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

Optimal. Leaf size=15 \[ -\frac {d}{a+b x+c x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643} \begin {gather*} -\frac {d}{a+b x+c x^2} \end {gather*}

Antiderivative was successfully verified.

[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*} \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}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} -\frac {d}{a+x (b+c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(d/(a + x*(b + c*x)))

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Maple [A]
time = 0.74, size = 16, normalized size = 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	\(-\frac {d}{c \,x^{2}+b x +a}\)	\(16\)
norman	\(\frac {\frac {b x d}{a}+\frac {c d \,x^{2}}{a}}{c \,x^{2}+b x +a}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.28, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{c x^{2} + b x + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 1.54, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{c x^{2} + b x + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.21, size = 12, normalized size = 0.80 \begin {gather*} - \frac {d}{a + b x + c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/(a + b*x + c*x**2)

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Giac [A]
time = 1.07, size = 22, normalized size = 1.47 \begin {gather*} -\frac {d}{a + \frac {c d x^{2} + b d x}{d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.42, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{c\,x^2+b\,x+a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/(a + b*x + c*x^2)

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