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

Optimal. Leaf size=16 \[ \frac {(b x)^{1+n}}{b (1+n)} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {32} \begin {gather*} \frac {(b x)^{n+1}}{b (n+1)} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (b x)^n \, dx &=\frac {(b x)^{1+n}}{b (1+n)}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 0.75 \begin {gather*} \frac {x (b x)^n}{1+n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x)^n)/(1 + n)

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Maple [A]
time = 0.01, size = 17, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (b x\right )^{n + 1}}{b {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x)^(n + 1)/(b*(n + 1))

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Fricas [A]
time = 0.90, size = 12, normalized size = 0.75 \begin {gather*} \frac {\left (b x\right )^{n} x}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x)^n*x/(n + 1)

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Sympy [A]
time = 0.01, size = 17, normalized size = 1.06 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.60, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (b x\right )^{n + 1}}{b {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x)^(n + 1)/(b*(n + 1))

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Mupad [B]
time = 0.18, size = 12, normalized size = 0.75 \begin {gather*} \frac {x\,{\left (b\,x\right )}^n}{n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(b*x)^n)/(n + 1)

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