Optimal. Leaf size=61 \[ \frac {x \sqrt {a+b \log \left (c x^n\right )}}{d (d+e x)}-\frac {b n \text {Int}\left (\frac {1}{(d+e x) \sqrt {a+b \log \left (c x^n\right )}},x\right )}{2 d} \]
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Rubi [A] time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx &=\frac {x \sqrt {a+b \log \left (c x^n\right )}}{d (d+e x)}-\frac {(b n) \int \frac {1}{(d+e x) \sqrt {a+b \log \left (c x^n\right )}} \, dx}{2 d}\\ \end {align*}
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Mathematica [A] time = 6.80, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \ln \left (c \,x^{n}\right )+a}}{\left (e x +d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+b\,\ln \left (c\,x^n\right )}}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b \log {\left (c x^{n} \right )}}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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