3.1.47 \(\int \frac {a+b x}{x} \, dx\) [47]

Optimal. Leaf size=8 \[ b x+a \log (x) \]

[Out]

b*x+a*ln(x)

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Rubi [A]
time = 0.00, antiderivative size = 8, 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 = {45} \begin {gather*} a \log (x)+b x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x,x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} b x+a \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x,x]

[Out]

b*x + a*Log[x]

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Mathics [A]
time = 1.59, size = 8, normalized size = 1.00 \begin {gather*} a \text {Log}\left [x\right ]+b x \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)/x^1,x]')

[Out]

a Log[x] + b x

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Maple [A]
time = 0.01, size = 9, normalized size = 1.12

method	result	size
default	\(b x +a \ln \left (x \right )\)	\(9\)
norman	\(b x +a \ln \left (x \right )\)	\(9\)
risch	\(b x +a \ln \left (x \right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x+a*ln(x)

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Maxima [A]
time = 0.25, size = 8, normalized size = 1.00 \begin {gather*} b x + a \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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Fricas [A]
time = 0.32, size = 8, normalized size = 1.00 \begin {gather*} b x + a \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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Sympy [A]
time = 0.04, size = 7, normalized size = 0.88 \begin {gather*} a \log {\left (x \right )} + b x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) + b*x

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Giac [A]
time = 0.00, size = 9, normalized size = 1.12 \begin {gather*} x b+a \ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(abs(x))

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Mupad [B]
time = 0.02, size = 8, normalized size = 1.00 \begin {gather*} b\,x+a\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x,x)

[Out]

b*x + a*log(x)

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