3.2454 \(\int (a+b x^n) \, dx\)

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

[Out]

a*x + (b*x^(1 + n))/(1 + n)

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Rubi [A]  time = 0.0035536, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ a x+\frac{b x^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[a + b*x^n,x]

[Out]

a*x + (b*x^(1 + n))/(1 + n)

Rubi steps

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

Mathematica [A]  time = 0.0059723, size = 16, normalized size = 1. \[ a x+\frac{b x^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*x^n,x]

[Out]

a*x + (b*x^(1 + n))/(1 + n)

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Maple [A]  time = 0., size = 17, normalized size = 1.1 \begin{align*} ax+{\frac{b{x}^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+b*x^(1+n)/(1+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(a+b*x^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61279, size = 45, normalized size = 2.81 \begin{align*} \frac{b x x^{n} +{\left (a n + a\right )} x}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x*x^n + (a*n + a)*x)/(n + 1)

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Sympy [A]  time = 0.054022, size = 17, normalized size = 1.06 \begin{align*} a x + b \left (\begin{cases} \frac{x^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.82322, size = 22, normalized size = 1.38 \begin{align*} a x + \frac{b x^{n + 1}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x^(n + 1)/(n + 1)