3.1.72 \(\int (b x^2)^p \, dx\)

Optimal. Leaf size=16 \[ \frac {x \left (b x^2\right )^p}{2 p+1} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30} \begin {gather*} \frac {x \left (b x^2\right )^p}{2 p+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x \left (b x^2\right )^p}{2 p+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x^2)^p)/(1 + 2*p)

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IntegrateAlgebraic [F]  time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x^2\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x^2)^p,x]

[Out]

Defer[IntegrateAlgebraic][(b*x^2)^p, x]

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fricas [A]  time = 1.25, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x}{2 \, p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2)^p*x/(2*p + 1)

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giac [A]  time = 0.19, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x}{2 \, p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2)^p*x/(2*p + 1)

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maple [A]  time = 0.00, size = 17, normalized size = 1.06 \begin {gather*} \frac {x \left (b \,x^{2}\right )^{p}}{2 p +1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 17, normalized size = 1.06 \begin {gather*} \frac {b^{p} x x^{2 \, p}}{2 \, p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^p*x*x^(2*p)/(2*p + 1)

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mupad [B]  time = 1.10, size = 16, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (b\,x^2\right )}^p}{2\,p+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(b*x^2)^p)/(2*p + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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