3.1805 \(\int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx\)

Optimal. Leaf size=86 \[ -\frac{a d+b c+2 b d x}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3} \]

[Out]

-((b*c + a*d + 2*b*d*x)/((b*c - a*d)^2*(a*c + (b*c + a*d)*x + b*d*x^2))) - (2*b*
d*Log[a + b*x])/(b*c - a*d)^3 + (2*b*d*Log[c + d*x])/(b*c - a*d)^3

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Rubi [A]  time = 0.0928303, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{a d+b c+2 b d x}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c + a*d + 2*b*d*x)/((b*c - a*d)^2*(a*c + (b*c + a*d)*x + b*d*x^2))) - (2*b*
d*Log[a + b*x])/(b*c - a*d)^3 + (2*b*d*Log[c + d*x])/(b*c - a*d)^3

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Rubi in Sympy [A]  time = 9.42887, size = 75, normalized size = 0.87 \[ \frac{4 b d \operatorname{atanh}{\left (\frac{a d + b c + 2 b d x}{a d - b c} \right )}}{\left (a d - b c\right )^{3}} - \frac{a d + b c + 2 b d x}{\left (a d - b c\right )^{2} \left (a c + b d x^{2} + x \left (a d + b c\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*b*d*atanh((a*d + b*c + 2*b*d*x)/(a*d - b*c))/(a*d - b*c)**3 - (a*d + b*c + 2*b
*d*x)/((a*d - b*c)**2*(a*c + b*d*x**2 + x*(a*d + b*c)))

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Mathematica [A]  time = 0.115812, size = 66, normalized size = 0.77 \[ \frac{\frac{b (a d-b c)}{a+b x}+\frac{d (a d-b c)}{c+d x}-2 b d \log (a+b x)+2 b d \log (c+d x)}{(b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

((b*(-(b*c) + a*d))/(a + b*x) + (d*(-(b*c) + a*d))/(c + d*x) - 2*b*d*Log[a + b*x
] + 2*b*d*Log[c + d*x])/(b*c - a*d)^3

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Maple [A]  time = 0.016, size = 82, normalized size = 1. \[ -{\frac{b}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}+2\,{\frac{\ln \left ( bx+a \right ) bd}{ \left ( ad-bc \right ) ^{3}}}-{\frac{d}{ \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-2\,{\frac{bd\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(a*d-b*c)^2/(b*x+a)+2*b/(a*d-b*c)^3*d*ln(b*x+a)-d/(a*d-b*c)^2/(d*x+c)-2*b/(a*
d-b*c)^3*d*ln(d*x+c)

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Maxima [A]  time = 0.786799, size = 281, normalized size = 3.27 \[ -\frac{2 \, b d \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, b d \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, b d x + b c + a d}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b*d*log(b*x + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 2*b*d*
log(d*x + c)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - (2*b*d*x + b*
c + a*d)/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a
^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x)

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Fricas [A]  time = 0.217953, size = 325, normalized size = 3.78 \[ -\frac{b^{2} c^{2} - a^{2} d^{2} + 2 \,{\left (b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b^2*c^2 - a^2*d^2 + 2*(b^2*c*d - a*b*d^2)*x + 2*(b^2*d^2*x^2 + a*b*c*d + (b^2*
c*d + a*b*d^2)*x)*log(b*x + a) - 2*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*
x)*log(d*x + c))/(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3 + (b
^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + (b^4*c^4 - 2*a*b
^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x)

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Sympy [A]  time = 4.77284, size = 405, normalized size = 4.71 \[ - \frac{2 b d \log{\left (x + \frac{- \frac{2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac{8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} - \frac{2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac{2 b d \log{\left (x + \frac{\frac{2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac{8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} + \frac{2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} - \frac{a d + b c + 2 b d x}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b*d*log(x + (-2*a**4*b*d**5/(a*d - b*c)**3 + 8*a**3*b**2*c*d**4/(a*d - b*c)**
3 - 12*a**2*b**3*c**2*d**3/(a*d - b*c)**3 + 8*a*b**4*c**3*d**2/(a*d - b*c)**3 +
2*a*b*d**2 - 2*b**5*c**4*d/(a*d - b*c)**3 + 2*b**2*c*d)/(4*b**2*d**2))/(a*d - b*
c)**3 + 2*b*d*log(x + (2*a**4*b*d**5/(a*d - b*c)**3 - 8*a**3*b**2*c*d**4/(a*d -
b*c)**3 + 12*a**2*b**3*c**2*d**3/(a*d - b*c)**3 - 8*a*b**4*c**3*d**2/(a*d - b*c)
**3 + 2*a*b*d**2 + 2*b**5*c**4*d/(a*d - b*c)**3 + 2*b**2*c*d)/(4*b**2*d**2))/(a*
d - b*c)**3 - (a*d + b*c + 2*b*d*x)/(a**3*c*d**2 - 2*a**2*b*c**2*d + a*b**2*c**3
+ x**2*(a**2*b*d**3 - 2*a*b**2*c*d**2 + b**3*c**2*d) + x*(a**3*d**3 - a**2*b*c*
d**2 - a*b**2*c**2*d + b**3*c**3))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError