3.1931 \(\int \frac {a+b x}{a^2+2 a b x+b^2 x^2} \, dx\)

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

[Out]

ln(b*x+a)/b

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 31} \[ \frac {\log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*x]/b

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {\log (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*x]/b

________________________________________________________________________________________

fricas [A]  time = 1.03, size = 10, normalized size = 1.00 \[ \frac {\log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x + a)/b

________________________________________________________________________________________

giac [B]  time = 0.15, size = 22, normalized size = 2.20 \[ \frac {\log \left (a^{2} + {\left (b x^{2} + 2 \, a x\right )} b\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a^2 + (b*x^2 + 2*a*x)*b)/b

________________________________________________________________________________________

maple [A]  time = 0.04, size = 11, normalized size = 1.10 \[ \frac {\ln \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*ln(b*x+a)

________________________________________________________________________________________

maxima [B]  time = 0.53, size = 22, normalized size = 2.20 \[ \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b^2*x^2 + 2*a*b*x + a^2)/b

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 10, normalized size = 1.00 \[ \frac {\ln \left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*x)/b

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 7, normalized size = 0.70 \[ \frac {\log {\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + b*x)/b

________________________________________________________________________________________