Integrand size = 41, antiderivative size = 281 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {B d^2 i n (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b B d i n (c+d x)^3}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i n (c+d x)^4}{16 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4} \]
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Time = 0.15 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2561, 45, 2372, 12, 14} \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {b^2 i (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac {d^2 i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b d i (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B i n (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^3}-\frac {B d^2 i n (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b B d i n (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^3} \]
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Rule 12
Rule 14
Rule 45
Rule 2372
Rule 2561
Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \frac {(b-d x)^2 \left (A+B \log \left (e x^n\right )\right )}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5} \\ & = -\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4}-\frac {(B i n) \text {Subst}\left (\int \frac {-3 b^2+8 b d x-6 d^2 x^2}{12 x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5} \\ & = -\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4}-\frac {(B i n) \text {Subst}\left (\int \frac {-3 b^2+8 b d x-6 d^2 x^2}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{12 (b c-a d)^3 g^5} \\ & = -\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4}-\frac {(B i n) \text {Subst}\left (\int \left (-\frac {3 b^2}{x^5}+\frac {8 b d}{x^4}-\frac {6 d^2}{x^3}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{12 (b c-a d)^3 g^5} \\ & = -\frac {B d^2 i n (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b B d i n (c+d x)^3}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i n (c+d x)^4}{16 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.78 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {i \left (\frac {36 A b c}{(a+b x)^4}-\frac {36 a A d}{(a+b x)^4}+\frac {9 b B c n}{(a+b x)^4}-\frac {9 a B d n}{(a+b x)^4}+\frac {48 A d}{(a+b x)^3}+\frac {4 B d n}{(a+b x)^3}-\frac {6 B d^2 n}{(b c-a d) (a+b x)^2}+\frac {12 B d^3 n}{(b c-a d)^2 (a+b x)}+\frac {12 B d^4 n \log (a+b x)}{(b c-a d)^3}+\frac {12 B (3 b c+a d+4 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4}-\frac {12 B d^4 n \log (c+d x)}{(b c-a d)^3}\right )}{144 b^2 g^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(985\) vs. \(2(269)=538\).
Time = 27.71 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.51
method | result | size |
parallelrisch | \(\frac {48 A \,x^{3} a^{7} b c \,d^{4} i n -288 A \,x^{3} a^{5} b^{3} c^{3} d^{2} i n -144 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} b \,c^{3} d^{2} i n +48 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b^{2} c^{4} d i n +384 A \,x^{3} a^{4} b^{4} c^{4} d i n +72 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{8} c \,d^{4} i n +48 B \,x^{2} a^{7} b \,c^{2} d^{3} i \,n^{2}-222 B \,x^{2} a^{6} b^{2} c^{3} d^{2} i \,n^{2}+192 B \,x^{2} a^{5} b^{3} c^{4} d i \,n^{2}-432 A \,x^{2} a^{6} b^{2} c^{3} d^{2} i n +576 A \,x^{2} a^{5} b^{3} c^{4} d i n +144 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{8} c^{2} d^{3} i n -168 B x \,a^{7} b \,c^{3} d^{2} i \,n^{2}+132 B x \,a^{6} b^{2} c^{4} d i \,n^{2}-432 A x \,a^{7} b \,c^{3} d^{2} i n +432 A x \,a^{6} b^{2} c^{4} d i n -96 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} b \,c^{4} d i n +12 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b^{2} c \,d^{4} i n +48 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} b c \,d^{4} i n +13 B \,x^{4} a^{6} b^{2} c \,d^{4} i \,n^{2}-36 B \,x^{4} a^{4} b^{4} c^{3} d^{2} i \,n^{2}+32 B \,x^{4} a^{3} b^{5} c^{4} d i \,n^{2}+12 A \,x^{4} a^{6} b^{2} c \,d^{4} i n -72 A \,x^{4} a^{4} b^{4} c^{3} d^{2} i n +96 A \,x^{4} a^{3} b^{5} c^{4} d i n +40 B \,x^{3} a^{7} b c \,d^{4} i \,n^{2}+12 B \,x^{3} a^{6} b^{2} c^{2} d^{3} i \,n^{2}-144 B \,x^{3} a^{5} b^{3} c^{3} d^{2} i \,n^{2}+128 B \,x^{3} a^{4} b^{4} c^{4} d i \,n^{2}-9 B \,x^{4} a^{2} b^{6} c^{5} i \,n^{2}-36 A \,x^{4} a^{2} b^{6} c^{5} i n -36 B \,x^{3} a^{3} b^{5} c^{5} i \,n^{2}-144 A \,x^{3} a^{3} b^{5} c^{5} i n +36 B \,x^{2} a^{8} c \,d^{4} i \,n^{2}-54 B \,x^{2} a^{4} b^{4} c^{5} i \,n^{2}+72 A \,x^{2} a^{8} c \,d^{4} i n -216 A \,x^{2} a^{4} b^{4} c^{5} i n +72 B x \,a^{8} c^{2} d^{3} i \,n^{2}-36 B x \,a^{5} b^{3} c^{5} i \,n^{2}+144 A x \,a^{8} c^{2} d^{3} i n -144 A x \,a^{5} b^{3} c^{5} i n +72 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{8} c^{3} d^{2} i n +36 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b^{2} c^{5} i n}{144 g^{5} \left (b x +a \right )^{4} n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a^{6} c}\) | \(986\) |
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Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (269) = 538\).
