Integrand size = 28, antiderivative size = 557 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {21 b^2 n^2 \sqrt {x}}{4 d^3 f^3}+\frac {a b n x}{2 d^2 f^2}-\frac {7 b^2 n^2 x}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2}}{108 d f}-\frac {3}{16} b^2 n^2 x^2-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 d^4 f^4}+\frac {1}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+\frac {b^2 n x \log \left (c x^n\right )}{2 d^2 f^2}-\frac {5 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 d f}+\frac {1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}-\frac {1}{2} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^4 f^4} \]
1/2*a*b*n*x/d^2/f^2-7/8*b^2*n^2*x/d^2/f^2+37/108*b^2*n^2*x^(3/2)/d/f-3/16* b^2*n^2*x^2+1/2*b^2*n*x*ln(c*x^n)/d^2/f^2+1/4*b*n*x*(a+b*ln(c*x^n))/d^2/f^ 2-7/18*b*n*x^(3/2)*(a+b*ln(c*x^n))/d/f+1/4*b*n*x^2*(a+b*ln(c*x^n))-1/4*x*( a+b*ln(c*x^n))^2/d^2/f^2+1/6*x^(3/2)*(a+b*ln(c*x^n))^2/d/f-1/8*x^2*(a+b*ln (c*x^n))^2-1/4*b^2*n^2*ln(1+d*f*x^(1/2))/d^4/f^4+1/4*b^2*n^2*x^2*ln(1+d*f* x^(1/2))+1/2*b*n*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))/d^4/f^4-1/2*b*n*x^2*(a+ b*ln(c*x^n))*ln(1+d*f*x^(1/2))-1/2*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))/d^4 /f^4+1/2*x^2*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))+b^2*n^2*polylog(2,-d*f*x^ (1/2))/d^4/f^4-2*b*n*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))/d^4/f^4+4*b^2 *n^2*polylog(3,-d*f*x^(1/2))/d^4/f^4+21/4*b^2*n^2*x^(1/2)/d^3/f^3-5/2*b*n* (a+b*ln(c*x^n))*x^(1/2)/d^3/f^3+1/2*(a+b*ln(c*x^n))^2*x^(1/2)/d^3/f^3
Time = 0.29 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.38 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {216 a^2 d f \sqrt {x}-1080 a b d f n \sqrt {x}+2268 b^2 d f n^2 \sqrt {x}-108 a^2 d^2 f^2 x+324 a b d^2 f^2 n x-378 b^2 d^2 f^2 n^2 x+72 a^2 d^3 f^3 x^{3/2}-168 a b d^3 f^3 n x^{3/2}+148 b^2 d^3 f^3 n^2 x^{3/2}-54 a^2 d^4 f^4 x^2+108 a b d^4 f^4 n x^2-81 b^2 d^4 f^4 n^2 x^2-216 a^2 \log \left (1+d f \sqrt {x}\right )+216 a b n \log \left (1+d f \sqrt {x}\right )-108 b^2 n^2 \log \left (1+d f \sqrt {x}\right )+216 a^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right )-216 a b d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right )+108 b^2 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+432 a b d f \sqrt {x} \log \left (c x^n\right )-1080 b^2 d f n \sqrt {x} \log \left (c x^n\right )-216 a b d^2 f^2 x \log \left (c x^n\right )+324 b^2 d^2 f^2 n x \log \left (c x^n\right )+144 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )-168 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )-108 a b d^4 f^4 x^2 \log \left (c x^n\right )+108 b^2 d^4 f^4 n x^2 \log \left (c x^n\right )-432 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+216 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+432 a b d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-216 b^2 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+216 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-108 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+72 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )-54 b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right )-216 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+216 b^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+432 b n \left (-2 a+b n-2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+1728 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{432 d^4 f^4} \]
