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

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

[Out]

(x^3*(b*x^2)^p)/(3 + 2*p)

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Rubi [A]  time = 0.0043131, antiderivative size = 18, 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} \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]

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*}

Mathematica [A]  time = 0.0020921, size = 18, normalized size = 1. \[ \frac{x^3 \left (b x^2\right )^p}{2 p+3} \]

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 = 0.001, size = 19, normalized size = 1.1 \begin{align*}{\frac{{x}^{3} \left ( b{x}^{2} \right ) ^{p}}{3+2\,p}} \end{align*}

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/(3+2*p)

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Maxima [A]  time = 1.05078, size = 26, normalized size = 1.44 \begin{align*} \frac{b^{p} x^{3} x^{2 \, p}}{2 \, p + 3} \end{align*}

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 = 1.97141, size = 34, normalized size = 1.89 \begin{align*} \frac{\left (b x^{2}\right )^{p} x^{3}}{2 \, p + 3} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.14206, size = 24, normalized size = 1.33 \begin{align*} \frac{\left (b x^{2}\right )^{p} x^{3}}{2 \, p + 3} \end{align*}

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)