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 = 0.0142773, 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 \[ \frac{(d+e x)^{m+1}}{e (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m,x]

[Out]

(d + e*x)^(1 + m)/(e*(1 + m))

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Rubi in Sympy [A]  time = 1.78324, size = 12, normalized size = 0.67 \[ \frac{\left (d + e x\right )^{m + 1}}{e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m,x)

[Out]

(d + e*x)**(m + 1)/(e*(m + 1))

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Mathematica [A]  time = 0.0113396, 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 = 0.002, size = 19, normalized size = 1.1 \[{\frac{ \left ( ex+d \right ) ^{1+m}}{e \left ( 1+m \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.233574, size = 27, normalized size = 1.5 \[ \frac{{\left (e x + d\right )}{\left (e x + d\right )}^{m}}{e m + e} \]

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 = 0.077394, size = 20, normalized size = 1.11 \[ \frac{\begin{cases} \frac{\left (d + e x\right )^{m + 1}}{m + 1} & \text{for}\: m \neq -1 \\\log{\left (d + e x \right )} & \text{otherwise} \end{cases}}{e} \]

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/XCAS [A]  time = 0.204059, size = 24, normalized size = 1.33 \[ \frac{{\left (x e + d\right )}^{m + 1} e^{\left (-1\right )}}{m + 1} \]

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)