(216*a^2*d*f*Sqrt[x] - 1080*a*b*d*f*n*Sqrt[x] + 2268*b^2*d*f*n^2*Sqrt[x] - 108*a^2*d^2*f^2*x + 324*a*b*d^2*f^2*n*x - 378*b^2*d^2*f^2*n^2*x + 72*a^2* d^3*f^3*x^(3/2) - 168*a*b*d^3*f^3*n*x^(3/2) + 148*b^2*d^3*f^3*n^2*x^(3/2) - 54*a^2*d^4*f^4*x^2 + 108*a*b*d^4*f^4*n*x^2 - 81*b^2*d^4*f^4*n^2*x^2 - 21 6*a^2*Log[1 + d*f*Sqrt[x]] + 216*a*b*n*Log[1 + d*f*Sqrt[x]] - 108*b^2*n^2* Log[1 + d*f*Sqrt[x]] + 216*a^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]] - 216*a*b* d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] + 108*b^2*d^4*f^4*n^2*x^2*Log[1 + d*f*S qrt[x]] + 432*a*b*d*f*Sqrt[x]*Log[c*x^n] - 1080*b^2*d*f*n*Sqrt[x]*Log[c*x^ n] - 216*a*b*d^2*f^2*x*Log[c*x^n] + 324*b^2*d^2*f^2*n*x*Log[c*x^n] + 144*a *b*d^3*f^3*x^(3/2)*Log[c*x^n] - 168*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] - 108 *a*b*d^4*f^4*x^2*Log[c*x^n] + 108*b^2*d^4*f^4*n*x^2*Log[c*x^n] - 432*a*b*L og[1 + d*f*Sqrt[x]]*Log[c*x^n] + 216*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 432*a*b*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 216*b^2*d^4*f^4*n *x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 216*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 108*b^2*d^2*f^2*x*Log[c*x^n]^2 + 72*b^2*d^3*f^3*x^(3/2)*Log[c*x^n]^2 - 54* b^2*d^4*f^4*x^2*Log[c*x^n]^2 - 216*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 216*b^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 432*b*n*(-2*a + b *n - 2*b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 1728*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(432*d^4*f^4)
Time = 0.74 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -2 b n \int \left (-\frac {1}{8} x \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4 x}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{6 d f}+\frac {a+b \log \left (c x^n\right )}{2 d^3 f^3 \sqrt {x}}-\frac {a+b \log \left (c x^n\right )}{4 d^2 f^2}\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}-2 b n \left (\frac {\operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4 f^4}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^4 f^4}+\frac {5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {7 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}+\frac {1}{4} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {a x}{4 d^2 f^2}-\frac {b x \log \left (c x^n\right )}{4 d^2 f^2}-\frac {b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{2 d^4 f^4}-\frac {2 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^4 f^4}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{8 d^4 f^4}-\frac {21 b n \sqrt {x}}{8 d^3 f^3}+\frac {7 b n x}{16 d^2 f^2}-\frac {37 b n x^{3/2}}{216 d f}-\frac {1}{8} b n x^2 \log \left (d f \sqrt {x}+1\right )+\frac {3}{32} b n x^2\right )+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2\) |
(Sqrt[x]*(a + b*Log[c*x^n])^2)/(2*d^3*f^3) - (x*(a + b*Log[c*x^n])^2)/(4*d ^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n])^2)/(6*d*f) - (x^2*(a + b*Log[c*x^n]) ^2)/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*d^4*f^4) + (x^2*Log [1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/2 - 2*b*n*((-21*b*n*Sqrt[x])/(8*d^ 3*f^3) - (a*x)/(4*d^2*f^2) + (7*b*n*x)/(16*d^2*f^2) - (37*b*n*x^(3/2))/(21 6*d*f) + (3*b*n*x^2)/32 + (b*n*Log[1 + d*f*Sqrt[x]])/(8*d^4*f^4) - (b*n*x^ 2*Log[1 + d*f*Sqrt[x]])/8 - (b*x*Log[c*x^n])/(4*d^2*f^2) + (5*Sqrt[x]*(a + b*Log[c*x^n]))/(4*d^3*f^3) - (x*(a + b*Log[c*x^n]))/(8*d^2*f^2) + (7*x^(3 /2)*(a + b*Log[c*x^n]))/(36*d*f) - (x^2*(a + b*Log[c*x^n]))/8 - (Log[1 + d *f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*d^4*f^4) + (x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/4 - (b*n*PolyLog[2, -(d*f*Sqrt[x])])/(2*d^4*f^4) + ((a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^4*f^4) - (2*b*n*PolyLog[3, - (d*f*Sqrt[x])])/(d^4*f^4))
3.1.54.3.1 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]))
\[\int x {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )d x\]
\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int x\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]