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

Optimal. Leaf size=350 \[ -\frac{2 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{3 d^6 f^6}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{5 b n x^2}{72 d^2 f^2}-\frac{b n x^{3/2}}{9 d^3 f^3}+\frac{2 b n x}{9 d^4 f^4}-\frac{7 b n \sqrt{x}}{9 d^5 f^5}+\frac{b n \log \left (d f \sqrt{x}+1\right )}{9 d^6 f^6}-\frac{11 b n x^{5/2}}{225 d f}-\frac{1}{9} b n x^3 \log \left (d f \sqrt{x}+1\right )+\frac{1}{27} b n x^3 \]

[Out]

(-7*b*n*Sqrt[x])/(9*d^5*f^5) + (2*b*n*x)/(9*d^4*f^4) - (b*n*x^(3/2))/(9*d^3*f^3) + (5*b*n*x^2)/(72*d^2*f^2) -
(11*b*n*x^(5/2))/(225*d*f) + (b*n*x^3)/27 + (b*n*Log[1 + d*f*Sqrt[x]])/(9*d^6*f^6) - (b*n*x^3*Log[1 + d*f*Sqrt
[x]])/9 + (Sqrt[x]*(a + b*Log[c*x^n]))/(3*d^5*f^5) - (x*(a + b*Log[c*x^n]))/(6*d^4*f^4) + (x^(3/2)*(a + b*Log[
c*x^n]))/(9*d^3*f^3) - (x^2*(a + b*Log[c*x^n]))/(12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n]))/(15*d*f) - (x^3*(a
+ b*Log[c*x^n]))/18 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*d^6*f^6) + (x^3*Log[1 + d*f*Sqrt[x]]*(a +
b*Log[c*x^n]))/3 - (2*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(3*d^6*f^6)

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Rubi [A]  time = 0.276543, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2454, 2395, 43, 2376, 2391} \[ -\frac{2 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{3 d^6 f^6}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{5 b n x^2}{72 d^2 f^2}-\frac{b n x^{3/2}}{9 d^3 f^3}+\frac{2 b n x}{9 d^4 f^4}-\frac{7 b n \sqrt{x}}{9 d^5 f^5}+\frac{b n \log \left (d f \sqrt{x}+1\right )}{9 d^6 f^6}-\frac{11 b n x^{5/2}}{225 d f}-\frac{1}{9} b n x^3 \log \left (d f \sqrt{x}+1\right )+\frac{1}{27} b n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b*n*Sqrt[x])/(9*d^5*f^5) + (2*b*n*x)/(9*d^4*f^4) - (b*n*x^(3/2))/(9*d^3*f^3) + (5*b*n*x^2)/(72*d^2*f^2) -
(11*b*n*x^(5/2))/(225*d*f) + (b*n*x^3)/27 + (b*n*Log[1 + d*f*Sqrt[x]])/(9*d^6*f^6) - (b*n*x^3*Log[1 + d*f*Sqrt
[x]])/9 + (Sqrt[x]*(a + b*Log[c*x^n]))/(3*d^5*f^5) - (x*(a + b*Log[c*x^n]))/(6*d^4*f^4) + (x^(3/2)*(a + b*Log[
c*x^n]))/(9*d^3*f^3) - (x^2*(a + b*Log[c*x^n]))/(12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n]))/(15*d*f) - (x^3*(a
+ b*Log[c*x^n]))/18 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*d^6*f^6) + (x^3*Log[1 + d*f*Sqrt[x]]*(a +
b*Log[c*x^n]))/3 - (2*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(3*d^6*f^6)

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 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]

