3.1150 \(\int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=13 \[ d \log \left (a+b x+c x^2\right ) \]

[Out]

d*Log[a + b*x + c*x^2]

_______________________________________________________________________________________

Rubi [A]  time = 0.0135273, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ d \log \left (a+b x+c x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

d*Log[a + b*x + c*x^2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.81599, size = 12, normalized size = 0.92 \[ d \log{\left (a + b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(a + b*x + c*x**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00380972, size = 12, normalized size = 0.92 \[ d \log (a+x (b+c x)) \]

Antiderivative was successfully verified.

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

[Out]

d*Log[a + x*(b + c*x)]

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 14, normalized size = 1.1 \[ d\ln \left ( c{x}^{2}+bx+a \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.699025, size = 18, normalized size = 1.38 \[ d \log \left (c x^{2} + b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(c*x^2 + b*x + a)

_______________________________________________________________________________________

Fricas [A]  time = 0.204295, size = 18, normalized size = 1.38 \[ d \log \left (c x^{2} + b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(c*x^2 + b*x + a)

_______________________________________________________________________________________

Sympy [A]  time = 1.33729, size = 12, normalized size = 0.92 \[ d \log{\left (a + b x + c x^{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(a + b*x + c*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215676, size = 18, normalized size = 1.38 \[ d{\rm ln}\left (c x^{2} + b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*ln(c*x^2 + b*x + a)