3.1151 \(\int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=48 \[ \frac{\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac{2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]

[Out]

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

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Rubi [A]  time = 0.0820558, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac{2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.5479, size = 41, normalized size = 0.85 \[ - \frac{2 \log{\left (b + 2 c x \right )}}{d \left (- 4 a c + b^{2}\right )} + \frac{\log{\left (a + b x + c x^{2} \right )}}{d \left (- 4 a c + b^{2}\right )} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0391723, size = 34, normalized size = 0.71 \[ \frac{\log (a+x (b+c x))-2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 54, normalized size = 1.1 \[ 2\,{\frac{\ln \left ( 2\,cx+b \right ) }{d \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{d \left ( 4\,ac-{b}^{2} \right ) }} \]

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),x)

[Out]

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

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Maxima [A]  time = 0.685866, size = 65, normalized size = 1.35 \[ \frac{\log \left (c x^{2} + b x + a\right )}{{\left (b^{2} - 4 \, a c\right )} d} - \frac{2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \]

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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.210065, size = 47, normalized size = 0.98 \[ \frac{\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \]

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)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.54545, size = 42, normalized size = 0.88 \[ \frac{2 \log{\left (\frac{b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )} - \frac{\log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218229, size = 77, normalized size = 1.6 \[ -\frac{2 \, c^{2}{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{b^{2} c^{2} d - 4 \, a c^{3} d} + \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{b^{2} d - 4 \, a c d} \]

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)),x, algorithm="giac")

[Out]

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