3.1.46 \(\int x^2 \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n)) \, dx\) [46]

Optimal. Leaf size=350 \[ -\frac {7 b n \sqrt {x}}{9 d^5 f^5}+\frac {2 b n x}{9 d^4 f^4}-\frac {b n x^{3/2}}{9 d^3 f^3}+\frac {5 b n x^2}{72 d^2 f^2}-\frac {11 b n x^{5/2}}{225 d f}+\frac {1}{27} b n x^3+\frac {b n \log \left (1+d f \sqrt {x}\right )}{9 d^6 f^6}-\frac {1}{9} b n x^3 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{3 d^6 f^6} \]

[Out]

2/9*b*n*x/d^4/f^4-1/9*b*n*x^(3/2)/d^3/f^3+5/72*b*n*x^2/d^2/f^2-11/225*b*n*x^(5/2)/d/f+1/27*b*n*x^3-1/6*x*(a+b*
ln(c*x^n))/d^4/f^4+1/9*x^(3/2)*(a+b*ln(c*x^n))/d^3/f^3-1/12*x^2*(a+b*ln(c*x^n))/d^2/f^2+1/15*x^(5/2)*(a+b*ln(c
*x^n))/d/f-1/18*x^3*(a+b*ln(c*x^n))+1/9*b*n*ln(1+d*f*x^(1/2))/d^6/f^6-1/9*b*n*x^3*ln(1+d*f*x^(1/2))-1/3*(a+b*l
n(c*x^n))*ln(1+d*f*x^(1/2))/d^6/f^6+1/3*x^3*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))-2/3*b*n*polylog(2,-d*f*x^(1/2))/
d^6/f^6-7/9*b*n*x^(1/2)/d^5/f^5+1/3*(a+b*ln(c*x^n))*x^(1/2)/d^5/f^5

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Rubi [A]
time = 0.19, antiderivative size = 350, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 45, 2423, 2438} \begin {gather*} -\frac {2 b n \text {PolyLog}\left (2,-d f \sqrt {x}\right )}{3 d^6 f^6}-\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 {\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}{3} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (d f \sqrt {x}+1\right )}{9 d^6 f^6}-\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}{9} b n x^3 \log \left (d f \sqrt {x}+1\right )+\frac {1}{27} b n x^3 \end {gather*}

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 45

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 2423

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 2438

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 2442

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 2504

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])

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) \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.22, size = 263, normalized size = 0.75 \begin {gather*} \frac {600 \left (-1+d^6 f^6 x^3\right ) \log \left (1+d f \sqrt {x}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )+d f \sqrt {x} \left (-30 a \left (-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}\right )+b n \left (-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}\right )-30 b \left (-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}\right ) \log \left (c x^n\right )\right )-3600 b n \text {Li}_2\left (-d f \sqrt {x}\right )}{5400 d^6 f^6} \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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