### 3.437 $$\int (d+e x)^m \, dx$$

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

[Out]

(d + e*x)^(1 + m)/(e*(1 + m))

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Rubi [A]  time = 0.0031172, antiderivative size = 18, normalized size of antiderivative = 1., 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} $\frac{(d+e x)^{m+1}}{e (m+1)}$

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*}

Mathematica [A]  time = 0.0092179, size = 17, normalized size = 0.94 $\frac{(d+e x)^{m+1}}{e m+e}$

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 = 0.043, size = 19, normalized size = 1.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{1+m}}{e \left ( 1+m \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.9878, size = 45, normalized size = 2.5 \begin{align*} \frac{{\left (e x + d\right )}{\left (e x + d\right )}^{m}}{e m + e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x + d)*(e*x + d)^m/(e*m + e)

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Sympy [A]  time = 0.06186, size = 20, normalized size = 1.11 \begin{align*} \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{align*}

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 = 1.23295, size = 24, normalized size = 1.33 \begin{align*} \frac{{\left (x e + d\right )}^{m + 1} e^{\left (-1\right )}}{m + 1} \end{align*}

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)