3.174 \(\int x^2 (b x^n)^p \, dx\)

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

[Out]

(x^3*(b*x^n)^p)/(3 + n*p)

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Rubi [A]  time = 0.0053697, 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} \[ \frac{x^3 \left (b x^n\right )^p}{n p+3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(b*x^n)^p)/(3 + 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^2 \left (b x^n\right )^p \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{2+n p} \, dx\\ &=\frac{x^3 \left (b x^n\right )^p}{3+n p}\\ \end{align*}

Mathematica [A]  time = 0.0026344, size = 18, normalized size = 1. \[ \frac{x^3 \left (b x^n\right )^p}{n p+3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(b*x^n)^p)/(3 + n*p)

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Maple [A]  time = 0.003, size = 19, normalized size = 1.1 \begin{align*}{\frac{{x}^{3} \left ( b{x}^{n} \right ) ^{p}}{np+3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*(b*x^n)^p/(n*p+3)

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Maxima [A]  time = 0.981741, size = 26, normalized size = 1.44 \begin{align*} \frac{b^{p} x^{3}{\left (x^{n}\right )}^{p}}{n p + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^p*x^3*(x^n)^p/(n*p + 3)

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Fricas [A]  time = 1.81083, size = 55, normalized size = 3.06 \begin{align*} \frac{x^{3} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*e^(n*p*log(x) + p*log(b))/(n*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**n)**p,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.14393, size = 30, normalized size = 1.67 \begin{align*} \frac{x^{3} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*e^(n*p*log(x) + p*log(b))/(n*p + 3)