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

Optimal. Leaf size=117 \[ -2 k \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b k n \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{2 b n}-\frac{k \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

[Out]

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

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Rubi [A]  time = 0.146134, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2375, 2337, 2374, 6589} \[ -2 k \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+4 b k n \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{2 b n}-\frac{k \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2375

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :
> Simp[(Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[(f*m*r)/(b*n*(p + 1)), Int[(
x^(m - 1)*(a + b*Log[c*x^n])^(p + 1))/(e + f*x^m), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,
0] && NeQ[d*e, 1]

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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} \, dx &=\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{(f k) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{4 b n}\\ &=\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{k \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+k \int \frac{\log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{k \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-2 k \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )+(2 b k n) \int \frac{\text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx\\ &=\frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{k \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-2 k \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )+4 b k n \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )\\ \end{align*}

Mathematica [A]  time = 0.165733, size = 186, normalized size = 1.59 \[ \frac{1}{2} \left (4 a k \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )-4 b k \log \left (c x^n\right ) \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )+8 b k n \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )+4 a \log \left (-\frac{f \sqrt{x}}{e}\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+2 b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-2 b k \log (x) \log \left (c x^n\right ) \log \left (\frac{f \sqrt{x}}{e}+1\right )-b n \log ^2(x) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+b k n \log ^2(x) \log \left (\frac{f \sqrt{x}}{e}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*Log[d*(e + f*Sqrt[x])^k]*Log[-((f*Sqrt[x])/e)] - b*n*Log[d*(e + f*Sqrt[x])^k]*Log[x]^2 + b*k*n*Log[1 + (f
*Sqrt[x])/e]*Log[x]^2 + 2*b*Log[d*(e + f*Sqrt[x])^k]*Log[x]*Log[c*x^n] - 2*b*k*Log[1 + (f*Sqrt[x])/e]*Log[x]*L
og[c*x^n] + 4*a*k*PolyLog[2, 1 + (f*Sqrt[x])/e] - 4*b*k*Log[c*x^n]*PolyLog[2, -((f*Sqrt[x])/e)] + 8*b*k*n*Poly
Log[3, -((f*Sqrt[x])/e)])/2

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{x}\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,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b e n \log \left (d\right ) \log \left (x\right )^{2} - 2 \, b e \log \left (d\right ) \log \left (x\right ) \log \left (x^{n}\right ) +{\left (b e n \log \left (x\right )^{2} - 2 \, b e \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b e \log \left (c\right ) + a e\right )} \log \left (x\right )\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) - 2 \,{\left (b e \log \left (c\right ) \log \left (d\right ) + a e \log \left (d\right )\right )} \log \left (x\right ) - \frac{b f k n x \log \left (x\right )^{2} - 2 \,{\left (b f k \log \left (c\right ) + a f k\right )} x \log \left (x\right ) + 4 \,{\left (a f k -{\left (2 \, f k n - f k \log \left (c\right )\right )} b\right )} x - 2 \,{\left (b f k x \log \left (x\right ) - 2 \, b f k x\right )} \log \left (x^{n}\right )}{\sqrt{x}}}{2 \, e} + \int -\frac{b f^{2} k n \log \left (x\right )^{2} - 2 \, b f^{2} k \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b f^{2} k \log \left (c\right ) + a f^{2} k\right )} \log \left (x\right )}{4 \,{\left (e f \sqrt{x} + e^{2}\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,x, algorithm="maxima")

[Out]

-1/2*(b*e*n*log(d)*log(x)^2 - 2*b*e*log(d)*log(x)*log(x^n) + (b*e*n*log(x)^2 - 2*b*e*log(x)*log(x^n) - 2*(b*e*
log(c) + a*e)*log(x))*log((f*sqrt(x) + e)^k) - 2*(b*e*log(c)*log(d) + a*e*log(d))*log(x) - (b*f*k*n*x*log(x)^2
- 2*(b*f*k*log(c) + a*f*k)*x*log(x) + 4*(a*f*k - (2*f*k*n - f*k*log(c))*b)*x - 2*(b*f*k*x*log(x) - 2*b*f*k*x)
*log(x^n))/sqrt(x))/e + integrate(-1/4*(b*f^2*k*n*log(x)^2 - 2*b*f^2*k*log(x)*log(x^n) - 2*(b*f^2*k*log(c) + a
*f^2*k)*log(x))/(e*f*sqrt(x) + e^2), 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}, 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,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x, 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,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}\,{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,x, algorithm="giac")

[Out]

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