3.2.78 \(\int \frac {(b x^n)^p}{x^2} \, dx\) [178]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\left (b x^n\right )^p}{x (1-n p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^n)^p/x^2,x]

[Out]

-((b*x^n)^p/((1 - n*p)*x))

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^2} \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{-2+n p} \, dx\\ &=-\frac {\left (b x^n\right )^p}{(1-n p) x}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 0.90 \begin {gather*} \frac {\left (b x^n\right )^p}{(-1+n p) x} \end {gather*}

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 = 0.02, size = 19, normalized size = 0.95

method	result	size
gosper	\(\frac {\left (b \,x^{n}\right )^{p}}{x \left (n p -1\right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^p/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 19, normalized size = 0.95 \begin {gather*} \frac {b^{p} {\left (x^{n}\right )}^{p}}{{\left (n p - 1\right )} x} \end {gather*}

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 = 0.36, size = 22, normalized size = 1.10 \begin {gather*} \frac {e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{{\left (n p - 1\right )} x} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {\left (b x^{n}\right )^{p}}{n p x - x} & \text {for}\: n \neq \frac {1}{p} \\\int \frac {\left (b x^{\frac {1}{p}}\right )^{p}}{x^{2}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**n)**p/x**2,x)

[Out]

Piecewise(((b*x**n)**p/(n*p*x - x), Ne(n, 1/p)), (Integral((b*x**(1/p))**p/x**2, x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\left (b\,x^n\right )}^p}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^p/x^2,x)

[Out]

int((b*x^n)^p/x^2, x)

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