3.74 \(\int \frac{(b x^2)^p}{x^2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (b x^2\right )^p}{(1-2 p) x} \]

[Out]

-((b*x^2)^p/((1 - 2*p)*x))

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Rubi [A]  time = 0.0059464, antiderivative size = 19, 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} \[ -\frac{\left (b x^2\right )^p}{(1-2 p) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0031188, size = 18, normalized size = 0.95 \[ \frac{\left (b x^2\right )^p}{(2 p-1) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2)^p/((-1 + 2*p)*x)

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \begin{align*}{\frac{ \left ( b{x}^{2} \right ) ^{p}}{x \left ( -1+2\,p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x/(-1+2*p)*(b*x^2)^p

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Maxima [A]  time = 1.00302, size = 26, normalized size = 1.37 \begin{align*} \frac{b^{p} x^{2 \, p}}{{\left (2 \, p - 1\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^p/x^2,x, algorithm="maxima")

[Out]

b^p*x^(2*p)/((2*p - 1)*x)

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Fricas [A]  time = 1.79112, size = 34, normalized size = 1.79 \begin{align*} \frac{\left (b x^{2}\right )^{p}}{{\left (2 \, p - 1\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^p/x^2,x, algorithm="fricas")

[Out]

(b*x^2)^p/((2*p - 1)*x)

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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**2)**p/x**2,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^{2}\right )^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^p/x^2,x, algorithm="giac")

[Out]

integrate((b*x^2)^p/x^2, x)