\(\int \frac {b d+2 c d x}{(a+b x+c x^2)^2} \, dx\) [1173]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 22, antiderivative size = 15 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a+b x+c x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643} \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a+b x+c x^2} \]

[In]

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

[Out]

-(d/(a + b*x + c*x^2))

Rule 643

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*} \text {integral}& = -\frac {d}{a+b x+c x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a+x (b+c x)} \]

[In]

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

[Out]

-(d/(a + x*(b + c*x)))

Maple [A] (verified)

Time = 2.81 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method result size
gosper \(-\frac {d}{c \,x^{2}+b x +a}\) \(16\)
default \(-\frac {d}{c \,x^{2}+b x +a}\) \(16\)
risch \(-\frac {d}{c \,x^{2}+b x +a}\) \(16\)
parallelrisch \(\frac {c d \,x^{2}+b d x}{a \left (c \,x^{2}+b x +a \right )}\) \(28\)
norman \(\frac {\frac {b x d}{a}+\frac {c d \,x^{2}}{a}}{c \,x^{2}+b x +a}\) \(31\)

[In]

int((2*c*d*x+b*d)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{c x^{2} + b x + a} \]

[In]

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

[Out]

-d/(c*x^2 + b*x + a)

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=- \frac {d}{a + b x + c x^{2}} \]

[In]

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

[Out]

-d/(a + b*x + c*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{c x^{2} + b x + a} \]

[In]

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

[Out]

-d/(c*x^2 + b*x + a)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a + \frac {c d x^{2} + b d x}{d}} \]

[In]

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

[Out]

-d/(a + (c*d*x^2 + b*d*x)/d)

Mupad [B] (verification not implemented)

Time = 9.52 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{c\,x^2+b\,x+a} \]

[In]

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

[Out]

-d/(a + b*x + c*x^2)