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

Optimal. Leaf size=25 \[ -\frac{\left (a \left (b x^n\right )^p\right )^q}{x (1-n p q)} \]

[Out]

-((a*(b*x^n)^p)^q/((1 - n*p*q)*x))

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Rubi [A]  time = 0.0829921, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\left (a \left (b x^n\right )^p\right )^q}{x (1-n p q)} \]

Antiderivative was successfully verified.

[In]  Int[(a*(b*x^n)^p)^q/x^2,x]

[Out]

-((a*(b*x^n)^p)^q/((1 - n*p*q)*x))

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Rubi in Sympy [A]  time = 6.61968, size = 32, normalized size = 1.28 \[ - \frac{x^{- n p q} x^{n p q - 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{- n p q + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0101476, size = 23, normalized size = 0.92 \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{x (n p q-1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*(b*x^n)^p)^q/x^2,x]

[Out]

(a*(b*x^n)^p)^q/((-1 + n*p*q)*x)

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Maple [A]  time = 0.003, size = 24, normalized size = 1. \[{\frac{ \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{x \left ( npq-1 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*(b*x^n)^p)^q/x^2,x)

[Out]

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

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Maxima [A]  time = 1.64015, size = 36, normalized size = 1.44 \[ \frac{a^{q}{\left (b^{p}\right )}^{q}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{{\left (n p q - 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.274529, size = 39, normalized size = 1.56 \[ \frac{e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{{\left (n p q - 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a \left (b x^{n}\right )^{p}\right )^{q}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*(b*x**n)**p)**q/x**2,x)

[Out]

Integral((a*(b*x**n)**p)**q/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((b*x^n)^p*a)^q/x^2, x)