Optimal. Leaf size=201 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 b n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {4 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}} \]
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Rubi [A] time = 0.32, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2347, 63, 208, 2348, 12, 5984, 5918, 2402, 2315, 2319} \[ -\frac {2 b n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {4 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 2315
Rule 2319
Rule 2347
Rule 2348
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^{3/2}} \, dx &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^{3/2}} \, dx}{d}\\ &=\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {(b n) \int -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d} x} \, dx}{d}-\frac {(2 b n) \int \frac {1}{x \sqrt {d+e x}} \, dx}{d}\\ &=\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac {(2 b n) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx}{d^{3/2}}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{d e}\\ &=\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}+\frac {(4 b n) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x}\right )}{d^{3/2}}\\ &=\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x}\right )}{d^2}\\ &=\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}+\frac {(4 b n) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x}\right )}{d^2}\\ &=\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )}{d^{3/2}}\\ &=\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}+\frac {2 \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}-\frac {2 b n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )}{d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 295, normalized size = 1.47 \[ \frac {\frac {4 \sqrt {d} \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x}}+2 \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (a+b \log \left (c x^n\right )\right )-2 \log \left (\sqrt {d+e x}+\sqrt {d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \left (2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right )\right )\right )+b n \left (2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right )+\log \left (\sqrt {d+e x}+\sqrt {d}\right ) \left (\log \left (\sqrt {d+e x}+\sqrt {d}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )\right )+8 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{2 d^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} b \log \left (c x^{n}\right ) + \sqrt {e x + d} a}{e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e x +d \right )^{\frac {3}{2}} x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {\log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2}{\sqrt {e x + d} d}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{{\left (e x^{2} + d x\right )} \sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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