3.183 \(\int \frac{(a+b \log (c x^n))^p (d+e \log (f x^r))}{x^2} \, dx\)

Optimal. Leaf size=260 \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c x^n\right )}{b n}\right )}{x}-\frac{e r e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+2,\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{x}+\frac{e r e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{b n x} \]

[Out]

-((e*E^(a/(b*n))*r*(c*x^n)^n^(-1)*Gamma[2 + p, a/(b*n) + Log[c*x^n]/n]*(a + b*Log[c*x^n])^p)/(x*((a + b*Log[c*
x^n])/(b*n))^p)) + (e*E^(a/(b*n))*r*(c*x^n)^n^(-1)*Gamma[1 + p, a/(b*n) + Log[c*x^n]/n]*(a + b*Log[c*x^n])^(1
+ p))/(b*n*x*((a + b*Log[c*x^n])/(b*n))^p) - (E^(a/(b*n))*(c*x^n)^n^(-1)*Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)
]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(x*((a + b*Log[c*x^n])/(b*n))^p)

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Rubi [A]  time = 0.226611, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2310, 2181, 2366, 12, 15, 19, 6557} \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c x^n\right )}{b n}\right )}{x}-\frac{e r e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+2,\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{x}+\frac{e r e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{b n x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/x^2,x]

[Out]

-((e*E^(a/(b*n))*r*(c*x^n)^n^(-1)*Gamma[2 + p, a/(b*n) + Log[c*x^n]/n]*(a + b*Log[c*x^n])^p)/(x*((a + b*Log[c*
x^n])/(b*n))^p)) + (e*E^(a/(b*n))*r*(c*x^n)^n^(-1)*Gamma[1 + p, a/(b*n) + Log[c*x^n]/n]*(a + b*Log[c*x^n])^(1
+ p))/(b*n*x*((a + b*Log[c*x^n])/(b*n))^p) - (E^(a/(b*n))*(c*x^n)^n^(-1)*Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)
]*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/(x*((a + b*Log[c*x^n])/(b*n))^p)

Rule 2310

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

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 19

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + n)*(b*v)^n)/(a*v)^n, Int[u*v^(m + n),
x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[m + n]

Rule 6557

Int[Gamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*Gamma[n, a + b*x])/b, x] - Simp[Gamma[n + 1, a
+ b*x]/b, x] /; FreeQ[{a, b, n}, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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} \left (d+e \log \left (f x^r\right )\right )}{x}+(e r) \int \frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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}}{x^2} \, dx\\ &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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} \left (d+e \log \left (f x^r\right )\right )}{x}+\left (e e^{\frac{a}{b n}} r\right ) \int \frac{\left (c x^n\right )^{\frac{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}}{x^2} \, dx\\ &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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} \left (d+e \log \left (f x^r\right )\right )}{x}+\frac{\left (e e^{\frac{a}{b n}} r \left (c x^n\right )^{\frac{1}{n}}\right ) \int \frac{\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}}{x} \, dx}{x}\\ &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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} \left (d+e \log \left (f x^r\right )\right )}{x}+\frac{\left (e e^{\frac{a}{b n}} r \left (c x^n\right )^{\frac{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}\right ) \int \frac{\Gamma \left (1+p,\frac{a+b \log \left (c x^n\right )}{b n}\right )}{x} \, dx}{x}\\ &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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} \left (d+e \log \left (f x^r\right )\right )}{x}+\frac{\left (e e^{\frac{a}{b n}} r \left (c x^n\right )^{\frac{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}\right ) \operatorname{Subst}\left (\int \Gamma \left (1+p,\frac{a+b x}{b n}\right ) \, dx,x,\log \left (c x^n\right )\right )}{n x}\\ &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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} \left (d+e \log \left (f x^r\right )\right )}{x}+\frac{\left (e e^{\frac{a}{b n}} r \left (c x^n\right )^{\frac{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}\right ) \operatorname{Subst}\left (\int \Gamma (1+p,x) \, dx,x,\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{x}\\ &=-\frac{e e^{\frac{a}{b n}} r \left (c x^n\right )^{\frac{1}{n}} \Gamma \left (2+p,\frac{a}{b n}+\frac{\log \left (c x^n\right )}{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}}{x}+\frac{e e^{\frac{a}{b n}} r \left (c x^n\right )^{\frac{1}{n}} \Gamma \left (1+p,\frac{a}{b n}+\frac{\log \left (c x^n\right )}{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} \left (\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{x}-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{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} \left (d+e \log \left (f x^r\right )\right )}{x}\\ \end{align*}

Mathematica [A]  time = 0.316715, size = 141, normalized size = 0.54 \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^{p-1} \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (\text{Gamma}\left (p+1,\frac{a+b \log \left (c x^n\right )}{b n}\right ) \left (-a e r-b e r \log \left (c x^n\right )+b d n+b e n \log \left (f x^r\right )\right )+b e n r \text{Gamma}\left (p+2,\frac{a+b \log \left (c x^n\right )}{b n}\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]))/x^2,x]

[Out]

-((E^(a/(b*n))*(c*x^n)^n^(-1)*(a + b*Log[c*x^n])^(-1 + p)*((a + b*Log[c*x^n])/(b*n))^(1 - p)*(b*e*n*r*Gamma[2
+ p, (a + b*Log[c*x^n])/(b*n)] + Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)]*(b*d*n - a*e*r - b*e*r*Log[c*x^n] + b*
e*n*Log[f*x^r])))/x)

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Maple [F]  time = 0.5, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p} \left ( d+e\ln \left ( f{x}^{r} \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/x^2,x)

[Out]

int((a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/x^2,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*(d+e*log(f*x^r))/x^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e \log \left (f x^{r}\right ) + d\right )}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^2,x, algorithm="fricas")

[Out]

integral((e*log(f*x^r) + d)*(b*log(c*x^n) + a)^p/x^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**p*(d+e*ln(f*x**r))/x**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e \log \left (f x^{r}\right ) + d\right )}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^p*(d+e*log(f*x^r))/x^2,x, algorithm="giac")

[Out]

integrate((e*log(f*x^r) + d)*(b*log(c*x^n) + a)^p/x^2, x)