\(\int (b x)^n \, dx\) [35]

Optimal result

Integrand size = 5, antiderivative size = 16 \[ \int (b x)^n \, dx=\frac {(b x)^{1+n}}{b (1+n)} \]

[Out]

(b*x)^(1+n)/b/(1+n)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {32} \[ \int (b x)^n \, dx=\frac {(b x)^{n+1}}{b (n+1)} \]

[In]

Int[(b*x)^n,x]

[Out]

(b*x)^(1 + n)/(b*(1 + n))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(b x)^{1+n}}{b (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (b x)^n \, dx=\frac {x (b x)^n}{1+n} \]

[In]

Integrate[(b*x)^n,x]

[Out]

(x*(b*x)^n)/(1 + n)

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {x \left (b x \right )^{n}}{1+n}\)	\(13\)
risch	\(\frac {x \left (b x \right )^{n}}{1+n}\)	\(13\)
parallelrisch	\(\frac {x \left (b x \right )^{n}}{1+n}\)	\(13\)
norman	\(\frac {x \,{\mathrm e}^{n \ln \left (b x \right )}}{1+n}\)	\(15\)
default	\(\frac {\left (b x \right )^{1+n}}{b \left (1+n \right )}\)	\(17\)

[In]

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

[Out]

x/(1+n)*(b*x)^n

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (b x)^n \, dx=\frac {\left (b x\right )^{n} x}{n + 1} \]

[In]

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

[Out]

(b*x)^n*x/(n + 1)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int (b x)^n \, dx=\frac {\begin {cases} \frac {\left (b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (b x \right )} & \text {otherwise} \end {cases}}{b} \]

[In]

integrate((b*x)**n,x)

[Out]

Piecewise(((b*x)**(n + 1)/(n + 1), Ne(n, -1)), (log(b*x), True))/b

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int (b x)^n \, dx=\frac {\left (b x\right )^{n + 1}}{b {\left (n + 1\right )}} \]

[In]

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

[Out]

(b*x)^(n + 1)/(b*(n + 1))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int (b x)^n \, dx=\frac {\left (b x\right )^{n + 1}}{b {\left (n + 1\right )}} \]

[In]

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

[Out]

(b*x)^(n + 1)/(b*(n + 1))

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (b x)^n \, dx=\frac {x\,{\left (b\,x\right )}^n}{n+1} \]

[In]

int((b*x)^n,x)

[Out]

(x*(b*x)^n)/(n + 1)