3.2712 \(\int x^m (b x^n)^p \, dx\)

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

[Out]

(x^(1 + m)*(b*x^n)^p)/(1 + m + n*p)

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Rubi [A]  time = 0.0062687, antiderivative size = 21, 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} \[ \frac{x^{m+1} \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.004, size = 21, normalized size = 1. \[ \frac{x^{m+1} \left (b x^n\right )^p}{m+n p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*(b*x^n)^p)/(1 + m + n*p)

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Maple [A]  time = 0.001, size = 22, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+m} \left ( b{x}^{n} \right ) ^{p}}{np+m+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959962, size = 34, normalized size = 1.62 \begin{align*} \frac{b^{p} x e^{\left (m \log \left (x\right ) + p \log \left (x^{n}\right )\right )}}{n p + m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42178, size = 63, normalized size = 3. \begin{align*} \frac{x x^{m} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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**m*(b*x**n)**p,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.12855, size = 32, normalized size = 1.52 \begin{align*} \frac{x x^{m} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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