Integrand size = 26, antiderivative size = 598 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {21 b^2 e^3 n^2 \sqrt {x}}{4 f^3}+\frac {a b e^2 n x}{2 f^2}-\frac {7 b^2 e^2 n^2 x}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2}}{108 f}-\frac {3}{16} b^2 n^2 x^2-\frac {b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right )}{4 f^4}+\frac {1}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )-\frac {b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}-\frac {5 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac {b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac {1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac {1}{2} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}-\frac {b^2 e^4 n^2 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {2 b e^4 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {4 b^2 e^4 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^4} \]
1/2*a*b*e^2*n*x/f^2-7/8*b^2*e^2*n^2*x/f^2+37/108*b^2*e*n^2*x^(3/2)/f-3/16* b^2*n^2*x^2+1/2*b^2*e^2*n*x*ln(c*x^n)/f^2+1/4*b*e^2*n*x*(a+b*ln(c*x^n))/f^ 2-7/18*b*e*n*x^(3/2)*(a+b*ln(c*x^n))/f+1/4*b*n*x^2*(a+b*ln(c*x^n))-1/4*e^2 *x*(a+b*ln(c*x^n))^2/f^2+1/6*e*x^(3/2)*(a+b*ln(c*x^n))^2/f-1/8*x^2*(a+b*ln (c*x^n))^2-1/4*b^2*e^4*n^2*ln(e+f*x^(1/2))/f^4+1/2*b*e^4*n*(a+b*ln(c*x^n)) *ln(e+f*x^(1/2))/f^4-b^2*e^4*n^2*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/f^4+1/4* b^2*n^2*x^2*ln(d*(e+f*x^(1/2)))-1/2*b*n*x^2*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1 /2)))+1/2*x^2*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))-1/2*e^4*(a+b*ln(c*x^n) )^2*ln(1+f*x^(1/2)/e)/f^4-2*b*e^4*n*(a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e )/f^4-b^2*e^4*n^2*polylog(2,1+f*x^(1/2)/e)/f^4+4*b^2*e^4*n^2*polylog(3,-f* x^(1/2)/e)/f^4+21/4*b^2*e^3*n^2*x^(1/2)/f^3-5/2*b*e^3*n*(a+b*ln(c*x^n))*x^ (1/2)/f^3+1/2*e^3*(a+b*ln(c*x^n))^2*x^(1/2)/f^3
Time = 0.33 (sec) , antiderivative size = 960, normalized size of antiderivative = 1.61 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {216 a^2 e^3 f \sqrt {x}-1080 a b e^3 f n \sqrt {x}+2268 b^2 e^3 f n^2 \sqrt {x}-108 a^2 e^2 f^2 x+324 a b e^2 f^2 n x-378 b^2 e^2 f^2 n^2 x+72 a^2 e f^3 x^{3/2}-168 a b e f^3 n x^{3/2}+148 b^2 e f^3 n^2 x^{3/2}-54 a^2 f^4 x^2+108 a b f^4 n x^2-81 b^2 f^4 n^2 x^2-216 a^2 e^4 \log \left (e+f \sqrt {x}\right )+216 a b e^4 n \log \left (e+f \sqrt {x}\right )-108 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right )+216 a^2 f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )-216 a b f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+108 b^2 f^4 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+432 a b e^4 n \log \left (e+f \sqrt {x}\right ) \log (x)-216 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log (x)-432 a b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+216 b^2 e^4 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-216 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x)+216 b^2 e^4 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+432 a b e^3 f \sqrt {x} \log \left (c x^n\right )-1080 b^2 e^3 f n \sqrt {x} \log \left (c x^n\right )-216 a b e^2 f^2 x \log \left (c x^n\right )+324 b^2 e^2 f^2 n x \log \left (c x^n\right )+144 a b e f^3 x^{3/2} \log \left (c x^n\right )-168 b^2 e f^3 n x^{3/2} \log \left (c x^n\right )-108 a b f^4 x^2 \log \left (c x^n\right )+108 b^2 f^4 n x^2 \log \left (c x^n\right )-432 a b e^4 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+216 b^2 e^4 n \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+432 a b f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )-216 b^2 f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+432 b^2 e^4 n \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )-432 b^2 e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )+216 b^2 e^3 f \sqrt {x} \log ^2\left (c x^n\right )-108 b^2 e^2 f^2 x \log ^2\left (c x^n\right )+72 b^2 e f^3 x^{3/2} \log ^2\left (c x^n\right )-54 b^2 f^4 x^2 \log ^2\left (c x^n\right )-216 b^2 e^4 \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )+216 b^2 f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )+432 b e^4 n \left (-2 a+b n-2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )+1728 b^2 e^4 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{432 f^4} \]
