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

Optimal. Leaf size=313 \[ \frac{b e^4 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^4}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{5 b e^3 k n \sqrt{x}}{4 f^3}+\frac{3 b e^2 k n x}{8 f^2}+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right )}{4 f^4}+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}-\frac{7 b e k n x^{3/2}}{36 f}+\frac{1}{8} b k n x^2 \]

[Out]

(-5*b*e^3*k*n*Sqrt[x])/(4*f^3) + (3*b*e^2*k*n*x)/(8*f^2) - (7*b*e*k*n*x^(3/2))/(36*f) + (b*k*n*x^2)/8 + (b*e^4
*k*n*Log[e + f*Sqrt[x]])/(4*f^4) - (b*n*x^2*Log[d*(e + f*Sqrt[x])^k])/4 + (b*e^4*k*n*Log[e + f*Sqrt[x]]*Log[-(
(f*Sqrt[x])/e)])/f^4 + (e^3*k*Sqrt[x]*(a + b*Log[c*x^n]))/(2*f^3) - (e^2*k*x*(a + b*Log[c*x^n]))/(4*f^2) + (e*
k*x^(3/2)*(a + b*Log[c*x^n]))/(6*f) - (k*x^2*(a + b*Log[c*x^n]))/8 - (e^4*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^
n]))/(2*f^4) + (x^2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/2 + (b*e^4*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])
/f^4

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Rubi [A]  time = 0.244333, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2454, 2395, 43, 2376, 2394, 2315} \[ \frac{b e^4 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{f^4}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{5 b e^3 k n \sqrt{x}}{4 f^3}+\frac{3 b e^2 k n x}{8 f^2}+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right )}{4 f^4}+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}-\frac{7 b e k n x^{3/2}}{36 f}+\frac{1}{8} b k n x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*e^3*k*n*Sqrt[x])/(4*f^3) + (3*b*e^2*k*n*x)/(8*f^2) - (7*b*e*k*n*x^(3/2))/(36*f) + (b*k*n*x^2)/8 + (b*e^4
*k*n*Log[e + f*Sqrt[x]])/(4*f^4) - (b*n*x^2*Log[d*(e + f*Sqrt[x])^k])/4 + (b*e^4*k*n*Log[e + f*Sqrt[x]]*Log[-(
(f*Sqrt[x])/e)])/f^4 + (e^3*k*Sqrt[x]*(a + b*Log[c*x^n]))/(2*f^3) - (e^2*k*x*(a + b*Log[c*x^n]))/(4*f^2) + (e*
k*x^(3/2)*(a + b*Log[c*x^n]))/(6*f) - (k*x^2*(a + b*Log[c*x^n]))/8 - (e^4*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^
n]))/(2*f^4) + (x^2*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/2 + (b*e^4*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])
/f^4

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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]

Rubi steps

\begin{align*} \int x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{e^2 k}{4 f^2}+\frac{e^3 k}{2 f^3 \sqrt{x}}+\frac{e k \sqrt{x}}{6 f}-\frac{k x}{8}-\frac{e^4 k \log \left (e+f \sqrt{x}\right )}{2 f^4 x}+\frac{1}{2} x \log \left (d \left (e+f \sqrt{x}\right )^k\right )\right ) \, dx\\ &=-\frac{b e^3 k n \sqrt{x}}{f^3}+\frac{b e^2 k n x}{4 f^2}-\frac{b e k n x^{3/2}}{9 f}+\frac{1}{16} b k n x^2+\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} (b n) \int x \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \, dx+\frac{\left (b e^4 k n\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{2 f^4}\\ &=-\frac{b e^3 k n \sqrt{x}}{f^3}+\frac{b e^2 k n x}{4 f^2}-\frac{b e k n x^{3/2}}{9 f}+\frac{1}{16} b k n x^2+\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \operatorname{Subst}\left (\int x^3 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt{x}\right )+\frac{\left (b e^4 k n\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{f^4}\\ &=-\frac{b e^3 k n \sqrt{x}}{f^3}+\frac{b e^2 k n x}{4 f^2}-\frac{b e k n x^{3/2}}{9 f}+\frac{1}{16} b k n x^2-\frac{1}{4} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\left (b e^4 k n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{f^3}+\frac{1}{4} (b f k n) \operatorname{Subst}\left (\int \frac{x^4}{e+f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{b e^3 k n \sqrt{x}}{f^3}+\frac{b e^2 k n x}{4 f^2}-\frac{b e k n x^{3/2}}{9 f}+\frac{1}{16} b k n x^2-\frac{1}{4} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{1}{4} (b f k n) \operatorname{Subst}\left (\int \left (-\frac{e^3}{f^4}+\frac{e^2 x}{f^3}-\frac{e x^2}{f^2}+\frac{x^3}{f}+\frac{e^4}{f^4 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{5 b e^3 k n \sqrt{x}}{4 f^3}+\frac{3 b e^2 k n x}{8 f^2}-\frac{7 b e k n x^{3/2}}{36 f}+\frac{1}{8} b k n x^2+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right )}{4 f^4}-\frac{1}{4} b n x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{b e^4 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{f^4}+\frac{e^3 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac{e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac{e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac{1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^4 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac{1}{2} x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{b e^4 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{f^4}\\ \end{align*}

