3.35 \(\int (b x)^n \, dx\)

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

[Out]

(b*x)^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.00327, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {32} \[ \frac{(b x)^{n+1}}{b (n+1)} \]

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*}

Mathematica [A]  time = 0.0065322, size = 12, normalized size = 0.75 \[ \frac{x (b x)^n}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x)^n)/(1 + n)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*}{\frac{x \left ( bx \right ) ^{n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73015, size = 26, normalized size = 1.62 \begin{align*} \frac{\left (b x\right )^{n} x}{n + 1} \end{align*}

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 = 0.054687, size = 17, normalized size = 1.06 \begin{align*} \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{align*}

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 = 1.22592, size = 22, normalized size = 1.38 \begin{align*} \frac{\left (b x\right )^{n + 1}}{b{\left (n + 1\right )}} \end{align*}

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))