Optimal. Leaf size=215 \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d g^5 (c+d x)^4}+\frac {b^4 B n \log (a+b x)}{4 d g^5 (b c-a d)^4}-\frac {b^4 B n \log (c+d x)}{4 d g^5 (b c-a d)^4}+\frac {b^3 B n}{4 d g^5 (c+d x) (b c-a d)^3}+\frac {b^2 B n}{8 d g^5 (c+d x)^2 (b c-a d)^2}+\frac {b B n}{12 d g^5 (c+d x)^3 (b c-a d)}+\frac {B n}{16 d g^5 (c+d x)^4} \]
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Rubi [A] time = 0.18, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d g^5 (c+d x)^4}+\frac {b^3 B n}{4 d g^5 (c+d x) (b c-a d)^3}+\frac {b^2 B n}{8 d g^5 (c+d x)^2 (b c-a d)^2}+\frac {b^4 B n \log (a+b x)}{4 d g^5 (b c-a d)^4}-\frac {b^4 B n \log (c+d x)}{4 d g^5 (b c-a d)^4}+\frac {b B n}{12 d g^5 (c+d x)^3 (b c-a d)}+\frac {B n}{16 d g^5 (c+d x)^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^5} \, dx &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}+\frac {(B n) \int \frac {b c-a d}{g^4 (a+b x) (c+d x)^5} \, dx}{4 d g}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}+\frac {(B (b c-a d) n) \int \frac {1}{(a+b x) (c+d x)^5} \, dx}{4 d g^5}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}+\frac {(B (b c-a d) n) \int \left (\frac {b^5}{(b c-a d)^5 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^5}-\frac {b d}{(b c-a d)^2 (c+d x)^4}-\frac {b^2 d}{(b c-a d)^3 (c+d x)^3}-\frac {b^3 d}{(b c-a d)^4 (c+d x)^2}-\frac {b^4 d}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 d g^5}\\ &=\frac {B n}{16 d g^5 (c+d x)^4}+\frac {b B n}{12 d (b c-a d) g^5 (c+d x)^3}+\frac {b^2 B n}{8 d (b c-a d)^2 g^5 (c+d x)^2}+\frac {b^3 B n}{4 d (b c-a d)^3 g^5 (c+d x)}+\frac {b^4 B n \log (a+b x)}{4 d (b c-a d)^4 g^5}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 d g^5 (c+d x)^4}-\frac {b^4 B n \log (c+d x)}{4 d (b c-a d)^4 g^5}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 162, normalized size = 0.75 \[ \frac {\frac {B n \left (12 b^4 \log (a+b x)+\frac {12 b^3 (b c-a d)}{c+d x}+\frac {6 b^2 (b c-a d)^2}{(c+d x)^2}+\frac {4 b (b c-a d)^3}{(c+d x)^3}+\frac {3 (b c-a d)^4}{(c+d x)^4}-12 b^4 \log (c+d x)\right )}{12 (b c-a d)^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(c+d x)^4}}{4 d g^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 735, normalized size = 3.42 \[ -\frac {12 \, A b^{4} c^{4} - 48 \, A a b^{3} c^{3} d + 72 \, A a^{2} b^{2} c^{2} d^{2} - 48 \, A a^{3} b c d^{3} + 12 \, A a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n x^{3} - 6 \, {\left (7 \, B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} n x^{2} - 4 \, {\left (13 \, B b^{4} c^{3} d - 18 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} n x - {\left (25 \, B b^{4} c^{4} - 48 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 16 \, B a^{3} b c d^{3} + 3 \, B a^{4} d^{4}\right )} n + 12 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} \log \relax (e) - 12 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B b^{4} c d^{3} n x^{3} + 6 \, B b^{4} c^{2} d^{2} n x^{2} + 4 \, B b^{4} c^{3} d n x + {\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{48 \, {\left ({\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} g^{5} x^{4} + 4 \, {\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} g^{5} x^{3} + 6 \, {\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} g^{5} x^{2} + 4 \, {\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} g^{5} x + {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )} g^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 11.42, size = 676, normalized size = 3.14 \[ \frac {1}{48} \, {\left (12 \, {\left (\frac {4 \, {\left (b x + a\right )} B b^{3} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}} - \frac {6 \, {\left (b x + a\right )}^{2} B b^{2} d n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{2}} + \frac {4 \, {\left (b x + a\right )}^{3} B b d^{2} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{4} B d^{3} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{4}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) + \frac {3 \, {\left (B d^{3} n - 4 \, A d^{3} - 4 \, B d^{3}\right )} {\left (b x + a\right )}^{4}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{4}} - \frac {16 \, {\left (B b d^{2} n - 3 \, A b d^{2} - 3 \, B b d^{2}\right )} {\left (b x + a\right )}^{3}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{3}} + \frac {36 \, {\left (B b^{2} d n - 2 \, A b^{2} d - 2 \, B b^{2} d\right )} {\left (b x + a\right )}^{2}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{2}} - \frac {48 \, {\left (B b^{3} n - A b^{3} - B b^{3}\right )} {\left (b x + a\right )}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (d g x +c g \right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 652, normalized size = 3.03 \[ \frac {1}{48} \, B n {\left (\frac {12 \, b^{3} d^{3} x^{3} + 25 \, b^{3} c^{3} - 23 \, a b^{2} c^{2} d + 13 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3} + 6 \, {\left (7 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (13 \, b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{{\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} g^{5} x^{4} + 4 \, {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} g^{5} x^{3} + 6 \, {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} g^{5} x^{2} + 4 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} g^{5} x + {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} g^{5}} + \frac {12 \, b^{4} \log \left (b x + a\right )}{{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} g^{5}} - \frac {12 \, b^{4} \log \left (d x + c\right )}{{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} g^{5}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{4 \, {\left (d^{5} g^{5} x^{4} + 4 \, c d^{4} g^{5} x^{3} + 6 \, c^{2} d^{3} g^{5} x^{2} + 4 \, c^{3} d^{2} g^{5} x + c^{4} d g^{5}\right )}} - \frac {A}{4 \, {\left (d^{5} g^{5} x^{4} + 4 \, c d^{4} g^{5} x^{3} + 6 \, c^{2} d^{3} g^{5} x^{2} + 4 \, c^{3} d^{2} g^{5} x + c^{4} d g^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.99, size = 603, normalized size = 2.80 \[ \frac {B\,b^4\,n\,\mathrm {atanh}\left (\frac {4\,a^4\,d^5\,g^5-8\,a^3\,b\,c\,d^4\,g^5+8\,a\,b^3\,c^3\,d^2\,g^5-4\,b^4\,c^4\,d\,g^5}{4\,d\,g^5\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,d\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{4\,d\,\left (c^4\,g^5+4\,c^3\,d\,g^5\,x+6\,c^2\,d^2\,g^5\,x^2+4\,c\,d^3\,g^5\,x^3+d^4\,g^5\,x^4\right )}-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3-3\,B\,a^3\,d^3\,n+25\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2-23\,B\,a\,b^2\,c^2\,d\,n+13\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,x\,\left (B\,n\,a^2\,d^3-5\,B\,n\,a\,b\,c\,d^2+13\,B\,n\,b^2\,c^2\,d\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {b^2\,x^2\,\left (B\,a\,d^3\,n-7\,B\,b\,c\,d^2\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,c^4\,d\,g^5+16\,c^3\,d^2\,g^5\,x+24\,c^2\,d^3\,g^5\,x^2+16\,c\,d^4\,g^5\,x^3+4\,d^5\,g^5\,x^4} \]
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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