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

Optimal. Leaf size=14 \[ \frac{\left (b x^2\right )^p}{2 p} \]

[Out]

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

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Rubi [A]  time = 0.0028232, antiderivative size = 14, 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} \[ \frac{\left (b x^2\right )^p}{2 p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0009849, size = 14, normalized size = 1. \[ \frac{\left (b x^2\right )^p}{2 p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( b{x}^{2} \right ) ^{p}}{2\,p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.984661, size = 18, normalized size = 1.29 \begin{align*} \frac{b^{p}{\left (x^{2}\right )}^{p}}{2 \, p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.69649, size = 23, normalized size = 1.64 \begin{align*} \frac{\left (b x^{2}\right )^{p}}{2 \, p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.189575, size = 14, normalized size = 1. \begin{align*} \begin{cases} \frac{b^{p} \left (x^{2}\right )^{p}}{2 p} & \text{for}\: p \neq 0 \\\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,x)

[Out]

Piecewise((b**p*(x**2)**p/(2*p), Ne(p, 0)), (log(x), True))

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Giac [A]  time = 1.13755, size = 16, normalized size = 1.14 \begin{align*} \frac{\left (b x^{2}\right )^{p}}{2 \, p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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