Integrand size = 43, antiderivative size = 93 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {B i^2 n (c+d x)^3}{9 (b c-a d) g^4 (a+b x)^3}-\frac {i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d) g^4 (a+b x)^3} \]
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Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2561, 2341} \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)}-\frac {B i^2 n (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)} \]
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Rule 2341
Rule 2561
Rubi steps \begin{align*} \text {integral}& = \frac {i^2 \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g^4} \\ & = -\frac {B i^2 n (c+d x)^3}{9 (b c-a d) g^4 (a+b x)^3}-\frac {i^2 (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) g^4 (a+b x)^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(329\) vs. \(2(93)=186\).
Time = 0.19 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.54 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {i^2 \left (3 A b^3 c^3-3 a^3 A d^3+b^3 B c^3 n-a^3 B d^3 n+9 A b^3 c^2 d x-9 a^2 A b d^3 x+3 b^3 B c^2 d n x-3 a^2 b B d^3 n x+9 A b^3 c d^2 x^2-9 a A b^2 d^3 x^2+3 b^3 B c d^2 n x^2-3 a b^2 B d^3 n x^2+3 B d^3 n (a+b x)^3 \log (a+b x)+3 B (b c-a d) \left (a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-3 a^3 B d^3 n \log (c+d x)-9 a^2 b B d^3 n x \log (c+d x)-9 a b^2 B d^3 n x^2 \log (c+d x)-3 b^3 B d^3 n x^3 \log (c+d x)\right )}{9 b^3 (b c-a d) g^4 (a+b x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(375\) vs. \(2(89)=178\).
Time = 10.00 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.04
method | result | size |
parallelrisch | \(-\frac {B \,a^{3} b^{2} d^{4} i^{2} n^{2}-B \,b^{5} c^{3} d \,i^{2} n^{2}+3 A \,a^{3} b^{2} d^{4} i^{2} n -3 A \,b^{5} c^{3} d \,i^{2} n -3 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} d^{4} i^{2} n +3 B \,x^{2} a \,b^{4} d^{4} i^{2} n^{2}-3 B \,x^{2} b^{5} c \,d^{3} i^{2} n^{2}+9 A \,x^{2} a \,b^{4} d^{4} i^{2} n -9 A \,x^{2} b^{5} c \,d^{3} i^{2} n +3 B x \,a^{2} b^{3} d^{4} i^{2} n^{2}-3 B x \,b^{5} c^{2} d^{2} i^{2} n^{2}+9 A x \,a^{2} b^{3} d^{4} i^{2} n -9 A x \,b^{5} c^{2} d^{2} i^{2} n -3 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{3} d \,i^{2} n -9 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c \,d^{3} i^{2} n -9 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{2} d^{2} i^{2} n}{9 g^{4} \left (b x +a \right )^{3} b^{5} d n \left (a d -c b \right )}\) | \(376\) |
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Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (89) = 178\).
Time = 0.33 (sec) , antiderivative size = 409, normalized size of antiderivative = 4.40 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {{\left (B b^{3} c^{3} - B a^{3} d^{3}\right )} i^{2} n + 3 \, {\left (A b^{3} c^{3} - A a^{3} d^{3}\right )} i^{2} + 3 \, {\left ({\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i^{2} n + 3 \, {\left (A b^{3} c d^{2} - A a b^{2} d^{3}\right )} i^{2}\right )} x^{2} + 3 \, {\left ({\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} i^{2} n + 3 \, {\left (A b^{3} c^{2} d - A a^{2} b d^{3}\right )} i^{2}\right )} x + 3 \, {\left (3 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \, {\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} i^{2} x + {\left (B b^{3} c^{3} - B a^{3} d^{3}\right )} i^{2}\right )} \log \left (e\right ) + 3 \, {\left (B b^{3} d^{3} i^{2} n x^{3} + 3 \, B b^{3} c d^{2} i^{2} n x^{2} + 3 \, B b^{3} c^{2} d i^{2} n x + B b^{3} c^{3} i^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{9 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \]
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Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1544 vs. \(2 (89) = 178\).
Time = 0.25 (sec) , antiderivative size = 1544, normalized size of antiderivative = 16.60 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\text {Too large to display} \]
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none
Time = 1.55 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {1}{9} \, {\left (\frac {3 \, {\left (d x + c\right )}^{3} B i^{2} n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{3} g^{4}} + \frac {{\left (B i^{2} n + 3 \, B i^{2} \log \left (e\right ) + 3 \, A i^{2}\right )} {\left (d x + c\right )}^{3}}{{\left (b x + a\right )}^{3} g^{4}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
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Time = 2.23 (sec) , antiderivative size = 421, normalized size of antiderivative = 4.53 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {x\,\left (3\,A\,a\,b\,d^2\,i^2+3\,A\,b^2\,c\,d\,i^2+B\,a\,b\,d^2\,i^2\,n+B\,b^2\,c\,d\,i^2\,n\right )+x^2\,\left (3\,A\,b^2\,d^2\,i^2+B\,b^2\,d^2\,i^2\,n\right )+A\,a^2\,d^2\,i^2+A\,b^2\,c^2\,i^2+\frac {B\,a^2\,d^2\,i^2\,n}{3}+\frac {B\,b^2\,c^2\,i^2\,n}{3}+A\,a\,b\,c\,d\,i^2+\frac {B\,a\,b\,c\,d\,i^2\,n}{3}}{3\,a^3\,b^3\,g^4+9\,a^2\,b^4\,g^4\,x+9\,a\,b^5\,g^4\,x^2+3\,b^6\,g^4\,x^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^3}+\frac {B\,c\,d\,i^2}{3\,b^2}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{3\,b^3}+\frac {B\,c\,d\,i^2}{3\,b^2}\right )+\frac {2\,B\,a\,d^2\,i^2}{3\,b^2}+\frac {2\,B\,c\,d\,i^2}{3\,b}\right )+\frac {B\,c^2\,i^2}{3\,b}+\frac {B\,d^2\,i^2\,x^2}{b}\right )}{a^3\,g^4+3\,a^2\,b\,g^4\,x+3\,a\,b^2\,g^4\,x^2+b^3\,g^4\,x^3}-\frac {B\,d^3\,i^2\,n\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )} \]
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