3.1536 \(\int \frac{b+2 c x}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{a+b x+c x^2} \]

[Out]

-(a + b*x + c*x^2)^(-1)

________________________________________________________________________________________

Rubi [A]  time = 0.0046043, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {629} \[ -\frac{1}{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*x + c*x^2)^(-1)

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{b+2 c x}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{1}{a+b x+c x^2}\\ \end{align*}

Mathematica [A]  time = 0.0038105, size = 13, normalized size = 0.93 \[ -\frac{1}{a+x (b+c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + x*(b + c*x))^(-1)

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 15, normalized size = 1.1 \begin{align*} - \left ( c{x}^{2}+bx+a \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/(c*x^2+b*x+a)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02049, size = 19, normalized size = 1.36 \begin{align*} -\frac{1}{c x^{2} + b x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(c*x^2 + b*x + a)

________________________________________________________________________________________

Fricas [A]  time = 1.2023, size = 30, normalized size = 2.14 \begin{align*} -\frac{1}{c x^{2} + b x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(c*x^2 + b*x + a)

________________________________________________________________________________________

Sympy [A]  time = 0.542564, size = 12, normalized size = 0.86 \begin{align*} - \frac{1}{a + b x + c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x**2+b*x+a)**2,x)

[Out]

-1/(a + b*x + c*x**2)

________________________________________________________________________________________

Giac [A]  time = 1.12618, size = 19, normalized size = 1.36 \begin{align*} -\frac{1}{c x^{2} + b x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(c*x^2 + b*x + a)