3.6.80 \(\int (c+d x^{-1+n}) \, dx\) [580]

Optimal. Leaf size=12 \[ c x+\frac {d x^n}{n} \]

[Out]

c*x+d*x^n/n

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} c x+\frac {d x^n}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[c + d*x^(-1 + n),x]

[Out]

c*x + (d*x^n)/n

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} c x+\frac {d x^n}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[c + d*x^(-1 + n),x]

[Out]

c*x + (d*x^n)/n

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Maple [A]
time = 0.02, size = 13, normalized size = 1.08

method	result	size
default	\(c x +\frac {d \,x^{n}}{n}\)	\(13\)
risch	\(c x +\frac {d x \,x^{-1+n}}{n}\)	\(16\)
norman	\(c x +\frac {d x \,{\mathrm e}^{\left (-1+n \right ) \ln \left (x \right )}}{n}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c+d*x^(-1+n),x,method=_RETURNVERBOSE)

[Out]

c*x+d*x^n/n

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Maxima [A]
time = 0.29, size = 12, normalized size = 1.00 \begin {gather*} c x + \frac {d x^{n}}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c+d*x^(-1+n),x, algorithm="maxima")

[Out]

c*x + d*x^n/n

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Fricas [A]
time = 0.39, size = 17, normalized size = 1.42 \begin {gather*} \frac {c n x + d x x^{n - 1}}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c+d*x^(-1+n),x, algorithm="fricas")

[Out]

(c*n*x + d*x*x^(n - 1))/n

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Sympy [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} c x + d \left (\begin {cases} \frac {x^{n}}{n} & \text {for}\: n \neq 0 \\\log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c+d*x**(-1+n),x)

[Out]

c*x + d*Piecewise((x**n/n, Ne(n, 0)), (log(x), True))

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Giac [A]
time = 1.68, size = 12, normalized size = 1.00 \begin {gather*} c x + \frac {d x^{n}}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c+d*x^(-1+n),x, algorithm="giac")

[Out]

c*x + d*x^n/n

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Mupad [B]
time = 5.01, size = 12, normalized size = 1.00 \begin {gather*} c\,x+\frac {d\,x^n}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c + d*x^(n - 1),x)

[Out]

c*x + (d*x^n)/n

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