Integrand size = 40, antiderivative size = 45 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{4 B (b c-a d) n} \]
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{4 (b B c n-a B d n)} \]
Time = 0.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2973, 2961, 2739, 15}
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.
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (c+d x)} \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (c+d x)}dx\) |
\(\Big \downarrow \) 2961 |
\(\displaystyle \frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{a+b x}d\frac {a+b x}{c+d x}}{b c-a d}\) |
\(\Big \downarrow \) 2739 |
\(\displaystyle \frac {\int \frac {(a+b x)^3}{(c+d x)^3}d\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n (b c-a d)}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {(a+b x)^4}{4 B n (c+d x)^4 (b c-a d)}\) |
3.3.28.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
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]
Time = 33.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(-\frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{4}}{4 n \left (a d -c b \right ) B}\) | \(44\) |
default | \(-\frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{4}}{4 n \left (a d -c b \right ) B}\) | \(44\) |
parallelrisch | \(-\frac {B^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{4} b^{2} d^{2}+4 A \,B^{2} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3} b^{2} d^{2}+6 A^{2} B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} b^{2} d^{2}+4 A^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{2} d^{2}}{4 n \,b^{2} d^{2} \left (a d -c b \right )}\) | \(150\) |
parts | \(\frac {A^{3} \ln \left (d x +c \right )}{a d -c b}-\frac {A^{3} \ln \left (b x +a \right )}{a d -c b}-\frac {B^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{4}}{4 n \left (a d -c b \right )}-\frac {A \,B^{2} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}}{n \left (a d -c b \right )}-\frac {3 A^{2} B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}}{2 n \left (a d -c b \right )}\) | \(162\) |
risch | \(\text {Expression too large to display}\) | \(64288\) |
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (43) = 86\).
Time = 0.32 (sec) , antiderivative size = 375, normalized size of antiderivative = 8.33 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\frac {B^{3} n^{3} \log \left (b x + a\right )^{4} + B^{3} n^{3} \log \left (d x + c\right )^{4} + 4 \, {\left (B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (b x + a\right )^{3} - 4 \, {\left (B^{3} n^{3} \log \left (b x + a\right ) + B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (d x + c\right )^{3} + 6 \, {\left (B^{3} n \log \left (e\right )^{2} + 2 \, A B^{2} n \log \left (e\right ) + A^{2} B n\right )} \log \left (b x + a\right )^{2} + 6 \, {\left (B^{3} n^{3} \log \left (b x + a\right )^{2} + B^{3} n \log \left (e\right )^{2} + 2 \, A B^{2} n \log \left (e\right ) + A^{2} B n + 2 \, {\left (B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} + 4 \, {\left (B^{3} \log \left (e\right )^{3} + 3 \, A B^{2} \log \left (e\right )^{2} + 3 \, A^{2} B \log \left (e\right ) + A^{3}\right )} \log \left (b x + a\right ) - 4 \, {\left (B^{3} n^{3} \log \left (b x + a\right )^{3} + B^{3} \log \left (e\right )^{3} + 3 \, A B^{2} \log \left (e\right )^{2} + 3 \, A^{2} B \log \left (e\right ) + A^{3} + 3 \, {\left (B^{3} n^{2} \log \left (e\right ) + A B^{2} n^{2}\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (B^{3} n \log \left (e\right )^{2} + 2 \, A B^{2} n \log \left (e\right ) + A^{2} B n\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{4 \, {\left (b c - a d\right )}} \]
1/4*(B^3*n^3*log(b*x + a)^4 + B^3*n^3*log(d*x + c)^4 + 4*(B^3*n^2*log(e) + A*B^2*n^2)*log(b*x + a)^3 - 4*(B^3*n^3*log(b*x + a) + B^3*n^2*log(e) + A* B^2*n^2)*log(d*x + c)^3 + 6*(B^3*n*log(e)^2 + 2*A*B^2*n*log(e) + A^2*B*n)* log(b*x + a)^2 + 6*(B^3*n^3*log(b*x + a)^2 + B^3*n*log(e)^2 + 2*A*B^2*n*lo g(e) + A^2*B*n + 2*(B^3*n^2*log(e) + A*B^2*n^2)*log(b*x + a))*log(d*x + c) ^2 + 4*(B^3*log(e)^3 + 3*A*B^2*log(e)^2 + 3*A^2*B*log(e) + A^3)*log(b*x + a) - 4*(B^3*n^3*log(b*x + a)^3 + B^3*log(e)^3 + 3*A*B^2*log(e)^2 + 3*A^2*B *log(e) + A^3 + 3*(B^3*n^2*log(e) + A*B^2*n^2)*log(b*x + a)^2 + 3*(B^3*n*l og(e)^2 + 2*A*B^2*n*log(e) + A^2*B*n)*log(b*x + a))*log(d*x + c))/(b*c - a *d)
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (43) = 86\).
