Optimal. Leaf size=112 \[ -\frac{a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac{b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac{b e^2 n \log (f+g x)}{2 g (e f-d g)^2}+\frac{b e n}{2 g (f+g x) (e f-d g)} \]
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Rubi [A] time = 0.0622702, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2395, 44} \[ -\frac{a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac{b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac{b e^2 n \log (f+g x)}{2 g (e f-d g)^2}+\frac{b e n}{2 g (f+g x) (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 2395
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac{(b e n) \int \frac{1}{(d+e x) (f+g x)^2} \, dx}{2 g}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac{(b e n) \int \left (\frac{e^2}{(e f-d g)^2 (d+e x)}-\frac{g}{(e f-d g) (f+g x)^2}-\frac{e g}{(e f-d g)^2 (f+g x)}\right ) \, dx}{2 g}\\ &=\frac{b e n}{2 g (e f-d g) (f+g x)}+\frac{b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}-\frac{b e^2 n \log (f+g x)}{2 g (e f-d g)^2}\\ \end{align*}
Mathematica [A] time = 0.103032, size = 83, normalized size = 0.74 \[ -\frac{a+b \log \left (c (d+e x)^n\right )-\frac{b e n (f+g x) (e (f+g x) \log (d+e x)-d g-e (f+g x) \log (f+g x)+e f)}{(e f-d g)^2}}{2 g (f+g x)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.374, size = 633, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1057, size = 225, normalized size = 2.01 \begin{align*} \frac{1}{2} \, b e n{\left (\frac{e \log \left (e x + d\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} - \frac{e \log \left (g x + f\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} + \frac{1}{e f^{2} g - d f g^{2} +{\left (e f g^{2} - d g^{3}\right )} x}\right )} - \frac{b \log \left ({\left (e x + d\right )}^{n} c\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 [B] time = 2.54747, size = 579, normalized size = 5.17 \begin{align*} -\frac{a e^{2} f^{2} - 2 \, a d e f g + a d^{2} g^{2} -{\left (b e^{2} f g - b d e g^{2}\right )} n x -{\left (b e^{2} f^{2} - b d e f g\right )} n -{\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x +{\left (2 \, b d e f g - b d^{2} g^{2}\right )} n\right )} \log \left (e x + d\right ) +{\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x + b e^{2} f^{2} n\right )} \log \left (g x + f\right ) +{\left (b e^{2} f^{2} - 2 \, b d e f g + b d^{2} g^{2}\right )} \log \left (c\right )}{2 \,{\left (e^{2} f^{4} g - 2 \, d e f^{3} g^{2} + d^{2} f^{2} g^{3} +{\left (e^{2} f^{2} g^{3} - 2 \, d e f g^{4} + d^{2} g^{5}\right )} x^{2} + 2 \,{\left (e^{2} f^{3} g^{2} - 2 \, d e f^{2} g^{3} + d^{2} f g^{4}\right )} x\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 = 1.22982, size = 408, normalized size = 3.64 \begin{align*} -\frac{b g^{2} n x^{2} e^{2} \log \left (g x + f\right ) - b g^{2} n x^{2} e^{2} \log \left (x e + d\right ) + b d g^{2} n x e + 2 \, b f g n x e^{2} \log \left (g x + f\right ) + b d^{2} g^{2} n \log \left (x e + d\right ) - 2 \, b f g n x e^{2} \log \left (x e + d\right ) - 2 \, b d f g n e \log \left (x e + d\right ) - b f g n x e^{2} + b d f g n e + b f^{2} n e^{2} \log \left (g x + f\right ) + b d^{2} g^{2} \log \left (c\right ) - 2 \, b d f g e \log \left (c\right ) + a d^{2} g^{2} - b f^{2} n e^{2} - 2 \, a d f g e + b f^{2} e^{2} \log \left (c\right ) + a f^{2} e^{2}}{2 \,{\left (d^{2} g^{5} x^{2} - 2 \, d f g^{4} x^{2} e + 2 \, d^{2} f g^{4} x + f^{2} g^{3} x^{2} e^{2} - 4 \, d f^{2} g^{3} x e + d^{2} f^{2} g^{3} + 2 \, f^{3} g^{2} x e^{2} - 2 \, d f^{3} g^{2} e + f^{4} g e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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