3.2709 \(\int \frac{\left (b x^n\right )^p}{x} \, dx\)

Optimal. Leaf size=14 \[ \frac{\left (b x^n\right )^p}{n p} \]

[Out]

(b*x^n)^p/(n*p)

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Rubi [A]  time = 0.0139109, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (b x^n\right )^p}{n p} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^p/x,x]

[Out]

(b*x^n)^p/(n*p)

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Rubi in Sympy [A]  time = 3.02343, size = 8, normalized size = 0.57 \[ \frac{\left (b x^{n}\right )^{p}}{n p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**n)**p/x,x)

[Out]

(b*x**n)**p/(n*p)

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Mathematica [A]  time = 0.0026197, size = 14, normalized size = 1. \[ \frac{\left (b x^n\right )^p}{n p} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^p/x,x]

[Out]

(b*x^n)^p/(n*p)

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Maple [A]  time = 0.002, size = 15, normalized size = 1.1 \[{\frac{ \left ( b{x}^{n} \right ) ^{p}}{np}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^p/x,x)

[Out]

(b*x^n)^p/n/p

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Maxima [A]  time = 1.36991, size = 20, normalized size = 1.43 \[ \frac{b^{p}{\left (x^{n}\right )}^{p}}{n p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^p/x,x, algorithm="maxima")

[Out]

b^p*(x^n)^p/(n*p)

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Fricas [A]  time = 0.23329, size = 24, normalized size = 1.71 \[ \frac{e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(n*p*log(x) + p*log(b))/(n*p)

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Sympy [A]  time = 0.770203, size = 22, normalized size = 1.57 \[ \begin{cases} \log{\left (x \right )} & \text{for}\: n = 0 \wedge p = 0 \\b^{p} \log{\left (x \right )} & \text{for}\: n = 0 \\\log{\left (x \right )} & \text{for}\: p = 0 \\\frac{b^{p} \left (x^{n}\right )^{p}}{n p} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**n)**p/x,x)

[Out]

Piecewise((log(x), Eq(n, 0) & Eq(p, 0)), (b**p*log(x), Eq(n, 0)), (log(x), Eq(p,
0)), (b**p*(x**n)**p/(n*p), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b x^{n}\right )^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n)^p/x, x)