3.2.13 \(\int \frac {b+2 c x}{-a+b x+c x^2} \, dx\) [113]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {642} \begin {gather*} \log \left (a-b x-c x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(-a + b*x + c*x^2),x]

[Out]

Log[a - b*x - c*x^2]

Rule 642

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} \log (-a+x (b+c x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-a + x*(b + c*x)]

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Maple [A]
time = 0.22, size = 14, normalized size = 1.08

method	result	size
derivativedivides	\(\ln \left (c \,x^{2}+b x -a \right )\)	\(14\)
default	\(\ln \left (-c \,x^{2}-b x +a \right )\)	\(14\)
norman	\(\ln \left (-c \,x^{2}-b x +a \right )\)	\(14\)
risch	\(\ln \left (-c \,x^{2}-b x +a \right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 13, normalized size = 1.00 \begin {gather*} \log \left (c x^{2} + b x - a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x - a)

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Fricas [A]
time = 0.34, size = 13, normalized size = 1.00 \begin {gather*} \log \left (c x^{2} + b x - a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^2 + b*x - a)

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Sympy [A]
time = 0.06, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (- a + b x + c x^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-a + b*x + c*x**2)

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Giac [A]
time = 2.77, size = 14, normalized size = 1.08 \begin {gather*} \log \left ({\left | c x^{2} + b x - a \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(c*x^2 + b*x - a))

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Mupad [B]
time = 0.05, size = 13, normalized size = 1.00 \begin {gather*} \ln \left (c\,x^2+b\,x-a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/(b*x - a + c*x^2),x)

[Out]

log(b*x - a + c*x^2)

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