Mathematica [A]  time = 0.348839, size = 336, normalized size = 1.07 \[ -\frac{72 b e^4 k n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )+18 e^4 k \log \left (e+f \sqrt{x}\right ) \left (2 a+2 b \log \left (c x^n\right )-2 b n \log (x)-b n\right )-36 a f^4 x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+18 a e^2 f^2 k x-36 a e^3 f k \sqrt{x}-12 a e f^3 k x^{3/2}+9 a f^4 k x^2-36 b f^4 x^2 \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+18 b e^2 f^2 k x \log \left (c x^n\right )-36 b e^3 f k \sqrt{x} \log \left (c x^n\right )-12 b e f^3 k x^{3/2} \log \left (c x^n\right )+9 b f^4 k x^2 \log \left (c x^n\right )+18 b f^4 n x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-27 b e^2 f^2 k n x+90 b e^3 f k n \sqrt{x}+36 b e^4 k n \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )+14 b e f^3 k n x^{3/2}-9 b f^4 k n x^2}{72 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-36*a*e^3*f*k*Sqrt[x] + 90*b*e^3*f*k*n*Sqrt[x] + 18*a*e^2*f^2*k*x - 27*b*e^2*f^2*k*n*x - 12*a*e*f^3*k*x^(3/2
) + 14*b*e*f^3*k*n*x^(3/2) + 9*a*f^4*k*x^2 - 9*b*f^4*k*n*x^2 - 36*a*f^4*x^2*Log[d*(e + f*Sqrt[x])^k] + 18*b*f^
4*n*x^2*Log[d*(e + f*Sqrt[x])^k] + 36*b*e^4*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 36*b*e^3*f*k*Sqrt[x]*Log[c*x^n
] + 18*b*e^2*f^2*k*x*Log[c*x^n] - 12*b*e*f^3*k*x^(3/2)*Log[c*x^n] + 9*b*f^4*k*x^2*Log[c*x^n] - 36*b*f^4*x^2*Lo
g[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 18*e^4*k*Log[e + f*Sqrt[x]]*(2*a - b*n - 2*b*n*Log[x] + 2*b*Log[c*x^n]) +
72*b*e^4*k*n*PolyLog[2, -((f*Sqrt[x])/e)])/(72*f^4)

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \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(x*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{50 \, b e x^{2} \log \left (d\right ) \log \left (x^{n}\right ) + 25 \,{\left (2 \, a e \log \left (d\right ) -{\left (e n \log \left (d\right ) - 2 \, e \log \left (c\right ) \log \left (d\right )\right )} b\right )} x^{2} + 25 \,{\left (2 \, b e x^{2} \log \left (x^{n}\right ) -{\left ({\left (e n - 2 \, e \log \left (c\right )\right )} b - 2 \, a e\right )} x^{2}\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) - \frac{10 \, b f k x^{3} \log \left (x^{n}\right ) +{\left (10 \, a f k -{\left (9 \, f k n - 10 \, f k \log \left (c\right )\right )} b\right )} x^{3}}{\sqrt{x}}}{100 \, e} + \int \frac{2 \, b f^{2} k x^{2} \log \left (x^{n}\right ) +{\left (2 \, a f^{2} k -{\left (f^{2} k n - 2 \, f^{2} k \log \left (c\right )\right )} b\right )} x^{2}}{8 \,{\left (e f \sqrt{x} + e^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/100*(50*b*e*x^2*log(d)*log(x^n) + 25*(2*a*e*log(d) - (e*n*log(d) - 2*e*log(c)*log(d))*b)*x^2 + 25*(2*b*e*x^2
*log(x^n) - ((e*n - 2*e*log(c))*b - 2*a*e)*x^2)*log((f*sqrt(x) + e)^k) - (10*b*f*k*x^3*log(x^n) + (10*a*f*k -
(9*f*k*n - 10*f*k*log(c))*b)*x^3)/sqrt(x))/e + integrate(1/8*(2*b*f^2*k*x^2*log(x^n) + (2*a*f^2*k - (f^2*k*n -
2*f^2*k*log(c))*b)*x^2)/(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 ({\left (b x \log \left (c x^{n}\right ) + a x\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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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 )} x \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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