Integrand size = 33, antiderivative size = 614 \[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {B^3 (b c-a d)^3 n^3 \log (c+d x)}{b d^3}+\frac {B^2 (b c-a d)^2 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b d^2}+\frac {4 B^2 (b c-a d)^3 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b d^3}+\frac {2 B (b c-a d)^2 n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b d^2}-\frac {b B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 d^3}+\frac {B (b c-a d)^3 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b d^3}+\frac {(a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 b}-\frac {B^2 (b c-a d)^3 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b d^3}+\frac {4 B^3 (b c-a d)^3 n^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^3}+\frac {2 B^2 (b c-a d)^3 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^3}+\frac {B^3 (b c-a d)^3 n^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b d^3}-\frac {2 B^3 (b c-a d)^3 n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b d^3} \]
-B^3*(-a*d+b*c)^3*n^3*ln(d*x+c)/b/d^3+B^2*(-a*d+b*c)^2*n^2*(b*x+a)*(A+B*ln (e*(b*x+a)^n/((d*x+c)^n)))/b/d^2+4*B^2*(-a*d+b*c)^3*n^2*ln((-a*d+b*c)/b/(d *x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^3+2*B*(-a*d+b*c)^2*n*(b*x+a)* (A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b/d^2-1/2*b*B*(-a*d+b*c)*n*(d*x+c)^2*( A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/d^3+B*(-a*d+b*c)^3*n*ln((-a*d+b*c)/b/(d *x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b/d^3+1/3*(b*x+a)^3*(A+B*ln(e*( b*x+a)^n/((d*x+c)^n)))^3/b-B^2*(-a*d+b*c)^3*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+ c)^n)))*ln(1-b*(d*x+c)/d/(b*x+a))/b/d^3+4*B^3*(-a*d+b*c)^3*n^3*polylog(2,d *(b*x+a)/b/(d*x+c))/b/d^3+2*B^2*(-a*d+b*c)^3*n^2*(A+B*ln(e*(b*x+a)^n/((d*x +c)^n)))*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^3+B^3*(-a*d+b*c)^3*n^3*polylog (2,b*(d*x+c)/d/(b*x+a))/b/d^3-2*B^3*(-a*d+b*c)^3*n^3*polylog(3,d*(b*x+a)/b /(d*x+c))/b/d^3
Leaf count is larger than twice the leaf count of optimal. \(4802\) vs. \(2(614)=1228\).
Time = 1.38 (sec) , antiderivative size = 4802, normalized size of antiderivative = 7.82 \[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Result too large to show} \]
(-6*a^3*A*B^2*n^2)/b - (2*a*A*b*B^2*c^2*n^2)/d^2 + (4*a^2*A*B^2*c*n^2)/d - (4*a^3*B^3*n^3)/b - (3*a*b*B^3*c^2*n^3)/d^2 + (7*a^2*B^3*c*n^3)/d + a^2*A ^3*x + 2*a^2*A^2*B*n*x + (A^2*b^2*B*c^2*n*x)/d^2 - (3*a*A^2*b*B*c*n*x)/d + a^2*A*B^2*n^2*x + (A*b^2*B^2*c^2*n^2*x)/d^2 - (2*a*A*b*B^2*c*n^2*x)/d + a *A^3*b*x^2 + (a*A^2*b*B*n*x^2)/2 - (A^2*b^2*B*c*n*x^2)/(2*d) + (A^3*b^2*x^ 3)/3 + (a^3*A^2*B*n*Log[a + b*x])/b + (3*a^3*A*B^2*n^2*Log[a + b*x])/b + ( 2*a*A*b*B^2*c^2*n^2*Log[a + b*x])/d^2 - (5*a^2*A*B^2*c*n^2*Log[a + b*x])/d + (7*a^3*B^3*n^3*Log[a + b*x])/b + (3*a*b*B^3*c^2*n^3*Log[a + b*x])/d^2 - (6*a^2*B^3*c*n^3*Log[a + b*x])/d - (a^3*A*B^2*n^2*Log[a + b*x]^2)/b - (3* a^3*B^3*n^3*Log[a + b*x]^2)/(2*b) - (a*b*B^3*c^2*n^3*Log[a + b*x]^2)/d^2 + (5*a^2*B^3*c*n^3*Log[a + b*x]^2)/(2*d) + (a^3*B^3*n^3*Log[a + b*x]^3)/(3* b) - (A^2*b^2*B*c^3*n*Log[c + d*x])/d^3 + (3*a*A^2*b*B*c^2*n*Log[c + d*x]) /d^2 - (3*a^2*A^2*B*c*n*Log[c + d*x])/d - (3*A*b^2*B^2*c^3*n^2*Log[c + d*x ])/d^3 + (7*a*A*b*B^2*c^2*n^2*Log[c + d*x])/d^2 - (4*a^2*A*B^2*c*n^2*Log[c + d*x])/d - (6*a^3*B^3*n^3*Log[c + d*x])/b - (b^2*B^3*c^3*n^3*Log[c + d*x ])/d^3 + (3*a^2*B^3*c*n^3*Log[c + d*x])/d + (2*a^3*A*B^2*n^2*Log[a + b*x]* Log[c + d*x])/b + (2*A*b^2*B^2*c^3*n^2*Log[a + b*x]*Log[c + d*x])/d^3 - (6 *a*A*b*B^2*c^2*n^2*Log[a + b*x]*Log[c + d*x])/d^2 + (6*a^2*A*B^2*c*n^2*Log [a + b*x]*Log[c + d*x])/d + (3*b^2*B^3*c^3*n^3*Log[a + b*x]*Log[c + d*x])/ d^3 - (7*a*b*B^3*c^2*n^3*Log[a + b*x]*Log[c + d*x])/d^2 + (4*a^2*B^3*c*...
