3.3.46 \(\int \frac {(a+b \log (c (d x^m)^n))^p}{x} \, dx\) [246]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2339, 30, 2495} \begin {gather*} \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{p+1}}{b m n (p+1)} \end {gather*}

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 30

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

Rule 2339

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 2495

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 &=\text {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 )\\ &=\text {Subst}\left (\frac {\text {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*}

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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^{1+p}}{b m n (1+p)} \end {gather*}

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 = 0.04, size = 34, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {\left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{1+p}}{b m n \left (1+p \right )}\)	\(34\)
default	\(\frac {\left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )^{1+p}}{b m n \left (1+p \right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 33, normalized size = 1.00 \begin {gather*} \frac {{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p + 1}}{b m n {\left (p + 1\right )}} \end {gather*}

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]

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

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Fricas [A]
time = 0.38, size = 49, normalized size = 1.48 \begin {gather*} \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 {gather*}

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 = 1.46, size = 80, normalized size = 2.42 \begin {gather*} - \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 {gather*}

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 = 5.62, size = 36, normalized size = 1.09 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 4.07, size = 33, normalized size = 1.00 \begin {gather*} \frac {{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^{p+1}}{b\,m\,n\,\left (p+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d*x^m)^n))^p/x,x)

[Out]

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

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