Time = 0.35 (sec) , antiderivative size = 773, normalized size of antiderivative = 2.75 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i n x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} i n x^{2} + {\left (9 \, B b^{4} c^{4} - 32 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 13 \, B a^{4} d^{4}\right )} i n + 12 \, {\left (3 \, A b^{4} c^{4} - 8 \, A a b^{3} c^{3} d + 6 \, A a^{2} b^{2} c^{2} d^{2} - A a^{4} d^{4}\right )} i + 4 \, {\left ({\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} i n + 12 \, {\left (A b^{4} c^{3} d - 3 \, A a b^{3} c^{2} d^{2} + 3 \, A a^{2} b^{2} c d^{3} - A a^{3} b d^{4}\right )} i\right )} x + 12 \, {\left (4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} i x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - B a^{4} d^{4}\right )} i\right )} \log \left (e\right ) + 12 \, {\left (B b^{4} d^{4} i n x^{4} + 4 \, B a b^{3} d^{4} i n x^{3} + 6 \, B a^{2} b^{2} d^{4} i n x^{2} + 4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3}\right )} i n x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \]
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Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (269) = 538\).
Time = 0.25 (sec) , antiderivative size = 1398, normalized size of antiderivative = 4.98 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
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none
Time = 1.36 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.40 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {1}{144} \, {\left (\frac {12 \, {\left (3 \, B b^{2} i n - \frac {8 \, {\left (b x + a\right )} B b d i n}{d x + c} + \frac {6 \, {\left (b x + a\right )}^{2} B d^{2} i n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {9 \, B b^{2} i n - \frac {32 \, {\left (b x + a\right )} B b d i n}{d x + c} + \frac {36 \, {\left (b x + a\right )}^{2} B d^{2} i n}{{\left (d x + c\right )}^{2}} + 36 \, B b^{2} i \log \left (e\right ) - \frac {96 \, {\left (b x + a\right )} B b d i \log \left (e\right )}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} B d^{2} i \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 36 \, A b^{2} i - \frac {96 \, {\left (b x + a\right )} A b d i}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} A d^{2} i}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
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Time = 2.43 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.17 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\frac {B\,d^4\,i\,n\,\mathrm {atanh}\left (\frac {12\,a^3\,b^2\,d^3\,g^5-12\,a^2\,b^3\,c\,d^2\,g^5-12\,a\,b^4\,c^2\,d\,g^5+12\,b^5\,c^3\,g^5}{12\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{6\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c\,i}{4\,b}+\frac {B\,a\,d\,i}{12\,b^2}+\frac {B\,d\,i\,x}{3\,b}\right )}{a^4\,g^5+4\,a^3\,b\,g^5\,x+6\,a^2\,b^2\,g^5\,x^2+4\,a\,b^3\,g^5\,x^3+b^4\,g^5\,x^4}-\frac {\frac {12\,A\,a^3\,d^3\,i+36\,A\,b^3\,c^3\,i+13\,B\,a^3\,d^3\,i\,n+9\,B\,b^3\,c^3\,i\,n-60\,A\,a\,b^2\,c^2\,d\,i+12\,A\,a^2\,b\,c\,d^2\,i-23\,B\,a\,b^2\,c^2\,d\,i\,n+13\,B\,a^2\,b\,c\,d^2\,i\,n}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i+12\,A\,b^3\,c^2\,d\,i-24\,A\,a\,b^2\,c\,d^2\,i+13\,B\,a^2\,b\,d^3\,i\,n+B\,b^3\,c^2\,d\,i\,n-5\,B\,a\,b^2\,c\,d^2\,i\,n\right )}{3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {d\,x^2\,\left (B\,b^3\,c\,d\,i\,n-7\,B\,a\,b^2\,d^2\,i\,n\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^3\,d^3\,i\,n\,x^3}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{12\,a^4\,b^2\,g^5+48\,a^3\,b^3\,g^5\,x+72\,a^2\,b^4\,g^5\,x^2+48\,a\,b^5\,g^5\,x^3+12\,b^6\,g^5\,x^4} \]
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