3.119 \(\int \frac{\log (d (e+f \sqrt{x})^k) (a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=248 \[ -\frac{2 b f^2 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x}+\frac{b f^2 k n \log ^2(x)}{4 e^2}+\frac{b f^2 k n \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{2 b f^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{b f^2 k n \log (x)}{2 e^2}-\frac{3 b f k n}{e \sqrt{x}} \]

[Out]

(-3*b*f*k*n)/(e*Sqrt[x]) + (b*f^2*k*n*Log[e + f*Sqrt[x]])/e^2 - (b*n*Log[d*(e + f*Sqrt[x])^k])/x - (2*b*f^2*k*
n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (b*f^2*k*n*Log[x])/(2*e^2) + (b*f^2*k*n*Log[x]^2)/(4*e^2) -
(f*k*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (f^2*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2 - (Log[d*(e + f*Sqrt[
x])^k]*(a + b*Log[c*x^n]))/x - (f^2*k*Log[x]*(a + b*Log[c*x^n]))/(2*e^2) - (2*b*f^2*k*n*PolyLog[2, 1 + (f*Sqrt
[x])/e])/e^2

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Rubi [A]  time = 0.210073, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2454, 2395, 44, 2376, 2394, 2315, 2301} \[ -\frac{2 b f^2 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x}+\frac{b f^2 k n \log ^2(x)}{4 e^2}+\frac{b f^2 k n \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{2 b f^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{b f^2 k n \log (x)}{2 e^2}-\frac{3 b f k n}{e \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*f*k*n)/(e*Sqrt[x]) + (b*f^2*k*n*Log[e + f*Sqrt[x]])/e^2 - (b*n*Log[d*(e + f*Sqrt[x])^k])/x - (2*b*f^2*k*
n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (b*f^2*k*n*Log[x])/(2*e^2) + (b*f^2*k*n*Log[x]^2)/(4*e^2) -
(f*k*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (f^2*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2 - (Log[d*(e + f*Sqrt[
x])^k]*(a + b*Log[c*x^n]))/x - (f^2*k*Log[x]*(a + b*Log[c*x^n]))/(2*e^2) - (2*b*f^2*k*n*PolyLog[2, 1 + (f*Sqrt
[x])/e])/e^2

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-(b n) \int \left (-\frac{f k}{e x^{3/2}}+\frac{f^2 k \log \left (e+f \sqrt{x}\right )}{e^2 x}-\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x^2}-\frac{f^2 k \log (x)}{2 e^2 x}\right ) \, dx\\ &=-\frac{2 b f k n}{e \sqrt{x}}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+(b n) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x^2} \, dx+\frac{\left (b f^2 k n\right ) \int \frac{\log (x)}{x} \, dx}{2 e^2}-\frac{\left (b f^2 k n\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{e^2}\\ &=-\frac{2 b f k n}{e \sqrt{x}}+\frac{b f^2 k n \log ^2(x)}{4 e^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+(2 b n) \operatorname{Subst}\left (\int \frac{\log \left (d (e+f x)^k\right )}{x^3} \, dx,x,\sqrt{x}\right )-\frac{\left (2 b f^2 k n\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{e^2}\\ &=-\frac{2 b f k n}{e \sqrt{x}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x}-\frac{2 b f^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{b f^2 k n \log ^2(x)}{4 e^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+(b f k n) \operatorname{Subst}\left (\int \frac{1}{x^2 (e+f x)} \, dx,x,\sqrt{x}\right )+\frac{\left (2 b f^3 k n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{e^2}\\ &=-\frac{2 b f k n}{e \sqrt{x}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x}-\frac{2 b f^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{b f^2 k n \log ^2(x)}{4 e^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{2 b f^2 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e^2}+(b f k n) \operatorname{Subst}\left (\int \left (\frac{1}{e x^2}-\frac{f}{e^2 x}+\frac{f^2}{e^2 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{3 b f k n}{e \sqrt{x}}+\frac{b f^2 k n \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x}-\frac{2 b f^2 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{b f^2 k n \log (x)}{2 e^2}+\frac{b f^2 k n \log ^2(x)}{4 e^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{f^2 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac{2 b f^2 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e^2}\\ \end{align*}

Mathematica [A]  time = 0.307658, size = 250, normalized size = 1.01 \[ -\frac{-8 b f^2 k n x \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )-4 f^2 k x \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)+b n\right )+4 a e^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+4 a e f k \sqrt{x}+2 a f^2 k x \log (x)+4 b e^2 \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+4 b e f k \sqrt{x} \log \left (c x^n\right )+2 b f^2 k x \log (x) \log \left (c x^n\right )+4 b e^2 n \log \left (d \left (e+f \sqrt{x}\right )^k\right )-4 b f^2 k n x \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )+12 b e f k n \sqrt{x}-b f^2 k n x \log ^2(x)+2 b f^2 k n x \log (x)}{4 e^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4*a*e*f*k*Sqrt[x] + 12*b*e*f*k*n*Sqrt[x] + 4*a*e^2*Log[d*(e + f*Sqrt[x])^k] + 4*b*e^2*n*Log[d*(e + f*Sqrt[x]
)^k] + 2*a*f^2*k*x*Log[x] + 2*b*f^2*k*n*x*Log[x] - 4*b*f^2*k*n*x*Log[1 + (f*Sqrt[x])/e]*Log[x] - b*f^2*k*n*x*L
og[x]^2 + 4*b*e*f*k*Sqrt[x]*Log[c*x^n] + 4*b*e^2*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 2*b*f^2*k*x*Log[x]*Log[
c*x^n] - 4*f^2*k*x*Log[e + f*Sqrt[x]]*(a + b*n - b*n*Log[x] + b*Log[c*x^n]) - 8*b*f^2*k*n*x*PolyLog[2, -((f*Sq
rt[x])/e)])/(4*e^2*x)

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2}}\ln \left ( d \left ( e+f\sqrt{x} \right ) ^{k} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^2,x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b e \log \left (d\right ) \log \left (x^{n}\right ) + a e \log \left (d\right ) +{\left (e n \log \left (d\right ) + e \log \left (c\right ) \log \left (d\right )\right )} b +{\left (b e \log \left (x^{n}\right ) +{\left (e n + e \log \left (c\right )\right )} b + a e\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) + \frac{b f k x \log \left (x^{n}\right ) +{\left (a f k +{\left (3 \, f k n + f k \log \left (c\right )\right )} b\right )} x}{\sqrt{x}}}{e x} - \int \frac{b f^{2} k \log \left (x^{n}\right ) + a f^{2} k +{\left (f^{2} k n + f^{2} k \log \left (c\right )\right )} b}{2 \,{\left (e f x^{\frac{3}{2}} + e^{2} x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^2,x, algorithm="maxima")

[Out]

-(b*e*log(d)*log(x^n) + a*e*log(d) + (e*n*log(d) + e*log(c)*log(d))*b + (b*e*log(x^n) + (e*n + e*log(c))*b + a
*e)*log((f*sqrt(x) + e)^k) + (b*f*k*x*log(x^n) + (a*f*k + (3*f*k*n + f*k*log(c))*b)*x)/sqrt(x))/(e*x) - integr
ate(1/2*(b*f^2*k*log(x^n) + a*f^2*k + (f^2*k*n + f^2*k*log(c))*b)/(e*f*x^(3/2) + e^2*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^2,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/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))*ln(d*(e+f*x**(1/2))**k)/x**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^2,x, algorithm="giac")

[Out]

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