Rubi steps

\begin{align*} \int x^2 \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \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 )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{1}{6 d^4 f^4}+\frac{1}{3 d^5 f^5 \sqrt{x}}+\frac{\sqrt{x}}{9 d^3 f^3}-\frac{x}{12 d^2 f^2}+\frac{x^{3/2}}{15 d f}-\frac{x^2}{18}-\frac{\log \left (1+d f \sqrt{x}\right )}{3 d^6 f^6 x}+\frac{1}{3} x^2 \log \left (1+d f \sqrt{x}\right )\right ) \, dx\\ &=-\frac{2 b n \sqrt{x}}{3 d^5 f^5}+\frac{b n x}{6 d^4 f^4}-\frac{2 b n x^{3/2}}{27 d^3 f^3}+\frac{b n x^2}{24 d^2 f^2}-\frac{2 b n x^{5/2}}{75 d f}+\frac{1}{54} b n x^3+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \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 )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (b n) \int x^2 \log \left (1+d f \sqrt{x}\right ) \, dx+\frac{(b n) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx}{3 d^6 f^6}\\ &=-\frac{2 b n \sqrt{x}}{3 d^5 f^5}+\frac{b n x}{6 d^4 f^4}-\frac{2 b n x^{3/2}}{27 d^3 f^3}+\frac{b n x^2}{24 d^2 f^2}-\frac{2 b n x^{5/2}}{75 d f}+\frac{1}{54} b n x^3+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \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 )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}-\frac{1}{3} (2 b n) \operatorname{Subst}\left (\int x^5 \log (1+d f x) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b n \sqrt{x}}{3 d^5 f^5}+\frac{b n x}{6 d^4 f^4}-\frac{2 b n x^{3/2}}{27 d^3 f^3}+\frac{b n x^2}{24 d^2 f^2}-\frac{2 b n x^{5/2}}{75 d f}+\frac{1}{54} b n x^3-\frac{1}{9} b n x^3 \log \left (1+d f \sqrt{x}\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \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 )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{1}{9} (b d f n) \operatorname{Subst}\left (\int \frac{x^6}{1+d f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b n \sqrt{x}}{3 d^5 f^5}+\frac{b n x}{6 d^4 f^4}-\frac{2 b n x^{3/2}}{27 d^3 f^3}+\frac{b n x^2}{24 d^2 f^2}-\frac{2 b n x^{5/2}}{75 d f}+\frac{1}{54} b n x^3-\frac{1}{9} b n x^3 \log \left (1+d f \sqrt{x}\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \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 )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}+\frac{1}{9} (b d f n) \operatorname{Subst}\left (\int \left (-\frac{1}{d^6 f^6}+\frac{x}{d^5 f^5}-\frac{x^2}{d^4 f^4}+\frac{x^3}{d^3 f^3}-\frac{x^4}{d^2 f^2}+\frac{x^5}{d f}+\frac{1}{d^6 f^6 (1+d f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{7 b n \sqrt{x}}{9 d^5 f^5}+\frac{2 b n x}{9 d^4 f^4}-\frac{b n x^{3/2}}{9 d^3 f^3}+\frac{5 b n x^2}{72 d^2 f^2}-\frac{11 b n x^{5/2}}{225 d f}+\frac{1}{27} b n x^3+\frac{b n \log \left (1+d f \sqrt{x}\right )}{9 d^6 f^6}-\frac{1}{9} b n x^3 \log \left (1+d f \sqrt{x}\right )+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac{x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac{x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac{x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac{1}{18} x^3 \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 )}{3 d^6 f^6}+\frac{1}{3} x^3 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n \text{Li}_2\left (-d f \sqrt{x}\right )}{3 d^6 f^6}\\ \end{align*}

Mathematica [A]  time = 0.297808, size = 263, normalized size = 0.75 \[ \frac{-3600 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right )+600 \left (d^6 f^6 x^3-1\right ) \log \left (d f \sqrt{x}+1\right ) \left (3 a+3 b \log \left (c x^n\right )-b n\right )+d f \sqrt{x} \left (-30 a \left (10 d^5 f^5 x^{5/2}-12 d^4 f^4 x^2+15 d^3 f^3 x^{3/2}-20 d^2 f^2 x+30 d f \sqrt{x}-60\right )-30 b \left (10 d^5 f^5 x^{5/2}-12 d^4 f^4 x^2+15 d^3 f^3 x^{3/2}-20 d^2 f^2 x+30 d f \sqrt{x}-60\right ) \log \left (c x^n\right )+b n \left (200 d^5 f^5 x^{5/2}-264 d^4 f^4 x^2+375 d^3 f^3 x^{3/2}-600 d^2 f^2 x+1200 d f \sqrt{x}-4200\right )\right )}{5400 d^6 f^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(600*(-1 + d^6*f^6*x^3)*Log[1 + d*f*Sqrt[x]]*(3*a - b*n + 3*b*Log[c*x^n]) + d*f*Sqrt[x]*(-30*a*(-60 + 30*d*f*S
qrt[x] - 20*d^2*f^2*x + 15*d^3*f^3*x^(3/2) - 12*d^4*f^4*x^2 + 10*d^5*f^5*x^(5/2)) + b*n*(-4200 + 1200*d*f*Sqrt
[x] - 600*d^2*f^2*x + 375*d^3*f^3*x^(3/2) - 264*d^4*f^4*x^2 + 200*d^5*f^5*x^(5/2)) - 30*b*(-60 + 30*d*f*Sqrt[x
] - 20*d^2*f^2*x + 15*d^3*f^3*x^(3/2) - 12*d^4*f^4*x^2 + 10*d^5*f^5*x^(5/2))*Log[c*x^n]) - 3600*b*n*PolyLog[2,
-(d*f*Sqrt[x])])/(5400*d^6*f^6)

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \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(x^2*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^(1/2))),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} \log \left (c x^{n}\right ) + a x^{2}\right )} \log \left (d f \sqrt{x} + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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