Optimal. Leaf size=190 \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{2 g (f+g x)^2}+\frac{b^2 B n \log (a+b x)}{2 g (b f-a g)^2}-\frac{B n (b c-a d)}{2 (f+g x) (b f-a g) (d f-c g)}+\frac{B n (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f)}{2 (b f-a g)^2 (d f-c g)^2}-\frac{B d^2 n \log (c+d x)}{2 g (d f-c g)^2} \]
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Rubi [A] time = 0.235799, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 72} \[ -\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{2 g (f+g x)^2}+\frac{b^2 B n \log (a+b x)}{2 g (b f-a g)^2}-\frac{B n (b c-a d)}{2 (f+g x) (b f-a g) (d f-c g)}+\frac{B n (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f)}{2 (b f-a g)^2 (d f-c g)^2}-\frac{B d^2 n \log (c+d x)}{2 g (d f-c g)^2} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 72
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(f+g x)^3} \, dx &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 g (f+g x)^2}+\frac{(B n) \int \frac{b c-a d}{(a+b x) (c+d x) (f+g x)^2} \, dx}{2 g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 g (f+g x)^2}+\frac{(B (b c-a d) n) \int \frac{1}{(a+b x) (c+d x) (f+g x)^2} \, dx}{2 g}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 g (f+g x)^2}+\frac{(B (b c-a d) n) \int \left (\frac{b^3}{(b c-a d) (b f-a g)^2 (a+b x)}-\frac{d^3}{(b c-a d) (-d f+c g)^2 (c+d x)}+\frac{g^2}{(b f-a g) (d f-c g) (f+g x)^2}-\frac{g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}\right ) \, dx}{2 g}\\ &=-\frac{B (b c-a d) n}{2 (b f-a g) (d f-c g) (f+g x)}+\frac{b^2 B n \log (a+b x)}{2 g (b f-a g)^2}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 g (f+g x)^2}-\frac{B d^2 n \log (c+d x)}{2 g (d f-c g)^2}+\frac{B (b c-a d) (2 b d f-b c g-a d g) n \log (f+g x)}{2 (b f-a g)^2 (d f-c g)^2}\\ \end{align*}
Mathematica [A] time = 0.69519, size = 173, normalized size = 0.91 \[ \frac{B n (b c-a d) \left (\frac{b^2 \log (a+b x)}{(b c-a d) (b f-a g)^2}+\frac{\frac{d^2 \log (c+d x)}{a d-b c}+\frac{g (c g-d f)}{(f+g x) (b f-a g)}-\frac{g \log (f+g x) (a d g+b c g-2 b d f)}{(b f-a g)^2}}{(d f-c g)^2}\right )-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{(f+g x)^2}}{2 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.513, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{3}} \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 [A] time = 1.29185, size = 479, normalized size = 2.52 \begin{align*} \frac{1}{2} \,{\left (\frac{b^{2} \log \left (b x + a\right )}{b^{2} f^{2} g - 2 \, a b f g^{2} + a^{2} g^{3}} - \frac{d^{2} \log \left (d x + c\right )}{d^{2} f^{2} g - 2 \, c d f g^{2} + c^{2} g^{3}} + \frac{{\left (2 \,{\left (b^{2} c d - a b d^{2}\right )} f -{\left (b^{2} c^{2} - a^{2} d^{2}\right )} g\right )} \log \left (g x + f\right )}{b^{2} d^{2} f^{4} + a^{2} c^{2} g^{4} - 2 \,{\left (b^{2} c d + a b d^{2}\right )} f^{3} g +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{2} - 2 \,{\left (a b c^{2} + a^{2} c d\right )} f g^{3}} - \frac{b c - a d}{b d f^{3} + a c f g^{2} -{\left (b c + a d\right )} f^{2} g +{\left (b d f^{2} g + a c g^{3} -{\left (b c + a d\right )} f g^{2}\right )} x}\right )} B n - \frac{B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{2 \,{\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} - \frac{A}{2 \,{\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 = 1.88225, size = 1165, normalized size = 6.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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