3.66 \(\int x^m (b x^2)^p \, dx\)

Optimal. Leaf size=21 \[ \frac {x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 21, 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} \[ \frac {x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*(b*x^2)^p)/(1 + m + 2*p)

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fricas [A]  time = 0.74, size = 24, normalized size = 1.14 \[ \frac {x x^{m} e^{\left (p \log \relax (b) + 2 \, p \log \relax (x)\right )}}{m + 2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*x^m*e^(p*log(b) + 2*p*log(x))/(m + 2*p + 1)

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giac [A]  time = 0.17, size = 24, normalized size = 1.14 \[ \frac {x x^{m} e^{\left (p \log \relax (b) + 2 \, p \log \relax (x)\right )}}{m + 2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*x^m*e^(p*log(b) + 2*p*log(x))/(m + 2*p + 1)

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maple [A]  time = 0.00, size = 22, normalized size = 1.05 \[ \frac {x^{m +1} \left (b \,x^{2}\right )^{p}}{m +2 p +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 24, normalized size = 1.14 \[ \frac {b^{p} x e^{\left (m \log \relax (x) + 2 \, p \log \relax (x)\right )}}{m + 2 \, p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^p*x*e^(m*log(x) + 2*p*log(x))/(m + 2*p + 1)

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mupad [B]  time = 1.07, size = 21, normalized size = 1.00 \[ \frac {x^{m+1}\,{\left (b\,x^2\right )}^p}{m+2\,p+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^(m + 1)*(b*x^2)^p)/(m + 2*p + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**p*x*x**m*(x**2)**p/(m + 2*p + 1), Ne(m, -2*p - 1)), (Integral(x**(-2*p - 1)*(b*x**2)**p, x), Tru
e))

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