Optimal. Leaf size=88 \[ \frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x) (e f-d g)}-\frac {b e n \text {Int}\left (\frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{2 (e f-d g)} \]
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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 = {} \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx &=\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{(e f-d g) (f+g x)}-\frac {(b e n) \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{2 (e f-d g)}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]
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 \[ \int \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \ln \left (c \left (e x +d \right )^{n}\right )+a}}{\left (g x +f \right )^{2}}\, dx \]
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 \[ \int \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}}{{\left (f+g\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}{\left (f + g x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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