3.12.61 \(\int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\) [1161]

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

[Out]

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

________________________________________________________________________________________

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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {642} \begin {gather*} d \log \left (a+b x+c x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} d \log (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),x]

[Out]

d*Log[a + x*(b + c*x)]

________________________________________________________________________________________

Maple [A]
time = 0.62, size = 14, normalized size = 1.08

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 13, normalized size = 1.00 \begin {gather*} d \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*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

d*log(c*x^2 + b*x + a)

________________________________________________________________________________________

Fricas [A]
time = 1.66, size = 13, normalized size = 1.00 \begin {gather*} d \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*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

d*log(c*x^2 + b*x + a)

________________________________________________________________________________________

Sympy [A]
time = 0.08, size = 12, normalized size = 0.92 \begin {gather*} d \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*d*x+b*d)/(c*x**2+b*x+a),x)

[Out]

d*log(a + b*x + c*x**2)

________________________________________________________________________________________

Giac [A]
time = 1.15, size = 21, normalized size = 1.62 \begin {gather*} d \log \left ({\left | a + \frac {c d x^{2} + b d x}{d} \right |}\right ) \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),x, algorithm="giac")

[Out]

d*log(abs(a + (c*d*x^2 + b*d*x)/d))

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 13, normalized size = 1.00 \begin {gather*} d\,\ln \left (c\,x^2+b\,x+a\right ) \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),x)

[Out]

d*log(a + b*x + c*x^2)

________________________________________________________________________________________