\(\int \frac {a+b x}{x^2} \, dx\) [48]

Optimal result

Integrand size = 9, antiderivative size = 11 \[ \int \frac {a+b x}{x^2} \, dx=-\frac {a}{x}+b \log (x) \]

[Out]

-a/x+b*ln(x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {a+b x}{x^2} \, dx=b \log (x)-\frac {a}{x} \]

[In]

Int[(a + b*x)/x^2,x]

[Out]

-(a/x) + b*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx \\ & = -\frac {a}{x}+b \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{x^2} \, dx=-\frac {a}{x}+b \log (x) \]

[In]

Integrate[(a + b*x)/x^2,x]

[Out]

-(a/x) + b*Log[x]

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
default	\(-\frac {a}{x}+b \ln \left (x \right )\)	\(12\)
norman	\(-\frac {a}{x}+b \ln \left (x \right )\)	\(12\)
risch	\(-\frac {a}{x}+b \ln \left (x \right )\)	\(12\)
parallelrisch	\(\frac {b \ln \left (x \right ) x -a}{x}\)	\(14\)

[In]

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

[Out]

-a/x+b*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \frac {a+b x}{x^2} \, dx=\frac {b x \log \left (x\right ) - a}{x} \]

[In]

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

[Out]

(b*x*log(x) - a)/x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int \frac {a+b x}{x^2} \, dx=- \frac {a}{x} + b \log {\left (x \right )} \]

[In]

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

[Out]

-a/x + b*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{x^2} \, dx=b \log \left (x\right ) - \frac {a}{x} \]

[In]

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

[Out]

b*log(x) - a/x

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {a+b x}{x^2} \, dx=b \log \left ({\left | x \right |}\right ) - \frac {a}{x} \]

[In]

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

[Out]

b*log(abs(x)) - a/x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{x^2} \, dx=b\,\ln \left (x\right )-\frac {a}{x} \]

[In]

int((a + b*x)/x^2,x)

[Out]

b*log(x) - a/x