Optimal. Leaf size=78 \[ \frac{2 \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{g}-\frac{b e n \text{Unintegrable}\left (\frac{\sqrt{f+g x}}{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right )}{g} \]
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Rubi [A] time = 0.243462, 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 = {} \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{f+g x}} \, dx \]
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 )}}{\sqrt{f+g x}} \, dx &=\frac{2 \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{g}-\frac{(b e n) \int \frac{\sqrt{f+g x}}{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{g}\\ \end{align*}
Mathematica [A] time = 1.52692, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{f+g x}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.799, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }{\frac{1}{\sqrt{gx+f}}}}\, 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}}{\sqrt{g x + f}}\,{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 )}}}{\sqrt{f + g x}}\, 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}}{\sqrt{g x + f}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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