3.1163 \(\int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{4 c \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^2}+\frac{8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \]

[Out]

-(1/((b^2 - 4*a*c)*d*(a + b*x + c*x^2))) + (8*c*Log[b + 2*c*x])/((b^2 - 4*a*c)^2
*d) - (4*c*Log[a + b*x + c*x^2])/((b^2 - 4*a*c)^2*d)

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Rubi [A]  time = 0.108892, antiderivative size = 78, 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 \[ -\frac{1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{4 c \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^2}+\frac{8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(1/((b^2 - 4*a*c)*d*(a + b*x + c*x^2))) + (8*c*Log[b + 2*c*x])/((b^2 - 4*a*c)^2
*d) - (4*c*Log[a + b*x + c*x^2])/((b^2 - 4*a*c)^2*d)

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Rubi in Sympy [A]  time = 37.6992, size = 71, normalized size = 0.91 \[ \frac{8 c \log{\left (b + 2 c x \right )}}{d \left (- 4 a c + b^{2}\right )^{2}} - \frac{4 c \log{\left (a + b x + c x^{2} \right )}}{d \left (- 4 a c + b^{2}\right )^{2}} - \frac{1}{d \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*c*log(b + 2*c*x)/(d*(-4*a*c + b**2)**2) - 4*c*log(a + b*x + c*x**2)/(d*(-4*a*c
+ b**2)**2) - 1/(d*(-4*a*c + b**2)*(a + b*x + c*x**2))

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Mathematica [A]  time = 0.111587, size = 59, normalized size = 0.76 \[ \frac{-\frac{b^2-4 a c}{a+x (b+c x)}-4 c \log (a+x (b+c x))+8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 119, normalized size = 1.5 \[ 8\,{\frac{c\ln \left ( 2\,cx+b \right ) }{d \left ( 4\,ac-{b}^{2} \right ) ^{2}}}+4\,{\frac{ac}{d \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{{b}^{2}}{d \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) }{d \left ( 4\,ac-{b}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.690562, size = 163, normalized size = 2.09 \[ -\frac{4 \, c \log \left (c x^{2} + b x + a\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d} + \frac{8 \, c \log \left (2 \, c x + b\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d} - \frac{1}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*c*log(c*x^2 + b*x + a)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*d) + 8*c*log(2*c*x + b
)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*d) - 1/((b^2*c - 4*a*c^2)*d*x^2 + (b^3 - 4*a*b
*c)*d*x + (a*b^2 - 4*a^2*c)*d)

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Fricas [A]  time = 0.213088, size = 190, normalized size = 2.44 \[ -\frac{b^{2} - 4 \, a c + 4 \,{\left (c^{2} x^{2} + b c x + a c\right )} \log \left (c x^{2} + b x + a\right ) - 8 \,{\left (c^{2} x^{2} + b c x + a c\right )} \log \left (2 \, c x + b\right )}{{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d x +{\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b^2 - 4*a*c + 4*(c^2*x^2 + b*c*x + a*c)*log(c*x^2 + b*x + a) - 8*(c^2*x^2 + b*
c*x + a*c)*log(2*c*x + b))/((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*x^2 + (b^5 - 8*
a*b^3*c + 16*a^2*b*c^2)*d*x + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*d)

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Sympy [A]  time = 6.91986, size = 102, normalized size = 1.31 \[ \frac{8 c \log{\left (\frac{b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )^{2}} - \frac{4 c \log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )^{2}} + \frac{1}{4 a^{2} c d - a b^{2} d + x^{2} \left (4 a c^{2} d - b^{2} c d\right ) + x \left (4 a b c d - b^{3} d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*c*log(b/(2*c) + x)/(d*(4*a*c - b**2)**2) - 4*c*log(a/c + b*x/c + x**2)/(d*(4*a
*c - b**2)**2) + 1/(4*a**2*c*d - a*b**2*d + x**2*(4*a*c**2*d - b**2*c*d) + x*(4*
a*b*c*d - b**3*d))

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GIAC/XCAS [A]  time = 0.218605, size = 146, normalized size = 1.87 \[ \frac{8 \, c^{2}{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{b^{4} c d - 8 \, a b^{2} c^{2} d + 16 \, a^{2} c^{3} d} - \frac{4 \, c{\rm ln}\left (c x^{2} + b x + a\right )}{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d} - \frac{1}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*c^2*ln(abs(2*c*x + b))/(b^4*c*d - 8*a*b^2*c^2*d + 16*a^2*c^3*d) - 4*c*ln(c*x^2
+ b*x + a)/(b^4*d - 8*a*b^2*c*d + 16*a^2*c^2*d) - 1/((c*x^2 + b*x + a)*(b^2 - 4
*a*c)*d)