(216*a^2*e^3*f*Sqrt[x] - 1080*a*b*e^3*f*n*Sqrt[x] + 2268*b^2*e^3*f*n^2*Sqr t[x] - 108*a^2*e^2*f^2*x + 324*a*b*e^2*f^2*n*x - 378*b^2*e^2*f^2*n^2*x + 7 2*a^2*e*f^3*x^(3/2) - 168*a*b*e*f^3*n*x^(3/2) + 148*b^2*e*f^3*n^2*x^(3/2) - 54*a^2*f^4*x^2 + 108*a*b*f^4*n*x^2 - 81*b^2*f^4*n^2*x^2 - 216*a^2*e^4*Lo g[e + f*Sqrt[x]] + 216*a*b*e^4*n*Log[e + f*Sqrt[x]] - 108*b^2*e^4*n^2*Log[ e + f*Sqrt[x]] + 216*a^2*f^4*x^2*Log[d*(e + f*Sqrt[x])] - 216*a*b*f^4*n*x^ 2*Log[d*(e + f*Sqrt[x])] + 108*b^2*f^4*n^2*x^2*Log[d*(e + f*Sqrt[x])] + 43 2*a*b*e^4*n*Log[e + f*Sqrt[x]]*Log[x] - 216*b^2*e^4*n^2*Log[e + f*Sqrt[x]] *Log[x] - 432*a*b*e^4*n*Log[1 + (f*Sqrt[x])/e]*Log[x] + 216*b^2*e^4*n^2*Lo g[1 + (f*Sqrt[x])/e]*Log[x] - 216*b^2*e^4*n^2*Log[e + f*Sqrt[x]]*Log[x]^2 + 216*b^2*e^4*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 + 432*a*b*e^3*f*Sqrt[x]* Log[c*x^n] - 1080*b^2*e^3*f*n*Sqrt[x]*Log[c*x^n] - 216*a*b*e^2*f^2*x*Log[c *x^n] + 324*b^2*e^2*f^2*n*x*Log[c*x^n] + 144*a*b*e*f^3*x^(3/2)*Log[c*x^n] - 168*b^2*e*f^3*n*x^(3/2)*Log[c*x^n] - 108*a*b*f^4*x^2*Log[c*x^n] + 108*b^ 2*f^4*n*x^2*Log[c*x^n] - 432*a*b*e^4*Log[e + f*Sqrt[x]]*Log[c*x^n] + 216*b ^2*e^4*n*Log[e + f*Sqrt[x]]*Log[c*x^n] + 432*a*b*f^4*x^2*Log[d*(e + f*Sqrt [x])]*Log[c*x^n] - 216*b^2*f^4*n*x^2*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 4 32*b^2*e^4*n*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] - 432*b^2*e^4*n*Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n] + 216*b^2*e^3*f*Sqrt[x]*Log[c*x^n]^2 - 10 8*b^2*e^2*f^2*x*Log[c*x^n]^2 + 72*b^2*e*f^3*x^(3/2)*Log[c*x^n]^2 - 54*b...
Time = 0.94 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2824, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -2 b n \int \left (-\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) e^4}{2 f^4 x}+\frac {\left (a+b \log \left (c x^n\right )\right ) e^3}{2 f^3 \sqrt {x}}-\frac {\left (a+b \log \left (c x^n\right )\right ) e^2}{4 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right ) e}{6 f}-\frac {1}{8} x \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )\right )dx+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b n \left (\frac {1}{4} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {e^4 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^4}-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 b f^4 n}+\frac {e^4 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 b f^4 n}-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {5 e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {7 e x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {a e^2 x}{4 f^2}-\frac {b e^2 x \log \left (c x^n\right )}{4 f^2}-\frac {1}{8} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {b e^4 n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{2 f^4}-\frac {2 b e^4 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {b e^4 n \log \left (e+f \sqrt {x}\right )}{8 f^4}+\frac {b e^4 n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 f^4}-\frac {21 b e^3 n \sqrt {x}}{8 f^3}+\frac {7 b e^2 n x}{16 f^2}-\frac {37 b e n x^{3/2}}{216 f}+\frac {3}{32} b n x^2\right )+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2\) |
(e^3*Sqrt[x]*(a + b*Log[c*x^n])^2)/(2*f^3) - (e^2*x*(a + b*Log[c*x^n])^2)/ (4*f^2) + (e*x^(3/2)*(a + b*Log[c*x^n])^2)/(6*f) - (x^2*(a + b*Log[c*x^n]) ^2)/8 - (e^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*f^4) + (x^2*Log[d *(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/2 - 2*b*n*((-21*b*e^3*n*Sqrt[x])/( 8*f^3) - (a*e^2*x)/(4*f^2) + (7*b*e^2*n*x)/(16*f^2) - (37*b*e*n*x^(3/2))/( 216*f) + (3*b*n*x^2)/32 + (b*e^4*n*Log[e + f*Sqrt[x]])/(8*f^4) - (b*n*x^2* Log[d*(e + f*Sqrt[x])])/8 + (b*e^4*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/ e)])/(2*f^4) - (b*e^2*x*Log[c*x^n])/(4*f^2) + (5*e^3*Sqrt[x]*(a + b*Log[c* x^n]))/(4*f^3) - (e^2*x*(a + b*Log[c*x^n]))/(8*f^2) + (7*e*x^(3/2)*(a + b* Log[c*x^n]))/(36*f) - (x^2*(a + b*Log[c*x^n]))/8 - (e^4*Log[e + f*Sqrt[x]] *(a + b*Log[c*x^n]))/(4*f^4) + (x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^ n]))/4 - (e^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(4*b*f^4*n) + (e^4* Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*b*f^4*n) + (b*e^4*n*PolyLo g[2, 1 + (f*Sqrt[x])/e])/(2*f^4) + (e^4*(a + b*Log[c*x^n])*PolyLog[2, -((f *Sqrt[x])/e)])/f^4 - (2*b*e^4*n*PolyLog[3, -((f*Sqrt[x])/e)])/f^4)
3.2.23.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ .))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)], x]}, Simp[(a + b*Log[c*x^n])^p u, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1)/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ [p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[ (q + 1)/m] && EqQ[d*e, 1]))
\[\int x {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (e +f \sqrt {x}\right )\right )d x\]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int x\,\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]