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

Optimal. Leaf size=19 \[ \frac{x^2 \left (b x^2\right )^p}{2 (p+1)} \]

[Out]

(x^2*(b*x^2)^p)/(2*(1 + p))

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Rubi [A]  time = 0.0046985, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{x^2 \left (b x^2\right )^p}{2 (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.001993, size = 18, normalized size = 0.95 \[ \frac{x^2 \left (b x^2\right )^p}{2 p+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(b*x^2)^p)/(2 + 2*p)

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Maple [A]  time = 0.002, size = 18, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( b{x}^{2} \right ) ^{p}}{2+2\,p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.82108, size = 36, normalized size = 1.89 \begin{align*} \frac{\left (b x^{2}\right )^{p} x^{2}}{2 \,{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2)^p*x^2/(p + 1)

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Sympy [A]  time = 0.24322, size = 22, normalized size = 1.16 \begin{align*} \begin{cases} \frac{b^{p} x^{2} \left (x^{2}\right )^{p}}{2 p + 2} & \text{for}\: p \neq -1 \\\frac{\log{\left (x \right )}}{b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15935, size = 26, normalized size = 1.37 \begin{align*} \frac{\left (b x^{2}\right )^{p + 1}}{2 \, b{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2)^(p + 1)/(b*(p + 1))