3.2447 \(\int \left (a+b x^n\right ) \, dx\)

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 = 0.0156328, 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. \[ a x+\frac{b x^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]  Int[a + b*x^n,x]

[Out]

a*x + (b*x^(1 + n))/(1 + n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b x^{n + 1}}{n + 1} + \int a\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a+b*x**n,x)

[Out]

b*x**(n + 1)/(n + 1) + Integral(a, x)

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Mathematica [A]  time = 0.00938126, 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 = 0.002, size = 17, normalized size = 1.1 \[ ax+{\frac{b{x}^{1+n}}{1+n}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237069, size = 27, normalized size = 1.69 \[ \frac{b x x^{n} +{\left (a n + a\right )} x}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x*x^n + (a*n + a)*x)/(n + 1)

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Sympy [A]  time = 0.065156, size = 17, normalized size = 1.06 \[ a x + b \left (\begin{cases} \frac{x^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) \]

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/XCAS [A]  time = 0.210342, size = 22, normalized size = 1.38 \[ a x + \frac{b x^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x + b*x^(n + 1)/(n + 1)