3.1932 \(\int \frac{a+b x}{(d+e x) (a^2+2 a b x+b^2 x^2)} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0078569, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {27, 36, 31} \[ \frac{\log (a+b x)}{b d-a e}-\frac{\log (d+e x)}{b d-a e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 36

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

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

Mathematica [A]  time = 0.0122795, size = 26, normalized size = 0.72 \[ \frac{\log (a+b x)-\log (d+e x)}{b d-a e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 37, normalized size = 1. \begin{align*}{\frac{\ln \left ( ex+d \right ) }{ae-bd}}-{\frac{\ln \left ( bx+a \right ) }{ae-bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.959161, size = 49, normalized size = 1.36 \begin{align*} \frac{\log \left (b x + a\right )}{b d - a e} - \frac{\log \left (e x + d\right )}{b d - a e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.50342, size = 58, normalized size = 1.61 \begin{align*} \frac{\log \left (b x + a\right ) - \log \left (e x + d\right )}{b d - a e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.323191, size = 128, normalized size = 3.56 \begin{align*} \frac{\log{\left (x + \frac{- \frac{a^{2} e^{2}}{a e - b d} + \frac{2 a b d e}{a e - b d} + a e - \frac{b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} - \frac{\log{\left (x + \frac{\frac{a^{2} e^{2}}{a e - b d} - \frac{2 a b d e}{a e - b d} + a e + \frac{b^{2} d^{2}}{a e - b d} + b d}{2 b e} \right )}}{a e - b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1129, size = 93, normalized size = 2.58 \begin{align*} \frac{\log \left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{{\left | b d - a e \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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