3.35 \(\int (b x)^n \, dx\)

Optimal. Leaf size=16 \[ \frac{(b x)^{n+1}}{b (n+1)} \]

[Out]

(b*x)^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0110471, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(b x)^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(b*x)^n,x]

[Out]

(b*x)^(1 + n)/(b*(1 + n))

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Rubi in Sympy [A]  time = 1.68607, size = 10, normalized size = 0.62 \[ \frac{\left (b x\right )^{n + 1}}{b \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x)**n,x)

[Out]

(b*x)**(n + 1)/(b*(n + 1))

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Mathematica [A]  time = 0.00263634, size = 12, normalized size = 0.75 \[ \frac{x (b x)^n}{n+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x)^n,x]

[Out]

(x*(b*x)^n)/(1 + n)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \[{\frac{x \left ( bx \right ) ^{n}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x)^n,x)

[Out]

x/(1+n)*(b*x)^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((b*x)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209398, size = 16, normalized size = 1. \[ \frac{\left (b x\right )^{n} x}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^n,x, algorithm="fricas")

[Out]

(b*x)^n*x/(n + 1)

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Sympy [A]  time = 0.034792, size = 17, normalized size = 1.06 \[ \frac{\begin{cases} \frac{\left (b x\right )^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (b x \right )} & \text{otherwise} \end{cases}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)**n,x)

[Out]

Piecewise(((b*x)**(n + 1)/(n + 1), Ne(n, -1)), (log(b*x), True))/b

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GIAC/XCAS [A]  time = 0.205556, size = 22, normalized size = 1.38 \[ \frac{\left (b x\right )^{n + 1}}{b{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^n,x, algorithm="giac")

[Out]

(b*x)^(n + 1)/(b*(n + 1))