3.2710 \(\int \frac{\left (b x^n\right )^p}{x^2} \, dx\)

Optimal. Leaf size=20 \[ -\frac{\left (b x^n\right )^p}{x (1-n p)} \]

[Out]

-((b*x^n)^p/((1 - n*p)*x))

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Rubi [A]  time = 0.0201919, antiderivative size = 20, 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 \[ -\frac{\left (b x^n\right )^p}{x (1-n p)} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^p/x^2,x]

[Out]

-((b*x^n)^p/((1 - n*p)*x))

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Rubi in Sympy [A]  time = 2.98825, size = 24, normalized size = 1.2 \[ - \frac{x^{- n p} x^{n p - 1} \left (b x^{n}\right )^{p}}{- n p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**n)**p/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.00612895, size = 18, normalized size = 0.9 \[ \frac{\left (b x^n\right )^p}{x (n p-1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^p/x^2,x]

[Out]

(b*x^n)^p/((-1 + n*p)*x)

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Maple [A]  time = 0.001, size = 19, normalized size = 1. \[{\frac{ \left ( b{x}^{n} \right ) ^{p}}{x \left ( np-1 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^p/x^2,x)

[Out]

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

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Maxima [A]  time = 1.40111, size = 26, normalized size = 1.3 \[ \frac{b^{p}{\left (x^{n}\right )}^{p}}{{\left (n p - 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^p/x^2,x, algorithm="maxima")

[Out]

b^p*(x^n)^p/((n*p - 1)*x)

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Fricas [A]  time = 0.235588, size = 30, normalized size = 1.5 \[ \frac{e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{{\left (n p - 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^p/x^2,x, algorithm="fricas")

[Out]

e^(n*p*log(x) + p*log(b))/((n*p - 1)*x)

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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**2,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (b x^{n}\right )^{p}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^p/x^2,x, algorithm="giac")

[Out]

integrate((b*x^n)^p/x^2, x)