Optimal. Leaf size=82 \[ \frac{2 (f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 g}-\frac{b e n \text{Unintegrable}\left (\frac{(f+g x)^{3/2}}{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}},x\right )}{3 g} \]
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Rubi [A] time = 0.264688, 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 = {} \[ \int \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx &=\frac{2 (f+g x)^{3/2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{3 g}-\frac{(b e n) \int \frac{(f+g x)^{3/2}}{(d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}} \, dx}{3 g}\\ \end{align*}
Mathematica [A] time = 1.36404, size = 0, normalized size = 0. \[ \int \sqrt{f+g x} \sqrt{a+b \log \left (c (d+e x)^n\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.881, size = 0, normalized size = 0. \begin{align*} \int \sqrt{gx+f}\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 \sqrt{g x + f} \sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}\,{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 [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 \sqrt{g x + f} \sqrt{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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