Optimal. Leaf size=65 \[ \frac{b n \text{Unintegrable}\left (\frac{1}{x (d+e x)^2 \sqrt{a+b \log \left (c x^n\right )}},x\right )}{4 e}-\frac{\sqrt{a+b \log \left (c x^n\right )}}{2 e (d+e x)^2} \]
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Rubi [A] time = 0.1978, 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 = {} \[ \int \frac{\sqrt{a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx &=-\frac{\sqrt{a+b \log \left (c x^n\right )}}{2 e (d+e x)^2}+\frac{(b n) \int \frac{1}{x (d+e x)^2 \sqrt{a+b \log \left (c x^n\right )}} \, dx}{4 e}\\ \end{align*}
Mathematica [A] time = 12.5711, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.494, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{3}}\sqrt{a+b\ln \left ( c{x}^{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 (c x^{n}\right ) + a}}{{\left (e x + d\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \log{\left (c x^{n} \right )}}}{\left (d + e x\right )^{3}}\, 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 (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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