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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 14, 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} \[ \frac {\left (b x^2\right )^p}{2 p} \]

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ \frac {\left (b x^2\right )^p}{2 p} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.65, size = 12, normalized size = 0.86 \[ \frac {\left (b x^{2}\right )^{p}}{2 \, p} \]

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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giac [A]  time = 0.15, size = 12, normalized size = 0.86 \[ \frac {\left (b x^{2}\right )^{p}}{2 \, p} \]

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \[ \frac {\left (b \,x^{2}\right )^{p}}{2 p} \]

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 = 1.31, size = 13, normalized size = 0.93 \[ \frac {b^{p} {\left (x^{2}\right )}^{p}}{2 \, p} \]

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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mupad [B]  time = 0.95, size = 12, normalized size = 0.86 \[ \frac {{\left (b\,x^2\right )}^p}{2\,p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.20, size = 14, normalized size = 1.00 \[ \begin {cases} \frac {b^{p} \left (x^{2}\right )^{p}}{2 p} & \text {for}\: p \neq 0 \\\log {\relax (x )} & \text {otherwise} \end {cases} \]

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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