3.19.55 \(\int (c+d x)^n \, dx\) [1855]

Optimal. Leaf size=18 \[ \frac {(c+d x)^{1+n}}{d (1+n)} \]

[Out]

(d*x+c)^(1+n)/d/(1+n)

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} \frac {(c+d x)^{n+1}}{d (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^n,x]

[Out]

(c + d*x)^(1 + n)/(d*(1 + n))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 17, normalized size = 0.94 \begin {gather*} \frac {(c+d x)^{1+n}}{d+d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^n,x]

[Out]

(c + d*x)^(1 + n)/(d + d*n)

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Maple [A]
time = 0.15, size = 19, normalized size = 1.06

method	result	size
gosper	\(\frac {\left (d x +c \right )^{1+n}}{d \left (1+n \right )}\)	\(19\)
default	\(\frac {\left (d x +c \right )^{1+n}}{d \left (1+n \right )}\)	\(19\)
risch	\(\frac {\left (d x +c \right ) \left (d x +c \right )^{n}}{d \left (1+n \right )}\)	\(22\)
norman	\(\frac {x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{1+n}+\frac {c \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (1+n \right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x+c)^(1+n)/d/(1+n)

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Maxima [A]
time = 0.30, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (d x + c\right )}^{n + 1}}{d {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)^(n + 1)/(d*(n + 1))

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Fricas [A]
time = 1.15, size = 20, normalized size = 1.11 \begin {gather*} \frac {{\left (d x + c\right )} {\left (d x + c\right )}^{n}}{d n + d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)*(d*x + c)^n/(d*n + d)

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Sympy [A]
time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} \frac {\begin {cases} \frac {\left (c + d x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (c + d x \right )} & \text {otherwise} \end {cases}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((c + d*x)**(n + 1)/(n + 1), Ne(n, -1)), (log(c + d*x), True))/d

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Giac [A]
time = 4.33, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (d x + c\right )}^{n + 1}}{d {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x + c)^(n + 1)/(d*(n + 1))

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Mupad [B]
time = 0.38, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (c+d\,x\right )}^{n+1}}{d\,\left (n+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^n,x)

[Out]

(c + d*x)^(n + 1)/(d*(n + 1))

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