Integrand size = 40, antiderivative size = 45 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 B (b c-a d) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \]
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Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2573, 2561, 2339, 30} \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 B n (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2} \]
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Rule 30
Rule 2339
Rule 2561
Rule 2573
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3} \, dx,e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \frac {1}{x \left (A+B \log \left (e x^n\right )\right )^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{b c-a d},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \frac {1}{x^3} \, dx,x,A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B (b c-a d) n},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = -\frac {1}{2 B (b c-a d) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 (b B c n-a B d n) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2} \]
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Time = 163.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {1}{2 {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2} B n \left (a d -c b \right )}\) | \(44\) |
default | \(\frac {1}{2 {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2} B n \left (a d -c b \right )}\) | \(44\) |
parallelrisch | \(\frac {1}{2 {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{2} B n \left (a d -c b \right )}\) | \(44\) |
risch | \(\frac {2}{B n \left (a d -c b \right ) {\left (2 A +2 B \ln \left (e \right )+2 B \ln \left (\left (b x +a \right )^{n}\right )-2 B \ln \left (\left (d x +c \right )^{n}\right )-i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i B \pi \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}-i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )+i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3}\right )}^{2}}\) | \(366\) |
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (43) = 86\).
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 5.29 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right )^{2} + {\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (d x + c\right )^{2} + {\left (B^{3} b c - B^{3} a d\right )} n \log \left (e\right )^{2} + 2 \, {\left (A B^{2} b c - A B^{2} a d\right )} n \log \left (e\right ) + {\left (A^{2} B b c - A^{2} B a d\right )} n + 2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \left (e\right ) + {\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right ) + {\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \left (e\right ) + {\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (43) = 86\).
Time = 0.36 (sec) , antiderivative size = 220, normalized size of antiderivative = 4.89 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 \, {\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (b c n - a d n\right )} B^{3} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + {\left (b c n - a d n\right )} A^{2} B + 2 \, {\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} A B^{2} + {\left (b c n \log \left (e\right )^{2} - a d n \log \left (e\right )^{2}\right )} B^{3} + 2 \, {\left ({\left (b c n - a d n\right )} A B^{2} + {\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right ) + {\left (b c n - a d n\right )} A B^{2} + {\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{3}\right )} \log \left ({\left (d x + c\right )}^{n}\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (43) = 86\).
Time = 0.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 7.13 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=-\frac {1}{2 \, {\left (B^{3} b c n^{3} \log \left (b x + a\right )^{2} - B^{3} a d n^{3} \log \left (b x + a\right )^{2} - 2 \, B^{3} b c n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + 2 \, B^{3} a d n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + B^{3} b c n^{3} \log \left (d x + c\right )^{2} - B^{3} a d n^{3} \log \left (d x + c\right )^{2} + 2 \, B^{3} b c n^{2} \log \left (b x + a\right ) \log \left (e\right ) - 2 \, B^{3} a d n^{2} \log \left (b x + a\right ) \log \left (e\right ) - 2 \, B^{3} b c n^{2} \log \left (d x + c\right ) \log \left (e\right ) + 2 \, B^{3} a d n^{2} \log \left (d x + c\right ) \log \left (e\right ) + 2 \, A B^{2} b c n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} a d n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} b c n^{2} \log \left (d x + c\right ) + 2 \, A B^{2} a d n^{2} \log \left (d x + c\right ) + B^{3} b c n \log \left (e\right )^{2} - B^{3} a d n \log \left (e\right )^{2} + 2 \, A B^{2} b c n \log \left (e\right ) - 2 \, A B^{2} a d n \log \left (e\right ) + A^{2} B b c n - A^{2} B a d n\right )}} \]
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Time = 1.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.60 \[ \int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx=\frac {1}{2\,B\,n\,\left (a\,d-b\,c\right )\,\left (A^2+2\,A\,B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )+B^2\,{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\right )} \]
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