Optimal. Leaf size=120 \[ \frac{g^2 \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{(g h-f i)^2}-\frac{g i \text{Unintegrable}\left (\frac{1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{(g h-f i)^2}-\frac{i \text{Unintegrable}\left (\frac{1}{(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g h-f i} \]
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Rubi [A] time = 0.233583, 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 = {} \[ \int \frac{1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(h+237 x)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx &=\int \left (\frac{237}{(237 f-g h) (h+237 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{237 g}{(237 f-g h)^2 (h+237 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{g^2}{(237 f-g h)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=-\frac{(237 g) \int \frac{1}{(h+237 x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{(237 f-g h)^2}+\frac{g^2 \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{(237 f-g h)^2}+\frac{237 \int \frac{1}{(h+237 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx}{237 f-g h}\\ \end{align*}
Mathematica [A] time = 3.06919, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.711, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) \left ( ix+h \right ) ^{2} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) }}\, 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*} \int \frac{1}{{\left (g x + f\right )}{\left (i x + h\right )}^{2}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\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 g i^{2} x^{3} + a f h^{2} +{\left (2 \, a g h i + a f i^{2}\right )} x^{2} +{\left (a g h^{2} + 2 \, a f h i\right )} x +{\left (b g i^{2} x^{3} + b f h^{2} +{\left (2 \, b g h i + b f i^{2}\right )} x^{2} +{\left (b g h^{2} + 2 \, b f h 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 )}^{2}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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