3.25.54 \(\int (a+b x^n) \, dx\) [2454]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} a x+\frac {b x^{n+1}}{n+1} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} a x+\frac {b x^{1+n}}{1+n} \end {gather*}

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 = 0.01, size = 17, normalized size = 1.06

method	result	size
risch	\(a x +\frac {b x \,x^{n}}{1+n}\)	\(16\)
default	\(a x +\frac {b \,x^{1+n}}{1+n}\)	\(17\)
norman	\(a x +\frac {b x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 16, normalized size = 1.00 \begin {gather*} a x + \frac {b x^{n + 1}}{n + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x^(n + 1)/(n + 1)

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Fricas [A]
time = 0.38, size = 20, normalized size = 1.25 \begin {gather*} \frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \end {gather*}

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 = 0.01, size = 17, normalized size = 1.06 \begin {gather*} 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 {gather*}

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 = 1.16, size = 16, normalized size = 1.00 \begin {gather*} a x + \frac {b x^{n + 1}}{n + 1} \end {gather*}

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)

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Mupad [B]
time = 1.28, size = 15, normalized size = 0.94 \begin {gather*} a\,x+\frac {b\,x\,x^n}{n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*x^n,x)

[Out]

a*x + (b*x*x^n)/(n + 1)

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