3.170 \(\int (a+b \log (c x^n))^p \, dx\)

Optimal. Leaf size=80 \[ x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c x^n\right )}{b n}\right ) \]

[Out]

(x*Gamma[1 + p, -((a + b*Log[c*x^n])/(b*n))]*(a + b*Log[c*x^n])^p)/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-((a + b*Log[c
*x^n])/(b*n)))^p)

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Rubi [A]  time = 0.0375806, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2300, 2181} \[ x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c x^n\right )}{b n}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^p,x]

[Out]

(x*Gamma[1 + p, -((a + b*Log[c*x^n])/(b*n))]*(a + b*Log[c*x^n])^p)/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-((a + b*Log[c
*x^n])/(b*n)))^p)

Rule 2300

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

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

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

Mathematica [A]  time = 0.0675463, size = 80, normalized size = 1. \[ x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c x^n\right )}{b n}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^p,x]

[Out]

(x*Gamma[1 + p, -((a + b*Log[c*x^n])/(b*n))]*(a + b*Log[c*x^n])^p)/(E^(a/(b*n))*(c*x^n)^n^(-1)*(-((a + b*Log[c
*x^n])/(b*n)))^p)

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Maple [F]  time = 0.344, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.30775, size = 132, normalized size = 1.65 \begin{align*} e^{\left (-\frac{b n p \log \left (-\frac{1}{b n}\right ) + b \log \left (c\right ) + a}{b n}\right )} \Gamma \left (p + 1, -\frac{b n \log \left (x\right ) + b \log \left (c\right ) + a}{b n}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(-(b*n*p*log(-1/(b*n)) + b*log(c) + a)/(b*n))*gamma(p + 1, -(b*n*log(x) + b*log(c) + a)/(b*n))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c x^{n} \right )}\right )^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*x**n))**p, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)^p, x)