3.176 \(\int \left (b x^n\right )^p \, dx\)

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

[Out]

(x*(b*x^n)^p)/(1 + n*p)

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Rubi [A]  time = 0.0138719, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x \left (b x^n\right )^p}{n p+1} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^p,x]

[Out]

(x*(b*x^n)^p)/(1 + n*p)

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Rubi in Sympy [A]  time = 2.30183, size = 22, normalized size = 1.38 \[ \frac{x^{- n p} x^{n p + 1} \left (b x^{n}\right )^{p}}{n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**n)**p,x)

[Out]

x**(-n*p)*x**(n*p + 1)*(b*x**n)**p/(n*p + 1)

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Mathematica [A]  time = 0.0033931, size = 16, normalized size = 1. \[ \frac{x \left (b x^n\right )^p}{n p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^p,x]

[Out]

(x*(b*x^n)^p)/(1 + n*p)

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Maple [A]  time = 0.002, size = 17, normalized size = 1.1 \[{\frac{x \left ( b{x}^{n} \right ) ^{p}}{np+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^p,x)

[Out]

x*(b*x^n)^p/(n*p+1)

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Maxima [A]  time = 1.48493, size = 23, normalized size = 1.44 \[ \frac{b^{p} x{\left (x^{n}\right )}^{p}}{n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^p*x*(x^n)^p/(n*p + 1)

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Fricas [A]  time = 0.2296, size = 27, normalized size = 1.69 \[ \frac{x e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^(n*p*log(x) + p*log(b))/(n*p + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**n)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.229151, size = 27, normalized size = 1.69 \[ \frac{x e^{\left (n p{\rm ln}\left (x\right ) + p{\rm ln}\left (b\right )\right )}}{n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*e^(n*p*ln(x) + p*ln(b))/(n*p + 1)