Optimal. Leaf size=215 \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{4 d g^5 (c+d x)^4}+\frac{b^3 B n}{4 d g^5 (c+d x) (b c-a d)^3}+\frac{b^2 B n}{8 d g^5 (c+d x)^2 (b c-a d)^2}+\frac{b^4 B n \log (a+b x)}{4 d g^5 (b c-a d)^4}-\frac{b^4 B n \log (c+d x)}{4 d g^5 (b c-a d)^4}+\frac{b B n}{12 d g^5 (c+d x)^3 (b c-a d)}+\frac{B n}{16 d g^5 (c+d x)^4} \]
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Rubi [A] time = 0.175639, antiderivative size = 215, normalized size of antiderivative = 1., 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 2525
Rule 12
Rule 44
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*}
Mathematica [A] time = 0.237246, size = 162, normalized size = 0.75 \[ \frac{\frac{B n \left (\frac{12 b^3 (b c-a d)}{c+d x}+\frac{6 b^2 (b c-a d)^2}{(c+d x)^2}+12 b^4 \log (a+b x)+\frac{4 b (b c-a d)^3}{(c+d x)^3}+\frac{3 (b c-a d)^4}{(c+d x)^4}-12 b^4 \log (c+d x)\right )}{12 (b c-a d)^4}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{(c+d x)^4}}{4 d g^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.455, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{5}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.25488, size = 880, normalized size = 4.09 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.999619, size = 1505, normalized size = 7. \begin{align*} -\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 \left (e\right ) - 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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34134, size = 1038, normalized size = 4.83 \begin{align*} \frac{B b^{4} n \log \left (b x + a\right )}{4 \,{\left (b^{4} c^{4} d g^{5} - 4 \, a b^{3} c^{3} d^{2} g^{5} + 6 \, a^{2} b^{2} c^{2} d^{3} g^{5} - 4 \, a^{3} b c d^{4} g^{5} + a^{4} d^{5} g^{5}\right )}} - \frac{B b^{4} n \log \left (d x + c\right )}{4 \,{\left (b^{4} c^{4} d g^{5} - 4 \, a b^{3} c^{3} d^{2} g^{5} + 6 \, a^{2} b^{2} c^{2} d^{3} g^{5} - 4 \, a^{3} b c d^{4} g^{5} + a^{4} d^{5} g^{5}\right )}} - \frac{B n \log \left (\frac{b x + a}{d x + c}\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{12 \, B b^{3} d^{3} n x^{3} + 42 \, B b^{3} c d^{2} n x^{2} - 6 \, B a b^{2} d^{3} n x^{2} + 52 \, B b^{3} c^{2} d n x - 20 \, B a b^{2} c d^{2} n x + 4 \, B a^{2} b d^{3} n x + 25 \, B b^{3} c^{3} n - 23 \, B a b^{2} c^{2} d n + 13 \, B a^{2} b c d^{2} n - 3 \, 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^{3} c^{3} d^{5} g^{5} x^{4} - 3 \, a b^{2} c^{2} d^{6} g^{5} x^{4} + 3 \, a^{2} b c d^{7} g^{5} x^{4} - a^{3} d^{8} g^{5} x^{4} + 4 \, b^{3} c^{4} d^{4} g^{5} x^{3} - 12 \, a b^{2} c^{3} d^{5} g^{5} x^{3} + 12 \, a^{2} b c^{2} d^{6} g^{5} x^{3} - 4 \, a^{3} c d^{7} g^{5} x^{3} + 6 \, b^{3} c^{5} d^{3} g^{5} x^{2} - 18 \, a b^{2} c^{4} d^{4} g^{5} x^{2} + 18 \, a^{2} b c^{3} d^{5} g^{5} x^{2} - 6 \, a^{3} c^{2} d^{6} g^{5} x^{2} + 4 \, b^{3} c^{6} d^{2} g^{5} x - 12 \, a b^{2} c^{5} d^{3} g^{5} x + 12 \, a^{2} b c^{4} d^{4} g^{5} x - 4 \, a^{3} c^{3} d^{5} g^{5} x + b^{3} c^{7} d g^{5} - 3 \, a b^{2} c^{6} d^{2} g^{5} + 3 \, a^{2} b c^{5} d^{3} g^{5} - a^{3} c^{4} d^{4} g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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