3.2719 \(\int \frac{(b x^n)^p}{x^4} \, dx\)

Optimal. Leaf size=20 \[ -\frac{\left (b x^n\right )^p}{x^3 (3-n p)} \]

[Out]

-((b*x^n)^p/((3 - n*p)*x^3))

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Rubi [A]  time = 0.0065255, 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, Rules used = {15, 30} \[ -\frac{\left (b x^n\right )^p}{x^3 (3-n p)} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^n)^p/x^4,x]

[Out]

-((b*x^n)^p/((3 - n*p)*x^3))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (b x^n\right )^p}{x^4} \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{-4+n p} \, dx\\ &=-\frac{\left (b x^n\right )^p}{(3-n p) x^3}\\ \end{align*}

Mathematica [A]  time = 0.0026201, size = 18, normalized size = 0.9 \[ \frac{\left (b x^n\right )^p}{x^3 (n p-3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^n)^p/x^4,x]

[Out]

(b*x^n)^p/((-3 + n*p)*x^3)

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Maple [A]  time = 0., size = 19, normalized size = 1. \begin{align*}{\frac{ \left ( b{x}^{n} \right ) ^{p}}{{x}^{3} \left ( np-3 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^p/x^4,x)

[Out]

1/x^3/(n*p-3)*(b*x^n)^p

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Maxima [A]  time = 0.983631, size = 26, normalized size = 1.3 \begin{align*} \frac{b^{p}{\left (x^{n}\right )}^{p}}{{\left (n p - 3\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^p/x^4,x, algorithm="maxima")

[Out]

b^p*(x^n)^p/((n*p - 3)*x^3)

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Fricas [A]  time = 1.26428, size = 58, normalized size = 2.9 \begin{align*} \frac{e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{{\left (n p - 3\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^p/x^4,x, algorithm="fricas")

[Out]

e^(n*p*log(x) + p*log(b))/((n*p - 3)*x^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**n)**p/x**4,x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{n}\right )^{p}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^p/x^4,x, algorithm="giac")

[Out]

integrate((b*x^n)^p/x^4, x)