3.1.66 \(\int x^m (b x^2)^p \, dx\) [66]

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

[Out]

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

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Rubi [A]
time = 0.00, 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} \begin {gather*} \frac {x^{m+1} \left (b x^2\right )^p}{m+2 p+1} \end {gather*}

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 \begin {gather*} \frac {x^{1+m} \left (b x^2\right )^p}{1+m+2 p} \end {gather*}

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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Maple [A]
time = 0.06, size = 22, normalized size = 1.05

method	result	size
gosper	\(\frac {x^{1+m} \left (b \,x^{2}\right )^{p}}{1+m +2 p}\)	\(22\)
norman	\(\frac {x \,{\mathrm e}^{m \ln \left (x \right )} {\mathrm e}^{p \ln \left (b \,x^{2}\right )}}{1+m +2 p}\)	\(25\)
risch	\(\frac {x \,x^{m} {\mathrm e}^{\frac {p \left (-i \mathrm {csgn}\left (i x^{2}\right )^{3} \pi +2 i \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi -i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2} \pi +i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i b \,x^{2}\right )^{2} \pi -i \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i b \,x^{2}\right ) \mathrm {csgn}\left (i b \right ) \pi -i \mathrm {csgn}\left (i b \,x^{2}\right )^{3} \pi +i \mathrm {csgn}\left (i b \,x^{2}\right )^{2} \mathrm {csgn}\left (i b \right ) \pi +4 \ln \left (x \right )+2 \ln \left (b \right )\right )}{2}}}{1+m +2 p}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 24, normalized size = 1.14 \begin {gather*} \frac {b^{p} x e^{\left (m \log \left (x\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \end {gather*}

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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Fricas [A]
time = 0.36, size = 24, normalized size = 1.14 \begin {gather*} \frac {x x^{m} e^{\left (p \log \left (b\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x x^{m} \left (b 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.42, size = 24, normalized size = 1.14 \begin {gather*} \frac {x x^{m} e^{\left (p \log \left (b\right ) + 2 \, p \log \left (x\right )\right )}}{m + 2 \, p + 1} \end {gather*}

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

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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