\(\int (d+e x)^m \, dx\) [437]

Optimal result

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

[Out]

(e*x+d)^(1+m)/e/(1+m)

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} \[ \int (d+e x)^m \, dx=\frac {(d+e x)^{m+1}}{e (m+1)} \]

[In]

Int[(d + e*x)^m,x]

[Out]

(d + e*x)^(1 + m)/(e*(1 + m))

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 {(d+e x)^{1+m}}{e (1+m)} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[(d + e*x)^m,x]

[Out]

(d + e*x)^(1 + m)/(e*(1 + m))

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06

method	result	size
gosper	\(\frac {\left (e x +d \right )^{1+m}}{e \left (1+m \right )}\)	\(19\)
default	\(\frac {\left (e x +d \right )^{1+m}}{e \left (1+m \right )}\)	\(19\)
risch	\(\frac {\left (e x +d \right ) \left (e x +d \right )^{m}}{e \left (1+m \right )}\)	\(22\)
parallelrisch	\(\frac {\left (e x +d \right )^{m} d e x +\left (e x +d \right )^{m} d^{2}}{\left (1+m \right ) d e}\)	\(36\)
norman	\(\frac {x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{1+m}+\frac {d \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (1+m \right )}\)	\(37\)

[In]

int((e*x+d)^m,x,method=_RETURNVERBOSE)

[Out]

(e*x+d)^(1+m)/e/(1+m)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (d+e x)^m \, dx=\frac {{\left (e x + d\right )} {\left (e x + d\right )}^{m}}{e m + e} \]

[In]

integrate((e*x+d)^m,x, algorithm="fricas")

[Out]

(e*x + d)*(e*x + d)^m/(e*m + e)

Sympy [A] (verification not implemented)

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

[In]

integrate((e*x+d)**m,x)

[Out]

Piecewise(((d + e*x)**(m + 1)/(m + 1), Ne(m, -1)), (log(d + e*x), True))/e

Maxima [A] (verification not implemented)

none

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

[In]

integrate((e*x+d)^m,x, algorithm="maxima")

[Out]

(e*x + d)^(m + 1)/(e*(m + 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (d+e x)^m \, dx=\frac {{\left (e x + d\right )}^{m + 1}}{e {\left (m + 1\right )}} \]

[In]

integrate((e*x+d)^m,x, algorithm="giac")

[Out]

(e*x + d)^(m + 1)/(e*(m + 1))

Mupad [B] (verification not implemented)

Time = 9.62 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (d+e x)^m \, dx=\frac {{\left (d+e\,x\right )}^{m+1}}{e\,\left (m+1\right )} \]

[In]

int((d + e*x)^m,x)

[Out]

(d + e*x)^(m + 1)/(e*(m + 1))