3.2.89 \(\int x^m (a (b x^n)^p)^q \, dx\) [189]

Optimal. Leaf size=26 \[ \frac {x^{1+m} \left (a \left (b x^n\right )^p\right )^q}{1+m+n p q} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1971, 30} \begin {gather*} \frac {x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 1971

Int[(u_.)*((c_.)*((d_.)*((a_.) + (b_.)*(x_))^(n_))^(q_))^(p_), x_Symbol] :> Dist[(c*(d*(a + b*x)^n)^q)^p/(a +
b*x)^(n*p*q), Int[u*(a + b*x)^(n*p*q), x], x] /; FreeQ[{a, b, c, d, n, q, p}, x] &&  !IntegerQ[q] &&  !Integer
Q[p]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \left (a \left (b x^n\right )^p\right )^q}{1+m+n p q} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.24, size = 27, normalized size = 1.04

method	result	size
gosper	\(\frac {x^{1+m} \left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{n p q +m +1}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a*(b*x^n)^p)^q,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.38, size = 33, normalized size = 1.27 \begin {gather*} \frac {a^{q} b^{p q} x e^{\left (m \log \left (x\right ) + q \log \left ({\left (x^{n}\right )}^{p}\right )\right )}}{n p q + m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 31, normalized size = 1.19 \begin {gather*} \frac {x x^{m} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a*(b*x**n)**p)**q,x)

[Out]

Integral(x**m*(a*(b*x**n)**p)**q, x)

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Giac [A]
time = 2.85, size = 31, normalized size = 1.19 \begin {gather*} \frac {x x^{m} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.23, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^{m+1}\,{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{m+n\,p\,q+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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