Integrand size = 31, antiderivative size = 166 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {B n}{9 b (a+b x)^3}+\frac {B d n}{6 b (b c-a d) (a+b x)^2}-\frac {B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3} \]
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Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 46} \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{3 b (a+b x)^3}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {B d^2 n}{3 b (a+b x) (b c-a d)^2}+\frac {B d n}{6 b (a+b x)^2 (b c-a d)}-\frac {B n}{9 b (a+b x)^3} \]
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Rule 46
Rule 2548
Rubi steps \begin{align*} \text {integral}& = -\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b} \\ & = -\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac {(B (b c-a d) n) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b} \\ & = -\frac {B n}{9 b (a+b x)^3}+\frac {B d n}{6 b (b c-a d) (a+b x)^2}-\frac {B d^2 n}{3 b (b c-a d)^2 (a+b x)}-\frac {B d^3 n \log (a+b x)}{3 b (b c-a d)^3}+\frac {B d^3 n \log (c+d x)}{3 b (b c-a d)^3}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {\frac {6 A}{(a+b x)^3}+B n \left (\frac {2+\frac {3 d (a+b x)}{-b c+a d}+\frac {6 d^2 (a+b x)^2}{(b c-a d)^2}}{(a+b x)^3}+\frac {6 d^3 \log (a+b x)}{(b c-a d)^3}-\frac {6 d^3 \log (c+d x)}{(b c-a d)^3}\right )+\frac {6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3}}{18 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(503\) vs. \(2(157)=314\).
Time = 54.43 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.04
method | result | size |
parallelrisch | \(-\frac {-18 A \,a^{2} b^{5} c \,d^{3}+18 A a \,b^{6} c^{2} d^{2}-6 B \ln \left (b x +a \right ) x^{3} b^{7} d^{4} n +6 B \,x^{2} a \,b^{6} d^{4} n -6 B \,x^{2} b^{7} c \,d^{3} n +15 B x \,a^{2} b^{5} d^{4} n +3 B x \,b^{7} c^{2} d^{2} n +6 B \ln \left (d x +c \right ) x^{3} b^{7} d^{4} n -6 B \ln \left (b x +a \right ) a^{3} b^{4} d^{4} n +6 B \ln \left (d x +c \right ) a^{3} b^{4} d^{4} n -18 B x a \,b^{6} c \,d^{3} n -18 B \ln \left (b x +a \right ) x^{2} a \,b^{6} d^{4} n +18 B \ln \left (d x +c \right ) x^{2} a \,b^{6} d^{4} n -18 B \ln \left (b x +a \right ) x \,a^{2} b^{5} d^{4} n +18 B \ln \left (d x +c \right ) x \,a^{2} b^{5} d^{4} n -18 B \,a^{2} b^{5} c \,d^{3} n +9 B a \,b^{6} c^{2} d^{2} n -18 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{2} b^{5} c \,d^{3}+18 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a \,b^{6} c^{2} d^{2}+6 A \,a^{3} b^{4} d^{4}-6 A \,b^{7} c^{3} d +6 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) a^{3} b^{4} d^{4}-6 B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{7} c^{3} d +11 B \,a^{3} b^{4} d^{4} n -2 B \,b^{7} c^{3} d n}{18 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )^{3} b^{5} d}\) | \(504\) |
risch | \(\text {Expression too large to display}\) | \(1976\) |
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Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (154) = 308\).
Time = 0.29 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.25 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {6 \, A b^{3} c^{3} - 18 \, A a b^{2} c^{2} d + 18 \, A a^{2} b c d^{2} - 6 \, A a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} n x + {\left (2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} - 11 \, B a^{3} d^{3}\right )} n + 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (b x + a\right ) - 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (d x + c\right ) + 6 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} \log \left (e\right )}{18 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x\right )}} \]
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Timed out. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (154) = 308\).
Time = 0.20 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.41 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {{\left (\frac {6 \, d^{3} e n \log \left (b x + a\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {6 \, d^{3} e n \log \left (d x + c\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {6 \, b^{2} d^{2} e n x^{2} + 2 \, b^{2} c^{2} e n - 7 \, a b c d e n + 11 \, a^{2} d^{2} e n - 3 \, {\left (b^{2} c d e n - 5 \, a b d^{2} e n\right )} x}{a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x}\right )} B}{18 \, e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {A}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (154) = 308\).
Time = 0.29 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.73 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=-\frac {B d^{3} n \log \left (b x + a\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} + \frac {B d^{3} n \log \left (d x + c\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {B n \log \left (b x + a\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {B n \log \left (d x + c\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {6 \, B b^{2} d^{2} n x^{2} - 3 \, B b^{2} c d n x + 15 \, B a b d^{2} n x + 2 \, B b^{2} c^{2} n - 7 \, B a b c d n + 11 \, B a^{2} d^{2} n + 6 \, B b^{2} c^{2} \log \left (e\right ) - 12 \, B a b c d \log \left (e\right ) + 6 \, B a^{2} d^{2} \log \left (e\right ) + 6 \, A b^{2} c^{2} - 12 \, A a b c d + 6 \, A a^{2} d^{2}}{18 \, {\left (b^{6} c^{2} x^{3} - 2 \, a b^{5} c d x^{3} + a^{2} b^{4} d^{2} x^{3} + 3 \, a b^{5} c^{2} x^{2} - 6 \, a^{2} b^{4} c d x^{2} + 3 \, a^{3} b^{3} d^{2} x^{2} + 3 \, a^{2} b^{4} c^{2} x - 6 \, a^{3} b^{3} c d x + 3 \, a^{4} b^{2} d^{2} x + a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )}} \]
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Time = 1.58 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.91 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx=\frac {2\,A\,a\,c\,d}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{3\,b\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,c^2\,n}{9\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,a\,d^2\,n\,x}{6\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,d^2\,n\,x^2}{3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {7\,B\,a\,c\,d\,n}{18\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {11\,B\,a^2\,d^2\,n}{18\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,c\,d\,n\,x}{6\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,n\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,b\,{\left (a\,d-b\,c\right )}^3} \]
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