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

Optimal. Leaf size=10 \[ \frac {\log (c+d x)}{d} \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {626, 31} \begin {gather*} \frac {\log (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c + d*x]/d

Rule 31

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

Rule 626

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c + d*x]/d

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(d*x + c)/d

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giac [A]  time = 0.15, size = 11, normalized size = 1.10 \begin {gather*} \frac {\log \left ({\left | d x + c \right |}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(d*x + c))/d

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maple [A]  time = 0.04, size = 11, normalized size = 1.10 \begin {gather*} \frac {\ln \left (d x +c \right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(d*x+c)/d

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maxima [A]  time = 0.97, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(d*x + c)/d

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mupad [B]  time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} \frac {\ln \left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c + d*x)/d

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sympy [A]  time = 0.08, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (c + d x \right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c + d*x)/d

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