Integrand size = 26, antiderivative size = 598 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {21 b^2 e^3 n^2 \sqrt {x}}{4 f^3}+\frac {a b e^2 n x}{2 f^2}-\frac {7 b^2 e^2 n^2 x}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2}}{108 f}-\frac {3}{16} b^2 n^2 x^2-\frac {b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right )}{4 f^4}+\frac {1}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )-\frac {b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {b^2 e^2 n x \log \left (c x^n\right )}{2 f^2}-\frac {5 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}+\frac {b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{18 f}+\frac {1}{4} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}-\frac {1}{2} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}-\frac {b^2 e^4 n^2 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {2 b e^4 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {4 b^2 e^4 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^4} \] Output:
21/4*b^2*e^3*n^2*x^(1/2)/f^3+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^2-1/4*b^2*e^4*n^2*ln(e+f*x^(1/2))/f ^4+1/4*b^2*n^2*x^2*ln(d*(e+f*x^(1/2)))-b^2*e^4*n^2*ln(e+f*x^(1/2))*ln(-f*x ^(1/2)/e)/f^4+1/2*b^2*e^2*n*x*ln(c*x^n)/f^2-5/2*b*e^3*n*x^(1/2)*(a+b*ln(c* x^n))/f^3+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))+1/2*b*e^4*n*ln(e+f*x^(1/2))*(a+b*ln(c*x ^n))/f^4-1/2*b*n*x^2*ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))+1/2*e^3*x^(1/2)*( a+b*ln(c*x^n))^2/f^3-1/4*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+1/2*x^2*ln(d*(e+f*x^(1/2)))*(a+b*ln (c*x^n))^2-1/2*e^4*ln(1+f*x^(1/2)/e)*(a+b*ln(c*x^n))^2/f^4-b^2*e^4*n^2*pol ylog(2,1+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+4*b^2*e^4*n^2*polylog(3,-f*x^(1/2)/e)/f^4
Time = 0.59 (sec) , antiderivative size = 960, normalized size of antiderivative = 1.61 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:
Integrate[x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2,x]
Output:
(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 = 1.08 (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 |
\(\displaystyle -2 b n \left (\frac {1}{4} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {e^4 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^4}-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 b f^4 n}+\frac {e^4 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 b f^4 n}-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {5 e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {7 e x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {a e^2 x}{4 f^2}-\frac {b e^2 x \log \left (c x^n\right )}{4 f^2}-\frac {1}{8} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {b e^4 n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{2 f^4}-\frac {2 b e^4 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {b e^4 n \log \left (e+f \sqrt {x}\right )}{8 f^4}+\frac {b e^4 n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 f^4}-\frac {21 b e^3 n \sqrt {x}}{8 f^3}+\frac {7 b e^2 n x}{16 f^2}-\frac {37 b e n x^{3/2}}{216 f}+\frac {3}{32} b n x^2\right )+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2\) |
Input:
Int[x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2,x]
Output:
(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)
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]))
\[\int x \ln \left (d \left (e +f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}d x\]
Input:
int(x*ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))^2,x)
Output:
int(x*ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))^2,x)
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \] Input:
integrate(x*log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^2,x, algorithm="fricas")
Output:
integral((b^2*x*log(c*x^n)^2 + 2*a*b*x*log(c*x^n) + a^2*x)*log(d*f*sqrt(x) + d*e), x)
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate(x*ln(d*(e+f*x**(1/2)))*(a+b*ln(c*x**n))**2,x)
Output:
Timed out
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \] Input:
integrate(x*log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^2,x, algorithm="maxima")
Output:
integrate((b*log(c*x^n) + a)^2*x*log((f*sqrt(x) + e)*d), x)
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \] Input:
integrate(x*log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^2,x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)^2*x*log((f*sqrt(x) + e)*d), x)
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int x\,\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \] Input:
int(x*log(d*(e + f*x^(1/2)))*(a + b*log(c*x^n))^2,x)
Output:
int(x*log(d*(e + f*x^(1/2)))*(a + b*log(c*x^n))^2, x)
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:
int(x*log(d*(e+f*x^(1/2)))*(a+b*log(c*x^n))^2,x)
Output:
(216*sqrt(x)*log(x**n*c)**2*b**2*e**3*f*n + 72*sqrt(x)*log(x**n*c)**2*b**2 *e*f**3*n*x + 432*sqrt(x)*log(x**n*c)*a*b*e**3*f*n + 144*sqrt(x)*log(x**n* c)*a*b*e*f**3*n*x - 1080*sqrt(x)*log(x**n*c)*b**2*e**3*f*n**2 - 168*sqrt(x )*log(x**n*c)*b**2*e*f**3*n**2*x + 216*sqrt(x)*a**2*e**3*f*n + 72*sqrt(x)* a**2*e*f**3*n*x - 1080*sqrt(x)*a*b*e**3*f*n**2 - 168*sqrt(x)*a*b*e*f**3*n* *2*x + 2268*sqrt(x)*b**2*e**3*f*n**3 + 148*sqrt(x)*b**2*e*f**3*n**3*x + 10 8*int(log(x**n*c)**2/(e**2*x - f**2*x**2),x)*b**2*e**6*n + 216*int(log(x** n*c)/(e**2*x - f**2*x**2),x)*a*b*e**6*n - 108*int(log(x**n*c)/(e**2*x - f* *2*x**2),x)*b**2*e**6*n**2 - 108*int((sqrt(x)*log(x**n*c)**2)/(e**2*x - f* *2*x**2),x)*b**2*e**5*f*n - 216*int((sqrt(x)*log(x**n*c))/(e**2*x - f**2*x **2),x)*a*b*e**5*f*n + 108*int((sqrt(x)*log(x**n*c))/(e**2*x - f**2*x**2), x)*b**2*e**5*f*n**2 + 216*log(sqrt(x)*d*f + d*e)*log(x**n*c)**2*b**2*f**4* n*x**2 + 432*log(sqrt(x)*d*f + d*e)*log(x**n*c)*a*b*f**4*n*x**2 - 216*log( sqrt(x)*d*f + d*e)*log(x**n*c)*b**2*f**4*n**2*x**2 - 216*log(sqrt(x)*d*f + d*e)*a**2*e**4*n + 216*log(sqrt(x)*d*f + d*e)*a**2*f**4*n*x**2 + 216*log( sqrt(x)*d*f + d*e)*a*b*e**4*n**2 - 216*log(sqrt(x)*d*f + d*e)*a*b*f**4*n** 2*x**2 - 108*log(sqrt(x)*d*f + d*e)*b**2*e**4*n**3 + 108*log(sqrt(x)*d*f + d*e)*b**2*f**4*n**3*x**2 - 36*log(x**n*c)**3*b**2*e**4 - 108*log(x**n*c)* *2*a*b*e**4 + 54*log(x**n*c)**2*b**2*e**4*n - 108*log(x**n*c)**2*b**2*e**2 *f**2*n*x - 54*log(x**n*c)**2*b**2*f**4*n*x**2 - 216*log(x**n*c)*a*b*e*...