Optimal. Leaf size=215 \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3 n}{4 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2 n}{8 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 n \log (a+b x)}{4 b g^5 (b c-a d)^4}-\frac{B d^4 n \log (c+d x)}{4 b g^5 (b c-a d)^4}+\frac{B d n}{12 b g^5 (a+b x)^3 (b c-a d)}-\frac{B n}{16 b g^5 (a+b x)^4} \]
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Rubi [A] time = 0.189061, 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} \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{4 b g^5 (a+b x)^4}+\frac{B d^3 n}{4 b g^5 (a+b x) (b c-a d)^3}-\frac{B d^2 n}{8 b g^5 (a+b x)^2 (b c-a d)^2}+\frac{B d^4 n \log (a+b x)}{4 b g^5 (b c-a d)^4}-\frac{B d^4 n \log (c+d x)}{4 b g^5 (b c-a d)^4}+\frac{B d n}{12 b g^5 (a+b x)^3 (b c-a d)}-\frac{B n}{16 b g^5 (a+b x)^4} \]
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 )}{(a g+b g x)^5} \, dx &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 b g^5 (a+b x)^4}+\frac{(B n) \int \frac{b c-a d}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 b g^5 (a+b x)^4}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b g^5}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{4 b g^5 (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 g^5}\\ &=-\frac{B n}{16 b g^5 (a+b x)^4}+\frac{B d n}{12 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2 n}{8 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3 n}{4 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 n \log (a+b x)}{4 b (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 b g^5 (a+b x)^4}-\frac{B d^4 n \log (c+d x)}{4 b (b c-a d)^4 g^5}\\ \end{align*}
Mathematica [A] time = 0.259689, size = 162, normalized size = 0.75 \[ \frac{\frac{B n \left (\frac{12 d^3 (b c-a d)}{a+b x}-\frac{6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac{4 d (b c-a d)^3}{(a+b x)^3}-\frac{3 (b c-a d)^4}{(a+b x)^4}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^4}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{(a+b x)^4}}{4 b g^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.437, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \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 = 1.34799, size = 879, normalized size = 4.09 \begin{align*} \frac{1}{48} \, B n{\left (\frac{12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\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 )} g^{5} 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 )} g^{5} 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 )} g^{5} 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 )} g^{5} x +{\left (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 )} g^{5}} + \frac{12 \, d^{4} \log \left (b x + a\right )}{{\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 )} g^{5}} - \frac{12 \, d^{4} \log \left (d x + c\right )}{{\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 )} 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 (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} - \frac{A}{4 \,{\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.06627, 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 (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} 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 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 (\frac{b x + a}{d x + c}\right )}{48 \,{\left ({\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 )} g^{5} 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 )} g^{5} 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 )} g^{5} 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 )} g^{5} x +{\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}\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 = 1.42214, size = 1038, normalized size = 4.83 \begin{align*} \frac{B d^{4} n \log \left (b x + a\right )}{4 \,{\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} - \frac{B d^{4} n \log \left (d x + c\right )}{4 \,{\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} - \frac{B n \log \left (\frac{b x + a}{d x + c}\right )}{4 \,{\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\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} g^{5} x^{4} - 3 \, a b^{7} c^{2} d g^{5} x^{4} + 3 \, a^{2} b^{6} c d^{2} g^{5} x^{4} - a^{3} b^{5} d^{3} g^{5} x^{4} + 4 \, a b^{7} c^{3} g^{5} x^{3} - 12 \, a^{2} b^{6} c^{2} d g^{5} x^{3} + 12 \, a^{3} b^{5} c d^{2} g^{5} x^{3} - 4 \, a^{4} b^{4} d^{3} g^{5} x^{3} + 6 \, a^{2} b^{6} c^{3} g^{5} x^{2} - 18 \, a^{3} b^{5} c^{2} d g^{5} x^{2} + 18 \, a^{4} b^{4} c d^{2} g^{5} x^{2} - 6 \, a^{5} b^{3} d^{3} g^{5} x^{2} + 4 \, a^{3} b^{5} c^{3} g^{5} x - 12 \, a^{4} b^{4} c^{2} d g^{5} x + 12 \, a^{5} b^{3} c d^{2} g^{5} x - 4 \, a^{6} b^{2} d^{3} g^{5} x + a^{4} b^{4} c^{3} g^{5} - 3 \, a^{5} b^{3} c^{2} d g^{5} + 3 \, a^{6} b^{2} c d^{2} g^{5} - a^{7} b d^{3} g^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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