3.16.36 \(\int \frac {b+2 c x}{(a+b x+c x^2)^2} \, dx\) [1536]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {643} \begin {gather*} -\frac {1}{a+b x+c x^2} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} -\frac {1}{a+x (b+c x)} \end {gather*}

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)

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Maple [A]
time = 0.81, size = 15, normalized size = 1.07

method	result	size
gosper	\(-\frac {1}{c \,x^{2}+b x +a}\)	\(15\)
derivativedivides	\(-\frac {1}{c \,x^{2}+b x +a}\)	\(15\)
default	\(-\frac {1}{c \,x^{2}+b x +a}\)	\(15\)
risch	\(-\frac {1}{c \,x^{2}+b x +a}\)	\(15\)
norman	\(\frac {\frac {b x}{a}+\frac {c \,x^{2}}{a}}{c \,x^{2}+b x +a}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{c x^{2} + b x + a} \end {gather*}

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)

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Fricas [A]
time = 2.54, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{c x^{2} + b x + a} \end {gather*}

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)

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Sympy [A]
time = 0.20, size = 12, normalized size = 0.86 \begin {gather*} - \frac {1}{a + b x + c x^{2}} \end {gather*}

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)

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Giac [A]
time = 3.37, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{c x^{2} + b x + a} \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{c\,x^2+b\,x+a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a + b*x + c*x^2)

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