Time = 0.91 (sec) , antiderivative size = 573, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2949, 2781, 2795, 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.
\(\displaystyle \int (a+b x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3 \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int (a+b x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3dx\) |
\(\Big \downarrow \) 2949 |
\(\displaystyle (b c-a d)^3 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}\) |
\(\Big \downarrow \) 2781 |
\(\displaystyle (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}\right )\) |
\(\Big \downarrow \) 2795 |
\(\displaystyle (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )d\frac {a+b x}{c+d x}}{b}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle (b c-a d)^3 \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3}-\frac {4 B n \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3}+\frac {B n \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^3}-\frac {B n (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {4 B^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3}-\frac {B^2 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^3}+\frac {2 B^2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3}-\frac {B^2 n^2 \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{b}\right )\) |
(b*c - a*d)^3*(((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*b *(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (B*n*(-((B*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d^2*(c + d*x)*(b - (d*(a + b*x))/(c + d*x)))) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*d^3*(b - (d* (a + b*x))/(c + d*x))^2) - (2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x)) ^n])^2)/(d^2*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (B^2*n^2*Log[b - ( d*(a + b*x))/(c + d*x)])/d^3 - (4*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n ])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^3 - ((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^3 + (B*n*(A + B*Log [e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/d^3 - ( 4*B^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^3 - (2*B*n*(A + B*Log [e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^3 - (B^2*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/d^3 + (2*B^2*n^2*PolyL og[3, (d*(a + b*x))/(b*(c + d*x))])/d^3))/b)
3.2.65.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x) ^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^(m + 1)*(g/b)^m Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && Ne Q[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || Lt Q[m, -1])
Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && !Intege rQ[n]
\[\int \left (b x +a \right )^{2} {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}d x\]
\[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (b x + a\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
integral(A^3*b^2*x^2 + 2*A^3*a*b*x + A^3*a^2 + (B^3*b^2*x^2 + 2*B^3*a*b*x + B^3*a^2)*log((b*x + a)^n*e/(d*x + c)^n)^3 + 3*(A*B^2*b^2*x^2 + 2*A*B^2*a *b*x + A*B^2*a^2)*log((b*x + a)^n*e/(d*x + c)^n)^2 + 3*(A^2*B*b^2*x^2 + 2* A^2*B*a*b*x + A^2*B*a^2)*log((b*x + a)^n*e/(d*x + c)^n), x)
Exception generated. \[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (b x + a\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
A^2*B*b^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A^3*b^2*x^3 + 3*A^2*B*a *b*x^2*log((b*x + a)^n*e/(d*x + c)^n) + A^3*a*b*x^2 + 3*A^2*B*a^2*x*log((b *x + a)^n*e/(d*x + c)^n) + A^3*a^2*x + 3*(a*e*n*log(b*x + a)/b - c*e*n*log (d*x + c)/d)*A^2*B*a^2/e - 3*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*log(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))*A^2*B*a*b/e + 1/2*(2*a^3*e*n*log(b* x + a)/b^3 - 2*c^3*e*n*log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)*x^2 - 2*(b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*A^2*B*b^2/e - 1/6*(2*(B^3*b ^3*d^3*x^3 + 3*B^3*a*b^2*d^3*x^2 + 3*B^3*a^2*b*d^3*x)*log((d*x + c)^n)^3 - 3*(2*B^3*a^3*d^3*n*log(b*x + a) - 2*(b^3*c^3*n - 3*a*b^2*c^2*d*n + 3*a^2* b*c*d^2*n)*B^3*log(d*x + c) + 2*(B^3*b^3*d^3*log(e) + A*B^2*b^3*d^3)*x^3 + (6*A*B^2*a*b^2*d^3 + (a*b^2*d^3*(n + 6*log(e)) - b^3*c*d^2*n)*B^3)*x^2 + 2*(3*A*B^2*a^2*b*d^3 + (a^2*b*d^3*(2*n + 3*log(e)) + b^3*c^2*d*n - 3*a*b^2 *c*d^2*n)*B^3)*x + 2*(B^3*b^3*d^3*x^3 + 3*B^3*a*b^2*d^3*x^2 + 3*B^3*a^2*b* d^3*x)*log((b*x + a)^n))*log((d*x + c)^n)^2)/(b*d^3) - integrate(-(B^3*a^2 *b*c*d^2*log(e)^3 + 3*A*B^2*a^2*b*c*d^2*log(e)^2 + (B^3*b^3*d^3*log(e)^3 + 3*A*B^2*b^3*d^3*log(e)^2)*x^3 + (B^3*b^3*d^3*x^3 + B^3*a^2*b*c*d^2 + (b^3 *c*d^2 + 2*a*b^2*d^3)*B^3*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*B^3*x)*log((b* x + a)^n)^3 + (3*(b^3*c*d^2*log(e)^2 + 2*a*b^2*d^3*log(e)^2)*A*B^2 + (b^3* c*d^2*log(e)^3 + 2*a*b^2*d^3*log(e)^3)*B^3)*x^2 + 3*(B^3*a^2*b*c*d^2*log(e ) + A*B^2*a^2*b*c*d^2 + (B^3*b^3*d^3*log(e) + A*B^2*b^3*d^3)*x^3 + ((b^...
\[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (b x + a\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
Timed out. \[ \int (a+b x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int {\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3\,{\left (a+b\,x\right )}^2 \,d x \]