Time = 0.25 (sec) , antiderivative size = 766, normalized size of antiderivative = 17.02 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=B^{3} {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \, A B^{2} {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \, A^{2} B {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) - \frac {1}{4} \, B^{3} {\left (\frac {6 \, {\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2}}{{\left (b c - a d\right )} e} - \frac {\frac {4 \, {\left (e^{2} n^{2} \log \left (b x + a\right )^{3} - 3 \, e^{2} n^{2} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, e^{2} n^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - e^{2} n^{2} \log \left (d x + c\right )^{3}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{{\left (b c - a d\right )} e} - \frac {e^{3} n^{3} \log \left (b x + a\right )^{4} - 4 \, e^{3} n^{3} \log \left (b x + a\right )^{3} \log \left (d x + c\right ) + 6 \, e^{3} n^{3} \log \left (b x + a\right )^{2} \log \left (d x + c\right )^{2} - 4 \, e^{3} n^{3} \log \left (b x + a\right ) \log \left (d x + c\right )^{3} + e^{3} n^{3} \log \left (d x + c\right )^{4}}{{\left (b c - a d\right )} e^{2}}}{e}\right )} + A^{3} {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} - A B^{2} {\left (\frac {3 \, {\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{{\left (b c - a d\right )} e} - \frac {e^{2} n^{2} \log \left (b x + a\right )^{3} - 3 \, e^{2} n^{2} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, e^{2} n^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - e^{2} n^{2} \log \left (d x + c\right )^{3}}{{\left (b c - a d\right )} e^{2}}\right )} - \frac {3 \, {\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} A^{2} B}{2 \, {\left (b c - a d\right )} e} \]
B^3*(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d))*log((b*x + a)^n* e/(d*x + c)^n)^3 + 3*A*B^2*(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d))*log((b*x + a)^n*e/(d*x + c)^n)^2 + 3*A^2*B*(log(b*x + a)/(b*c - a*d ) - log(d*x + c)/(b*c - a*d))*log((b*x + a)^n*e/(d*x + c)^n) - 1/4*B^3*(6* (e*n*log(b*x + a)^2 - 2*e*n*log(b*x + a)*log(d*x + c) + e*n*log(d*x + c)^2 )*log((b*x + a)^n*e/(d*x + c)^n)^2/((b*c - a*d)*e) - (4*(e^2*n^2*log(b*x + a)^3 - 3*e^2*n^2*log(b*x + a)^2*log(d*x + c) + 3*e^2*n^2*log(b*x + a)*log (d*x + c)^2 - e^2*n^2*log(d*x + c)^3)*log((b*x + a)^n*e/(d*x + c)^n)/((b*c - a*d)*e) - (e^3*n^3*log(b*x + a)^4 - 4*e^3*n^3*log(b*x + a)^3*log(d*x + c) + 6*e^3*n^3*log(b*x + a)^2*log(d*x + c)^2 - 4*e^3*n^3*log(b*x + a)*log( d*x + c)^3 + e^3*n^3*log(d*x + c)^4)/((b*c - a*d)*e^2))/e) + A^3*(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d)) - A*B^2*(3*(e*n*log(b*x + a)^ 2 - 2*e*n*log(b*x + a)*log(d*x + c) + e*n*log(d*x + c)^2)*log((b*x + a)^n* e/(d*x + c)^n)/((b*c - a*d)*e) - (e^2*n^2*log(b*x + a)^3 - 3*e^2*n^2*log(b *x + a)^2*log(d*x + c) + 3*e^2*n^2*log(b*x + a)*log(d*x + c)^2 - e^2*n^2*l og(d*x + c)^3)/((b*c - a*d)*e^2)) - 3/2*(e*n*log(b*x + a)^2 - 2*e*n*log(b* x + a)*log(d*x + c) + e*n*log(d*x + c)^2)*A^2*B/((b*c - a*d)*e)
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
Time = 2.61 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.13 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x) (c+d x)} \, dx=-\frac {\frac {3\,A^2\,B\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2}{2}+A\,B^2\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^3+\frac {B^3\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^4}{4}}{n\,\left (a\,d-b\,c\right )}+\frac {A^3\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{a\,d-b\,c} \]