3.1.70 \(\int x^2 (b x^2)^p \, dx\) [70]

Optimal. Leaf size=18 \[ \frac {x^3 \left (b x^2\right )^p}{3+2 p} \]

[Out]

x^3*(b*x^2)^p/(3+2*p)

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Rubi [A]
time = 0.00, antiderivative size = 18, 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} \begin {gather*} \frac {x^3 \left (b x^2\right )^p}{2 p+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^3 \left (b x^2\right )^p}{3+2 p} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(b*x^2)^p)/(3 + 2*p)

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Maple [A]
time = 0.02, size = 19, normalized size = 1.06

method	result	size
gosper	\(\frac {x^{3} \left (b \,x^{2}\right )^{p}}{3+2 p}\)	\(19\)
risch	\(\frac {x^{3} \left (b \,x^{2}\right )^{p}}{3+2 p}\)	\(19\)
norman	\(\frac {x^{3} {\mathrm e}^{p \ln \left (b \,x^{2}\right )}}{3+2 p}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^2)^p,x,method=_RETURNVERBOSE)

[Out]

x^3*(b*x^2)^p/(3+2*p)

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Maxima [A]
time = 0.27, size = 19, normalized size = 1.06 \begin {gather*} \frac {b^{p} x^{3} x^{2 \, p}}{2 \, p + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^p*x^3*x^(2*p)/(2*p + 3)

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Fricas [A]
time = 0.35, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x^{3}}{2 \, p + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2)^p*x^3/(2*p + 3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x^{3} \left (b x^{2}\right )^{p}}{2 p + 3} & \text {for}\: p \neq - \frac {3}{2} \\\int \frac {x^{2}}{\left (b x^{2}\right )^{\frac {3}{2}}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*(b*x**2)**p/(2*p + 3), Ne(p, -3/2)), (Integral(x**2/(b*x**2)**(3/2), x), True))

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Giac [A]
time = 1.54, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (b x^{2}\right )^{p} x^{3}}{2 \, p + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x^2)^p*x^3/(2*p + 3)

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Mupad [B]
time = 0.98, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^3\,{\left (b\,x^2\right )}^p}{2\,p+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3*(b*x^2)^p)/(2*p + 3)

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