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

Optimal result

Integrand size = 7, antiderivative size = 16 \[ \int \left (a+b x^n\right ) \, dx=a x+\frac {b x^{1+n}}{1+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+b x^n\right ) \, dx=a x+\frac {b x^{n+1}}{n+1} \]

[In]

Int[a + b*x^n,x]

[Out]

a*x + (b*x^(1 + n))/(1 + n)

Rubi steps \begin{align*} \text {integral}& = a x+\frac {b x^{1+n}}{1+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right ) \, dx=a x+\frac {b x^{1+n}}{1+n} \]

[In]

Integrate[a + b*x^n,x]

[Out]

a*x + (b*x^(1 + n))/(1 + n)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \left (a+b x^n\right ) \, dx=\frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \]

[In]

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

[Out]

(b*x*x^n + (a*n + a)*x)/(n + 1)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^n\right ) \, dx=a x + b \left (\begin {cases} \frac {x^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right ) \, dx=a x + \frac {b x^{n + 1}}{n + 1} \]

[In]

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

[Out]

a*x + b*x^(n + 1)/(n + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^n\right ) \, dx=a x + \frac {b x^{n + 1}}{n + 1} \]

[In]

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

[Out]

a*x + b*x^(n + 1)/(n + 1)

Mupad [B] (verification not implemented)

Time = 5.62 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \left (a+b x^n\right ) \, dx=a\,x+\frac {b\,x\,x^n}{n+1} \]

[In]

int(a + b*x^n,x)

[Out]

a*x + (b*x*x^n)/(n + 1)