d/(2*c*d*x + b*d) - 4*a*c*d/(2*c*d*x + b*d))/sqrt(-b^2 + 4*a*c)
)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)*d^2) + 4*c/((b^4 - 8*a*b^2*
c + 16*a^2*c^2)*(2*c*d*x + b*d)*(b^2*d^2/(2*c*d*x + b*d)^2 - 4*a*c*d^2/(2*c*d*x
+ b*d)^2 - 1)*d)