Optimal. Leaf size=78 \[ \frac{g \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i}-\frac{i \text{Unintegrable}\left (\frac{1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i} \]
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Rubi [A] time = 0.182707, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(h+240 x) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\int \left (\frac{240}{(240 f-g h) (h+240 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{g}{(240 f-g h) (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx\\ &=\frac{240 \int \frac{1}{(h+240 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{240 f-g h}-\frac{g \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{240 f-g h}\\ \end{align*}
Mathematica [A] time = 12.6754, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) (h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 15.503, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ix+h \right ) \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e x + d}{b^{2} e f h n \log \left (c\right ) + a b e f h n +{\left (b^{2} e g i n \log \left (c\right ) + a b e g i n\right )} x^{2} +{\left ({\left (g h n + f i n\right )} b^{2} e \log \left (c\right ) +{\left (g h n + f i n\right )} a b e\right )} x +{\left (b^{2} e g i n x^{2} + b^{2} e f h n +{\left (g h n + f i n\right )} b^{2} e x\right )} \log \left ({\left (e x + d\right )}^{n}\right )} - \int \frac{e g i x^{2} + 2 \, d g i x - e f h +{\left (g h + f i\right )} d}{b^{2} e f^{2} h^{2} n \log \left (c\right ) + a b e f^{2} h^{2} n +{\left (b^{2} e g^{2} i^{2} n \log \left (c\right ) + a b e g^{2} i^{2} n\right )} x^{4} + 2 \,{\left ({\left (g^{2} h i n + f g i^{2} n\right )} b^{2} e \log \left (c\right ) +{\left (g^{2} h i n + f g i^{2} n\right )} a b e\right )} x^{3} +{\left ({\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} b^{2} e \log \left (c\right ) +{\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} a b e\right )} x^{2} + 2 \,{\left ({\left (f g h^{2} n + f^{2} h i n\right )} b^{2} e \log \left (c\right ) +{\left (f g h^{2} n + f^{2} h i n\right )} a b e\right )} x +{\left (b^{2} e g^{2} i^{2} n x^{4} + b^{2} e f^{2} h^{2} n + 2 \,{\left (g^{2} h i n + f g i^{2} n\right )} b^{2} e x^{3} +{\left (g^{2} h^{2} n + 4 \, f g h i n + f^{2} i^{2} n\right )} b^{2} e x^{2} + 2 \,{\left (f g h^{2} n + f^{2} h i n\right )} b^{2} e x\right )} \log \left ({\left (e x + d\right )}^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} g i x^{2} + a^{2} f h +{\left (b^{2} g i x^{2} + b^{2} f h +{\left (b^{2} g h + b^{2} f i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} +{\left (a^{2} g h + a^{2} f i\right )} x + 2 \,{\left (a b g i x^{2} + a b f h +{\left (a b g h + a b f i\right )} x\right )} \log \left ({\left (e x + d\right )}^{n} c\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}{\left (i x + h\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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