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

Optimal. Leaf size=36 \[ \frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \]

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {616, 31} \begin {gather*} \frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.72 \begin {gather*} \frac {\log (a+b x)-\log (c+d x)}{b c-a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a c+(b c+a d) x+b d x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*c + (b*c + a*d)*x + b*d*x^2)^(-1),x]

[Out]

IntegrateAlgebraic[(a*c + (b*c + a*d)*x + b*d*x^2)^(-1), x]

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fricas [A]  time = 0.39, size = 26, normalized size = 0.72 \begin {gather*} \frac {\log \left (b x + a\right ) - \log \left (d x + c\right )}{b c - a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 46, normalized size = 1.28 \begin {gather*} \frac {b \log \left ({\left | b x + a \right |}\right )}{b^{2} c - a b d} - \frac {d \log \left ({\left | d x + c \right |}\right )}{b c d - a d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 37, normalized size = 1.03 \begin {gather*} -\frac {\ln \left (b x +a \right )}{a d -b c}+\frac {\ln \left (d x +c \right )}{a d -b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.11, size = 36, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.08, size = 40, normalized size = 1.11 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,d-b\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*2i)/(a*d - b*c)

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sympy [B]  time = 0.35, size = 128, normalized size = 3.56 \begin {gather*} \frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{a d - b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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