3.50 \(\int \frac{\log (d (\frac{1}{d}+f \sqrt{x})) (a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=196 \[ 2 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{1}{4} b d^2 f^2 n \log ^2(x)+b d^2 f^2 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{2} b d^2 f^2 n \log (x)-\frac{3 b d f n}{\sqrt{x}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.151145, antiderivative size = 196, 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} \[ 2 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+d^2 f^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{\sqrt{x}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{1}{4} b d^2 f^2 n \log ^2(x)+b d^2 f^2 n \log \left (d f \sqrt{x}+1\right )-\frac{1}{2} b d^2 f^2 n \log (x)-\frac{3 b d f n}{\sqrt{x}}-\frac{b n \log \left (d f \sqrt{x}+1\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x^2,x]

[Out]

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

Mathematica [A]  time = 0.191486, size = 124, normalized size = 0.63 \[ 2 b d^2 f^2 n \text{PolyLog}\left (2,-d f \sqrt{x}\right )-\frac{1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )+b n\right )+\frac{\left (d^2 f^2 x-1\right ) \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{x}-\frac{d f \left (a+b \log \left (c x^n\right )+3 b n\right )}{\sqrt{x}}+\frac{1}{4} b d^2 f^2 n \log ^2(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x^2,x]

[Out]

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

[Out]

int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^(1/2)))/x^2,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^{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*(1/d+f*x^(1/2)))/x^2,x, algorithm="maxima")

[Out]

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

[Out]

integral((b*log(c*x^n) + a)*log(d*f*sqrt(x) + 1)/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*(1/d+f*x**(1/2)))/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} + \frac{1}{d}\right )} 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*(1/d+f*x^(1/2)))/x^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^2, x)