3.51 \(\int \frac{\log (d (\frac{1}{d}+f \sqrt{x})) (a+b \log (c x^n))}{x^3} \, dx\)

Optimal. Leaf size=289 \[ b d^4 f^4 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+\frac{1}{2} d^4 f^4 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{5 b d^3 f^3 n}{4 \sqrt{x}}+\frac{3 b d^2 f^2 n}{8 x}+\frac{1}{8} b d^4 f^4 n \log ^2(x)+\frac{1}{4} b d^4 f^4 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{8} b d^4 f^4 n \log (x)-\frac{7 b d f n}{36 x^{3/2}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{4 x^2} \]

[Out]

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

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Rubi [A]  time = 0.204623, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2454, 2395, 44, 2376, 2391, 2301} \[ b d^4 f^4 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+\frac{1}{2} d^4 f^4 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{5 b d^3 f^3 n}{4 \sqrt{x}}+\frac{3 b d^2 f^2 n}{8 x}+\frac{1}{8} b d^4 f^4 n \log ^2(x)+\frac{1}{4} b d^4 f^4 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{8} b d^4 f^4 n \log (x)-\frac{7 b d f n}{36 x^{3/2}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x^3,x]

[Out]

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

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 2391

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

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

Mathematica [A]  time = 0.250113, size = 207, normalized size = 0.72 \[ b d^4 f^4 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )-\frac{d f \left (9 d^3 f^3 x^{3/2} \log (x) \left (2 a+2 b \log \left (c x^n\right )+b n\right )+36 a d^2 f^2 x-18 a d f \sqrt{x}+12 a+6 b \left (6 d^2 f^2 x-3 d f \sqrt{x}+2\right ) \log \left (c x^n\right )-9 b d^3 f^3 n x^{3/2} \log ^2(x)+90 b d^2 f^2 n x-27 b d f n \sqrt{x}+14 b n\right )}{72 x^{3/2}}+\frac{\left (d^4 f^4 x^2-1\right ) \log \left (d f \sqrt{x}+1\right ) \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x^3,x]

[Out]

((-1 + d^4*f^4*x^2)*Log[1 + d*f*Sqrt[x]]*(2*a + b*n + 2*b*Log[c*x^n]))/(4*x^2) - (d*f*(12*a + 14*b*n - 18*a*d*
f*Sqrt[x] - 27*b*d*f*n*Sqrt[x] + 36*a*d^2*f^2*x + 90*b*d^2*f^2*n*x - 9*b*d^3*f^3*n*x^(3/2)*Log[x]^2 + 6*b*(2 -
3*d*f*Sqrt[x] + 6*d^2*f^2*x)*Log[c*x^n] + 9*d^3*f^3*x^(3/2)*Log[x]*(2*a + b*n + 2*b*Log[c*x^n])))/(72*x^(3/2)
) + b*d^4*f^4*n*PolyLog[2, -(d*f*Sqrt[x])]

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3}}\ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^(1/2)))/x^3,x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^(1/2)))/x^3,x)

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Maxima [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} + \frac{1}{d}\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^(1/2)))/x^3,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^3, 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 (d f \sqrt{x} + 1\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^(1/2)))/x^3,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log(d*f*sqrt(x) + 1)/x^3, 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*(1/d+f*x**(1/2)))/x**3,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} + \frac{1}{d}\right )} d\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(1/d+f*x^(1/2)))/x^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^3, x)