3.246 \(\int \frac{(a+b \log (c (d x^m)^n))^p}{x} \, dx\)

Optimal. Leaf size=33 \[ \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \]

[Out]

(a + b*Log[c*(d*x^m)^n])^(1 + p)/(b*m*n*(1 + p))

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Rubi [A]  time = 0.0939415, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2302, 30, 2445} \[ \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d*x^m)^n])^p/x,x]

[Out]

(a + b*Log[c*(d*x^m)^n])^(1 + p)/(b*m*n*(1 + p))

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

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

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^n x^{m n}\right )\right )^p}{x} \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int x^p \, dx,x,a+b \log \left (c d^n x^{m n}\right )\right )}{b m n},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{1+p}}{b m n (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.008967, size = 33, normalized size = 1. \[ \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d*x^m)^n])^p/x,x]

[Out]

(a + b*Log[c*(d*x^m)^n])^(1 + p)/(b*m*n*(1 + p))

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Maple [A]  time = 0.006, size = 34, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{n} \right ) \right ) ^{1+p}}{mnb \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d*x^m)^n))^p/x,x)

[Out]

(a+b*ln(c*(d*x^m)^n))^(1+p)/b/m/n/(1+p)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*x^m)^n))^p/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.89625, size = 144, normalized size = 4.36 \begin{align*} \frac{{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}^{p}}{b m n p + b m n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*x^m)^n))^p/x,x, algorithm="fricas")

[Out]

(b*m*n*log(x) + b*n*log(d) + b*log(c) + a)*(b*m*n*log(x) + b*n*log(d) + b*log(c) + a)^p/(b*m*n*p + b*m*n)

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Sympy [A]  time = 5.96086, size = 80, normalized size = 2.42 \begin{align*} - \begin{cases} - a^{p} \log{\left (x \right )} & \text{for}\: b = 0 \\- \left (a + b \log{\left (c d^{n} \right )}\right )^{p} \log{\left (x \right )} & \text{for}\: m = 0 \\- \left (a + b \log{\left (c \right )}\right )^{p} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\left (a + b \log{\left (c \left (d x^{m}\right )^{n} \right )}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + b \log{\left (c \left (d x^{m}\right )^{n} \right )} \right )} & \text{otherwise} \end{cases}}{b m n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d*x**m)**n))**p/x,x)

[Out]

-Piecewise((-a**p*log(x), Eq(b, 0)), (-(a + b*log(c*d**n))**p*log(x), Eq(m, 0)), (-(a + b*log(c))**p*log(x), E
q(n, 0)), (-Piecewise(((a + b*log(c*(d*x**m)**n))**(p + 1)/(p + 1), Ne(p, -1)), (log(a + b*log(c*(d*x**m)**n))
, True))/(b*m*n), True))

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Giac [A]  time = 1.31088, size = 49, normalized size = 1.48 \begin{align*} \frac{{\left (b m n \log \left (x\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a\right )}^{p + 1}}{b m n{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d*x^m)^n))^p/x,x, algorithm="giac")

[Out]

(b*m*n*log(x) + b*n*log(d) + b*log(c) + a)^(p + 1)/(b*m*n*(p + 1))