Optimal. Leaf size=208 \[ -\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A}{4 b g^5 (a+b x)^4}-\frac{B d^3}{2 b g^5 (a+b x) (b c-a d)^3}+\frac{B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}+\frac{B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}-\frac{B d}{6 b g^5 (a+b x)^3 (b c-a d)}+\frac{B}{8 b g^5 (a+b x)^4} \]
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Rubi [A] time = 0.142715, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A}{4 b g^5 (a+b x)^4}-\frac{B d^3}{2 b g^5 (a+b x) (b c-a d)^3}+\frac{B d^2}{4 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B d^4 \log (a+b x)}{2 b g^5 (b c-a d)^4}+\frac{B d^4 \log (c+d x)}{2 b g^5 (b c-a d)^4}-\frac{B d}{6 b g^5 (a+b x)^3 (b c-a d)}+\frac{B}{8 b g^5 (a+b x)^4} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a g+b g x)^5} \, dx &=-\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{2 (-b c+a d)}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}-\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}-\frac{(B (b c-a d)) \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}{2 b g^5}\\ &=\frac{B}{8 b g^5 (a+b x)^4}-\frac{B d}{6 b (b c-a d) g^5 (a+b x)^3}+\frac{B d^2}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac{B d^3}{2 b (b c-a d)^3 g^5 (a+b x)}-\frac{B d^4 \log (a+b x)}{2 b (b c-a d)^4 g^5}+\frac{B d^4 \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{4 b g^5 (a+b x)^4}\\ \end{align*}
Mathematica [A] time = 0.189059, size = 162, normalized size = 0.78 \[ \frac{\frac{B \left (6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a+b x)^3 (a d-b c)+12 d^4 (a+b x)^4 \log (c+d x)+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )}{(b c-a d)^4}-6 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{24 b g^5 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 587, normalized size = 2.8 \begin{align*} -{\frac{A}{4\,b \left ( bx+a \right ) ^{4}{g}^{5}}}-{\frac{B}{4\,b \left ( bx+a \right ) ^{4}{g}^{5}}\ln \left ({\frac{e}{{b}^{2}} \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) ^{2}} \right ) }+{\frac{B{a}^{4}{d}^{4}}{8\,b{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{4}}}-{\frac{B{a}^{3}{d}^{3}c}{2\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{4}}}+{\frac{3\,bB{a}^{2}{d}^{2}{c}^{2}}{4\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{4}}}-{\frac{{b}^{2}Bad{c}^{3}}{2\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{4}}}+{\frac{B{a}^{3}{d}^{4}}{6\,b{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{3}}}-{\frac{B{a}^{2}{d}^{3}c}{2\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{3}}}+{\frac{bBa{d}^{2}{c}^{2}}{2\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{3}}}+{\frac{B{a}^{2}{d}^{4}}{4\,b{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{2}}}-{\frac{Ba{d}^{3}c}{2\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{2}}}+{\frac{Ba{d}^{4}}{2\,b{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}+{\frac{Ba{d}^{5}}{2\,b{g}^{5} \left ( ad-bc \right ) ^{5}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) }+{\frac{{b}^{3}B{c}^{4}}{8\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{4}}}-{\frac{{b}^{2}B{c}^{3}d}{6\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{3}}}+{\frac{bB{c}^{2}{d}^{2}}{4\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) ^{2}}}-{\frac{Bc{d}^{3}}{2\,{g}^{5} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}-{\frac{Bc{d}^{4}}{2\,{g}^{5} \left ( ad-bc \right ) ^{5}}\ln \left ({\frac{ad}{bx+a}}-{\frac{bc}{bx+a}}-d \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.3591, size = 944, normalized size = 4.54 \begin{align*} -\frac{1}{24} \, B{\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{6 \, \log \left (\frac{d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac{c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{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}} + \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{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 = 1.09048, size = 1328, normalized size = 6.38 \begin{align*} -\frac{3 \,{\left (2 \, A - B\right )} b^{4} c^{4} - 8 \,{\left (3 \, A - 2 \, B\right )} a b^{3} c^{3} d + 36 \,{\left (A - B\right )} a^{2} b^{2} c^{2} d^{2} - 24 \,{\left (A - 2 \, B\right )} a^{3} b c d^{3} +{\left (6 \, A - 25 \, B\right )} a^{4} d^{4} + 12 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} 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 )} 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 )} x - 6 \,{\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - 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 )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{24 \,{\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 [B] time = 8.01469, size = 947, normalized size = 4.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39681, size = 562, normalized size = 2.7 \begin{align*} \frac{B d^{4} \log \left (-\frac{b c g}{b g x + a g} + \frac{a d g}{b g x + a g} - d\right )}{2 \,{\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^{3}}{2 \,{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )}{\left (b g x + a g\right )} b g} + \frac{B d^{2}}{4 \,{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )}{\left (b g x + a g\right )}^{2} b g^{2}} - \frac{B \log \left (\frac{\frac{b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac{2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{2 \, b c d g}{b g x + a g} - \frac{2 \, a d^{2} g}{b g x + a g} + d^{2}}{b^{2}}\right )}{4 \,{\left (b g x + a g\right )}^{4} b g} - \frac{B d}{6 \,{\left (b g x + a g\right )}^{3}{\left (b c - a d\right )} b g^{2}} - \frac{2 \, A b^{3} g^{3} + B b^{3} g^{3}}{8 \,{\left (b g x + a g\right )}^{4} b^{4} g^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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