Optimal. Leaf size=195 \[ -\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{4 b (a+b x)^4}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}+\frac {B d^3 n}{4 b (a+b x) (b c-a d)^3}-\frac {B d^2 n}{8 b (a+b x)^2 (b c-a d)^2}+\frac {B d n}{12 b (a+b x)^3 (b c-a d)}-\frac {B n}{16 b (a+b x)^4} \]
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Rubi [A] time = 0.19, antiderivative size = 207, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6742, 2492, 44} \[ -\frac {A}{4 b (a+b x)^4}+\frac {B d^3 n}{4 b (a+b x) (b c-a d)^3}-\frac {B d^2 n}{8 b (a+b x)^2 (b c-a d)^2}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {B d n}{12 b (a+b x)^3 (b c-a d)}-\frac {B n}{16 b (a+b x)^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2492
Rule 6742
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx &=\int \left (\frac {A}{(a+b x)^5}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}\right ) \, dx\\ &=-\frac {A}{4 b (a+b x)^4}+B \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx\\ &=-\frac {A}{4 b (a+b x)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b}\\ &=-\frac {A}{4 b (a+b x)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac {(B (b c-a d) n) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac {A}{4 b (a+b x)^4}-\frac {B n}{16 b (a+b x)^4}+\frac {B d n}{12 b (b c-a d) (a+b x)^3}-\frac {B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac {B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 165, normalized size = 0.85 \[ -\frac {\frac {12 A}{(a+b x)^4}+B n \left (-\frac {12 d^4 \log (a+b x)}{(b c-a d)^4}+\frac {12 d^4 \log (c+d x)}{(b c-a d)^4}+\frac {-\frac {12 d^3 (a+b x)^3}{(b c-a d)^3}+\frac {6 d^2 (a+b x)^2}{(b c-a d)^2}+\frac {4 d (a+b x)}{a d-b c}+3}{(a+b x)^4}\right )+\frac {12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}}{48 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.52, size = 820, normalized size = 4.21 \[ -\frac {12 \, A b^{4} c^{4} - 48 \, A a b^{3} c^{3} d + 72 \, A a^{2} b^{2} c^{2} d^{2} - 48 \, A a^{3} b c d^{3} + 12 \, A a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} 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 )} n x^{2} - 4 \, {\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 )} n x + {\left (3 \, B b^{4} c^{4} - 16 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 48 \, B a^{3} b c d^{3} + 25 \, B a^{4} d^{4}\right )} n - 12 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x - {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} n\right )} \log \left (b x + a\right ) + 12 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x - {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} n\right )} \log \left (d x + c\right ) + 12 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} \log \relax (e)}{48 \, {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4} + {\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 710, normalized size = 3.64 \[ \frac {B d^{4} n \log \left (b x + a\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B d^{4} n \log \left (d x + c\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B n \log \left (b x + a\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {B n \log \left (d x + c\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {12 \, B b^{3} d^{3} n x^{3} - 6 \, B b^{3} c d^{2} n x^{2} + 42 \, B a b^{2} d^{3} n x^{2} + 4 \, B b^{3} c^{2} d n x - 20 \, B a b^{2} c d^{2} n x + 52 \, B a^{2} b d^{3} n x - 3 \, B b^{3} c^{3} n + 13 \, B a b^{2} c^{2} d n - 23 \, B a^{2} b c d^{2} n + 25 \, B a^{3} d^{3} n - 12 \, A b^{3} c^{3} - 12 \, B b^{3} c^{3} + 36 \, A a b^{2} c^{2} d + 36 \, B a b^{2} c^{2} d - 36 \, A a^{2} b c d^{2} - 36 \, B a^{2} b c d^{2} + 12 \, A a^{3} d^{3} + 12 \, B a^{3} d^{3}}{48 \, {\left (b^{8} c^{3} x^{4} - 3 \, a b^{7} c^{2} d x^{4} + 3 \, a^{2} b^{6} c d^{2} x^{4} - a^{3} b^{5} d^{3} x^{4} + 4 \, a b^{7} c^{3} x^{3} - 12 \, a^{2} b^{6} c^{2} d x^{3} + 12 \, a^{3} b^{5} c d^{2} x^{3} - 4 \, a^{4} b^{4} d^{3} x^{3} + 6 \, a^{2} b^{6} c^{3} x^{2} - 18 \, a^{3} b^{5} c^{2} d x^{2} + 18 \, a^{4} b^{4} c d^{2} x^{2} - 6 \, a^{5} b^{3} d^{3} x^{2} + 4 \, a^{3} b^{5} c^{3} x - 12 \, a^{4} b^{4} c^{2} d x + 12 \, a^{5} b^{3} c d^{2} x - 4 \, a^{6} b^{2} d^{3} x + a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.67, size = 2583, normalized size = 13.25 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 618, normalized size = 3.17 \[ \frac {{\left (\frac {12 \, d^{4} e n \log \left (b x + a\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {12 \, d^{4} e n \log \left (d x + c\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {12 \, b^{3} d^{3} e n x^{3} - 3 \, b^{3} c^{3} e n + 13 \, a b^{2} c^{2} d e n - 23 \, a^{2} b c d^{2} e n + 25 \, a^{3} d^{3} e n - 6 \, {\left (b^{3} c d^{2} e n - 7 \, a b^{2} d^{3} e n\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d e n - 5 \, a b^{2} c d^{2} e n + 13 \, a^{2} b d^{3} e n\right )} x}{a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x}\right )} B}{48 \, e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {A}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.33, size = 555, normalized size = 2.85 \[ -\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3+25\,B\,a^3\,d^3\,n-3\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d\,n-23\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,n\,a^2\,b\,d^2-5\,B\,n\,a\,b^2\,c\,d+B\,n\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c\,n-7\,B\,a\,b^2\,d\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b+16\,a^3\,b^2\,x+24\,a^2\,b^3\,x^2+16\,a\,b^4\,x^3+4\,b^5\,x^4}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{4\,b\,\left (a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4\right )}-\frac {B\,d^4\,n\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4+8\,a^3\,b^2\,c\,d^3-8\,a\,b^4\,c^3\,d+4\,b^5\,c^4}{4\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,{\left (a\,d-b\,c\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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