3.1855 \(\int (c+d x)^n \, dx\)

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

[Out]

(c + d*x)^(1 + n)/(d*(1 + n))

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Rubi [A]  time = 0.0108916, 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 \[ \frac{(c+d x)^{n+1}}{d (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^n,x]

[Out]

(c + d*x)^(1 + n)/(d*(1 + n))

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Rubi in Sympy [A]  time = 1.66533, size = 12, normalized size = 0.67 \[ \frac{\left (c + d x\right )^{n + 1}}{d \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**n,x)

[Out]

(c + d*x)**(n + 1)/(d*(n + 1))

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Mathematica [A]  time = 0.010218, size = 17, normalized size = 0.94 \[ \frac{(c+d x)^{n+1}}{d n+d} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^n,x]

[Out]

(c + d*x)^(1 + n)/(d + d*n)

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Maple [A]  time = 0.002, size = 19, normalized size = 1.1 \[{\frac{ \left ( dx+c \right ) ^{1+n}}{d \left ( 1+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^n,x)

[Out]

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

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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((d*x + c)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217538, size = 27, normalized size = 1.5 \[ \frac{{\left (d x + c\right )}{\left (d x + c\right )}^{n}}{d n + d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x + c)*(d*x + c)^n/(d*n + d)

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Sympy [A]  time = 0.0396, size = 20, normalized size = 1.11 \[ \frac{\begin{cases} \frac{\left (c + d x\right )^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (c + d x \right )} & \text{otherwise} \end{cases}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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GIAC/XCAS [A]  time = 0.275069, size = 24, normalized size = 1.33 \[ \frac{{\left (d x + c\right )}^{n + 1}}{d{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x + c)^(n + 1)/(d*(n + 1))