3.75 \(\int \frac{(b x^2)^p}{x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (b x^2\right )^p}{2 (1-p) x^2} \]

[Out]

-(b*x^2)^p/(2*(1 - p)*x^2)

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Rubi [A]  time = 0.0051819, antiderivative size = 21, 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} \[ -\frac{\left (b x^2\right )^p}{2 (1-p) x^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2)^p/x^3,x]

[Out]

-(b*x^2)^p/(2*(1 - p)*x^2)

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^3} \, dx &=\left (x^{-2 p} \left (b x^2\right )^p\right ) \int x^{-3+2 p} \, dx\\ &=-\frac{\left (b x^2\right )^p}{2 (1-p) x^2}\\ \end{align*}

Mathematica [A]  time = 0.0020254, size = 18, normalized size = 0.86 \[ \frac{\left (b x^2\right )^p}{(2 p-2) x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2)^p/x^3,x]

[Out]

(b*x^2)^p/((-2 + 2*p)*x^2)

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Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \begin{align*}{\frac{ \left ( b{x}^{2} \right ) ^{p}}{2\,{x}^{2} \left ( p-1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^p/x^3,x)

[Out]

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

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Maxima [A]  time = 0.981526, size = 24, normalized size = 1.14 \begin{align*} \frac{b^{p}{\left (x^{2}\right )}^{p}}{2 \,{\left (p - 1\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^p/x^3,x, algorithm="maxima")

[Out]

1/2*b^p*(x^2)^p/((p - 1)*x^2)

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Fricas [A]  time = 1.87311, size = 39, normalized size = 1.86 \begin{align*} \frac{\left (b x^{2}\right )^{p}}{2 \,{\left (p - 1\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^p/x^3,x, algorithm="fricas")

[Out]

1/2*(b*x^2)^p/((p - 1)*x^2)

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Sympy [A]  time = 0.484649, size = 24, normalized size = 1.14 \begin{align*} \begin{cases} \frac{b^{p} \left (x^{2}\right )^{p}}{2 p x^{2} - 2 x^{2}} & \text{for}\: p \neq 1 \\b \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**p/x**3,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{2}\right )^{p}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^p/x^3,x, algorithm="giac")

[Out]

integrate((b*x^2)^p/x^3, x)