3.2.83 \(\int \frac {(a+b \log (c x^n))^p (d+e \log (f x^r))}{x^2} \, dx\) [183]

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

[Out]

-e*exp(a/b/n)*r*(c*x^n)^(1/n)*GAMMA(2+p,a/b/n+ln(c*x^n)/n)*(a+b*ln(c*x^n))^p/x/(((a+b*ln(c*x^n))/b/n)^p)+e*exp
(a/b/n)*r*(c*x^n)^(1/n)*GAMMA(1+p,a/b/n+ln(c*x^n)/n)*(a+b*ln(c*x^n))^(1+p)/b/n/x/(((a+b*ln(c*x^n))/b/n)^p)-exp
(a/b/n)*(c*x^n)^(1/n)*GAMMA(1+p,(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/x/(((a+b*ln(c*x^n))/b/n
)^p)

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Rubi [A]
time = 0.15, antiderivative size = 260, normalized size of antiderivative = 1.00, 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 = {2347, 2212, 2413, 12, 15, 19, 6692} \begin {gather*} -\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} \end {gather*}

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 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 2212

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

Rule 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2413

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 6692

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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.24, size = 141, normalized size = 0.54 \begin {gather*} -\frac {e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (a+b \log \left (c x^n\right )\right )^{-1+p} \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (b e n r \Gamma \left (2+p,\frac {a+b \log \left (c x^n\right )}{b n}\right )+\Gamma \left (1+p,\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (b d n-a e r-b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right )}{x} \end {gather*}

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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{p} \left (d +e \ln \left (f \,x^{r}\right )\right )}{x^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (d+e\,\ln \left (f\,x^r\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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