Optimal. Leaf size=87 \[ \frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x) (e f-d g)}-\frac{b e n \text{Unintegrable}\left (\frac{1}{(f+g x) \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right )}{2 (e f-d g)} \]
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Rubi [A] time = 0.102329, 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 = {} \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx &=\frac{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{(e f-d g) (f+g x)}-\frac{(b e n) \int \frac{1}{(f+g x) \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{2 (e f-d g)}\\ \end{align*}
Mathematica [A] time = 0.318953, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.889, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{2}}\sqrt{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \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{\sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\sqrt{a + b \log{\left (c \left (d + e x\right )^{n} \right )}}}{\left (f + g x\right )^{2}}\, 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{\sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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