Optimal. Leaf size=78 \[ \frac {b e n \text {Int}\left (\frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{g}-\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{g \sqrt {f+g x}} \]
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Rubi [A] time = 0.26, 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 = {} \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \]
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)^{3/2}} \, dx &=-\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{g \sqrt {f+g x}}+\frac {(b e n) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{g}\\ \end {align*}
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Mathematica [A] time = 1.32, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \]
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 \[ \int \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \ln \left (c \left (e x +d \right )^{n}\right )+a}}{\left (g x +f \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}}{{\left (f+g\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}{\left (f + g x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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