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

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

[Out]

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

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

Antiderivative was successfully verified.

[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*} \int (d+e x)^m \, dx &=\frac {(d+e x)^{1+m}}{e (1+m)}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m,x]

[Out]

(d + e*x)^(1 + m)/(e + e*m)

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Maple [A]
time = 0.40, size = 19, normalized size = 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	\(\frac {\left (e x +d \right ) \left (e x +d \right )^{m}}{e \left (1+m \right )}\)	\(22\)
norman	\(\frac {x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{1+m}+\frac {d \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (1+m \right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} e^{\left (-1\right )}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)^(m + 1)*e^(-1)/(m + 1)

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Fricas [A]
time = 1.70, size = 22, normalized size = 1.22 \begin {gather*} \frac {{\left (x e + d\right )} {\left (x e + d\right )}^{m} e^{\left (-1\right )}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)*(x*e + d)^m*e^(-1)/(m + 1)

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Sympy [A]
time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]
time = 1.19, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} e^{\left (-1\right )}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)^(m + 1)*e^(-1)/(m + 1)

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Mupad [B]
time = 0.36, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (d+e\,x\right )}^{m+1}}{e\,\left (m+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^m,x)

[Out]

(d + e*x)^(m + 1)/(e*(m + 1))

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