3.1162 \(\int \frac{1}{(b d+2 c d x) (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=48 \[ \frac{\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac{2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]

[Out]

(-2*Log[b + 2*c*x])/((b^2 - 4*a*c)*d) + Log[a + b*x + c*x^2]/((b^2 - 4*a*c)*d)

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Rubi [A]  time = 0.0233232, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {681, 31, 628} \[ \frac{\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac{2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)),x]

[Out]

(-2*Log[b + 2*c*x])/((b^2 - 4*a*c)*d) + Log[a + b*x + c*x^2]/((b^2 - 4*a*c)*d)

Rule 681

Int[1/(((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[(-4*b*c)/(d*(b^2 - 4*a*c)),
Int[1/(b + 2*c*x), x], x] + Dist[b^2/(d^2*(b^2 - 4*a*c)), Int[(d + e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 628

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{1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx &=\frac{\int \frac{b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) d^2}-\frac{(4 c) \int \frac{1}{b+2 c x} \, dx}{\left (b^2-4 a c\right ) d}\\ &=-\frac{2 \log (b+2 c x)}{\left (b^2-4 a c\right ) d}+\frac{\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right ) d}\\ \end{align*}

Mathematica [A]  time = 0.0203116, size = 34, normalized size = 0.71 \[ \frac{\log (a+x (b+c x))-2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)),x]

[Out]

(-2*Log[b + 2*c*x] + Log[a + x*(b + c*x)])/((b^2 - 4*a*c)*d)

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Maple [A]  time = 0.044, size = 54, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{d \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{\ln \left ( 2\,cx+b \right ) }{d \left ( 4\,ac-{b}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*c*d*x+b*d)/(c*x^2+b*x+a),x)

[Out]

-1/d/(4*a*c-b^2)*ln(c*x^2+b*x+a)+2/d/(4*a*c-b^2)*ln(2*c*x+b)

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Maxima [A]  time = 1.20289, size = 65, normalized size = 1.35 \begin{align*} \frac{\log \left (c x^{2} + b x + a\right )}{{\left (b^{2} - 4 \, a c\right )} d} - \frac{2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

log(c*x^2 + b*x + a)/((b^2 - 4*a*c)*d) - 2*log(2*c*x + b)/((b^2 - 4*a*c)*d)

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Fricas [A]  time = 1.64533, size = 82, normalized size = 1.71 \begin{align*} \frac{\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(log(c*x^2 + b*x + a) - 2*log(2*c*x + b))/((b^2 - 4*a*c)*d)

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Sympy [A]  time = 0.92411, size = 42, normalized size = 0.88 \begin{align*} \frac{2 \log{\left (\frac{b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )} - \frac{\log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a),x)

[Out]

2*log(b/(2*c) + x)/(d*(4*a*c - b**2)) - log(a/c + b*x/c + x**2)/(d*(4*a*c - b**2))

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Giac [A]  time = 1.19323, size = 77, normalized size = 1.6 \begin{align*} -\frac{2 \, c^{2} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{2} c^{2} d - 4 \, a c^{3} d} + \frac{\log \left (c x^{2} + b x + a\right )}{b^{2} d - 4 \, a c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

-2*c^2*log(abs(2*c*x + b))/(b^2*c^2*d - 4*a*c^3*d) + log(c*x^2 + b*x + a)/(b^2*d - 4*a*c*d)