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 = 0.00, 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 = {} \[ a x+\frac {b x^{n+1}}{n+1} \]

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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ a x+\frac {b x^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*x^n,x]

[Out]

a*x + (b*x^(1 + n))/(1 + n)

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fricas [A]  time = 0.77, size = 20, normalized size = 1.25 \[ \frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \]

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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giac [A]  time = 0.15, size = 16, normalized size = 1.00 \[ a x + \frac {b x^{n + 1}}{n + 1} \]

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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maple [A]  time = 0.00, size = 17, normalized size = 1.06 \[ a x +\frac {b \,x^{n +1}}{n +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.52, size = 16, normalized size = 1.00 \[ a x + \frac {b x^{n + 1}}{n + 1} \]

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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mupad [B]  time = 1.28, size = 15, normalized size = 0.94 \[ a\,x+\frac {b\,x\,x^n}{n+1} \]

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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sympy [A]  time = 0.06, size = 17, normalized size = 1.06 \[ a x + b \left (\begin {cases} \frac {x^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) \]

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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