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

Optimal result

Integrand size = 7, antiderivative size = 18 \[ \int (c+d x)^n \, dx=\frac {(c+d x)^{1+n}}{d (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (c+d x)^n \, dx=\frac {(c+d x)^{n+1}}{d (n+1)} \]

[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*} \text {integral}& = \frac {(c+d x)^{1+n}}{d (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (c+d x)^n \, dx=\frac {(c+d x)^{1+n}}{d (1+n)} \]

[In]

Integrate[(c + d*x)^n,x]

[Out]

(c + d*x)^(1 + n)/(d*(1 + n))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 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	\(\frac {\left (d x +c \right ) \left (d x +c \right )^{n}}{d \left (1+n \right )}\)	\(22\)
parallelrisch	\(\frac {x \left (d x +c \right )^{n} c d +\left (d x +c \right )^{n} c^{2}}{\left (1+n \right ) c d}\)	\(36\)
norman	\(\frac {x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{1+n}+\frac {c \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (1+n \right )}\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (c+d x)^n \, dx=\frac {{\left (d x + c\right )} {\left (d x + c\right )}^{n}}{d n + d} \]

[In]

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

[Out]

(d*x + c)*(d*x + c)^n/(d*n + d)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (c+d x)^n \, dx=\frac {\begin {cases} \frac {\left (c + d x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (c + d x \right )} & \text {otherwise} \end {cases}}{d} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (c+d x)^n \, dx=\frac {{\left (d x + c\right )}^{n + 1}}{d {\left (n + 1\right )}} \]

[In]

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

[Out]

(d*x + c)^(n + 1)/(d*(n + 1))

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (c+d x)^n \, dx=\frac {{\left (d x + c\right )}^{n + 1}}{d {\left (n + 1\right )}} \]

[In]

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

[Out]

(d*x + c)^(n + 1)/(d*(n + 1))

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (c+d x)^n \, dx=\frac {{\left (c+d\,x\right )}^{n+1}}{d\,\left (n+1\right )} \]

[In]

int((c + d*x)^n,x)

[Out]

(c + d*x)^(n + 1)/(d*(n + 1))