\(\int \frac {a+b x^n}{x} \, dx\) [2455]

Optimal result

Integrand size = 11, antiderivative size = 13 \[ \int \frac {a+b x^n}{x} \, dx=\frac {b x^n}{n}+a \log (x) \]

[Out]

b*x^n/n+a*ln(x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \frac {a+b x^n}{x} \, dx=a \log (x)+\frac {b x^n}{n} \]

[In]

Int[(a + b*x^n)/x,x]

[Out]

(b*x^n)/n + a*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{x}+b x^{-1+n}\right ) \, dx \\ & = \frac {b x^n}{n}+a \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {a+b x^n}{x} \, dx=\frac {b x^n}{n}+\frac {a \log \left (x^n\right )}{n} \]

[In]

Integrate[(a + b*x^n)/x,x]

[Out]

(b*x^n)/n + (a*Log[x^n])/n

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
risch	\(\frac {b \,x^{n}}{n}+a \ln \left (x \right )\)	\(14\)
norman	\(a \ln \left (x \right )+\frac {b \,{\mathrm e}^{n \ln \left (x \right )}}{n}\)	\(16\)
parallelrisch	\(\frac {a \ln \left (x \right ) n +b \,x^{n}}{n}\)	\(16\)
derivativedivides	\(\frac {b \,x^{n}+a \ln \left (x^{n}\right )}{n}\)	\(17\)
default	\(\frac {b \,x^{n}+a \ln \left (x^{n}\right )}{n}\)	\(17\)

[In]

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

[Out]

b*x^n/n+a*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x^n}{x} \, dx=\frac {a n \log \left (x\right ) + b x^{n}}{n} \]

[In]

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

[Out]

(a*n*log(x) + b*x^n)/n

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {a+b x^n}{x} \, dx=\begin {cases} a \log {\left (x \right )} + \frac {b x^{n}}{n} & \text {for}\: n \neq 0 \\\left (a + b\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*log(x) + b*x**n/n, Ne(n, 0)), ((a + b)*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {a+b x^n}{x} \, dx=\frac {b x^{n}}{n} + \frac {a \log \left (x^{n}\right )}{n} \]

[In]

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

[Out]

b*x^n/n + a*log(x^n)/n

Giac [F]

\[ \int \frac {a+b x^n}{x} \, dx=\int { \frac {b x^{n} + a}{x} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)/x, x)

Mupad [B] (verification not implemented)

Time = 5.58 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^n}{x} \, dx=a\,\ln \left (x\right )+\frac {b\,x^n}{n} \]

[In]

int((a + b*x^n)/x,x)

[Out]

a*log(x) + (b*x^n)/n