### 3.1804 $$\int \frac{a+b x}{a c+(b c+a d) x+b d x^2} \, dx$$

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

[Out]

Log[c + d*x]/d

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Rubi [A]  time = 0.0050334, antiderivative size = 10, normalized size of antiderivative = 1., 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} $\frac{\log (c+d x)}{d}$

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 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]

Rule 31

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

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*}

Mathematica [A]  time = 0.0011979, size = 10, normalized size = 1. $\frac{\log (c+d x)}{d}$

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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Maple [A]  time = 0.039, size = 11, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{d}} \end{align*}

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 = 1.04392, size = 14, normalized size = 1.4 \begin{align*} \frac{\log \left (d x + c\right )}{d} \end{align*}

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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Fricas [A]  time = 1.40938, size = 22, normalized size = 2.2 \begin{align*} \frac{\log \left (d x + c\right )}{d} \end{align*}

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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Sympy [A]  time = 0.139582, size = 7, normalized size = 0.7 \begin{align*} \frac{\log{\left (c + d x \right )}}{d} \end{align*}

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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Giac [A]  time = 1.18707, size = 15, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | d x + c \right |}\right )}{d} \end{align*}

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