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

Optimal. Leaf size=11 \[ b \log (x)-\frac {a}{x} \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} b \log (x)-\frac {a}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^2,x]

[Out]

-(a/x) + b*Log[x]

Rule 43

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} b \log (x)-\frac {a}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^2,x]

[Out]

-(a/x) + b*Log[x]

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)/x^2,x]

[Out]

IntegrateAlgebraic[(a + b*x)/x^2, x]

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fricas [A]  time = 1.41, size = 13, normalized size = 1.18 \begin {gather*} \frac {b x \log \relax (x) - a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*log(x) - a)/x

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giac [A]  time = 1.05, size = 12, normalized size = 1.09 \begin {gather*} b \log \left ({\left | x \right |}\right ) - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(abs(x)) - a/x

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maple [A]  time = 0.02, size = 12, normalized size = 1.09 \begin {gather*} b \ln \relax (x )-\frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x+b*ln(x)

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maxima [A]  time = 0.87, size = 11, normalized size = 1.00 \begin {gather*} b \log \relax (x) - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(x) - a/x

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mupad [B]  time = 0.03, size = 11, normalized size = 1.00 \begin {gather*} b\,\ln \relax (x)-\frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^2,x)

[Out]

b*log(x) - a/x

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sympy [A]  time = 0.11, size = 7, normalized size = 0.64 \begin {gather*} - \frac {a}{x} + b \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x + b*log(x)

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