Optimal. Leaf size=142 \[ \frac{(g h-f i) \text{Unintegrable}\left (\frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g}+\frac{i e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e g n^2}-\frac{i (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )} \]
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Rubi [A] time = 0.196317, 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 = {} \[ \int \frac{h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{h+238 x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\int \left (\frac{238}{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac{-238 f+g h}{g (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx\\ &=\frac{238 \int \frac{1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}+\frac{(-238 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}\\ &=\frac{238 \operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{e g}+\frac{(-238 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}\\ &=-\frac{238 (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{(-238 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}+\frac{238 \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e g n}\\ &=-\frac{238 (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{(-238 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}+\frac{\left (238 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e g n^2}\\ &=\frac{238 e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e g n^2}-\frac{238 (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{(-238 f+g h) \int \frac{1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}\\ \end{align*}
Mathematica [A] time = 1.19359, size = 0, normalized size = 0. \[ \int \frac{h+i x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 2.901, size = 0, normalized size = 0. \begin{align*} \int{\frac{ix+h}{ \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 i x^{2} + d h +{\left (e h + d i\right )} x}{b^{2} e f n \log \left (c\right ) + a b e f n +{\left (b^{2} e g n \log \left (c\right ) + a b e g n\right )} x +{\left (b^{2} e g n x + b^{2} e f n\right )} \log \left ({\left (e x + d\right )}^{n}\right )} + \int \frac{e g i x^{2} + 2 \, e f i x + e f h -{\left (g h - f i\right )} d}{b^{2} e f^{2} n \log \left (c\right ) + a b e f^{2} n +{\left (b^{2} e g^{2} n \log \left (c\right ) + a b e g^{2} n\right )} x^{2} + 2 \,{\left (b^{2} e f g n \log \left (c\right ) + a b e f g n\right )} x +{\left (b^{2} e g^{2} n x^{2} + 2 \, b^{2} e f g n x + b^{2} e f^{2} n\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{i x + h}{a^{2} g x + a^{2} f +{\left (b^{2} g x + b^{2} f\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \,{\left (a b g x + a b f\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{h + i x}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right )}\, dx \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{i x + h}{{\left (g x + f\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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