Optimal. Leaf size=78 \[ \frac{3 b e n \text{Unintegrable}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(d+e x) (f+g x)^2},x\right )}{4 g}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 g (f+g x)^2} \]
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Rubi [A] time = 0.207254, 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 = {} \[ \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx &=-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 g (f+g x)^2}+\frac{(3 b e n) \int \frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{(d+e x) (f+g x)^2} \, dx}{4 g}\\ \end{align*}
Mathematica [A] time = 0.831871, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.915, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{3}} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{{\frac{3}{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*} \int \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}{{\left (g x + f\right )}^{3}}\,{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 \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{3}{2}}}{{\left (g x + f\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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