Optimal. Leaf size=226 \[ \frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {b e n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b e n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b n \sqrt {d+e x}}{d x} \]
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Rubi [A] time = 0.27, antiderivative size = 226, 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 = {51, 63, 208, 2350, 14, 47, 5984, 5918, 2402, 2315} \[ \frac {b e n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{d^{3/2}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {2 b e n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b n \sqrt {d+e x}}{d x} \]
Antiderivative was successfully verified.
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Rule 14
Rule 47
Rule 51
Rule 63
Rule 208
Rule 2315
Rule 2350
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \sqrt {d+e x}} \, dx &=-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-(b n) \int \frac {-\frac {\sqrt {d+e x}}{d}+\frac {e x \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}}{x^2} \, dx\\ &=-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-(b n) \int \left (-\frac {\sqrt {d+e x}}{d x^2}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2} x}\right ) \, dx\\ &=-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \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 {\sqrt {d+e x}}{x^2} \, dx}{d}-\frac {(b e n) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx}{d^{3/2}}\\ &=-\frac {b n \sqrt {d+e x}}{d x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \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 e 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 {(b e n) \int \frac {1}{x \sqrt {d+e x}} \, dx}{2 d}\\ &=-\frac {b n \sqrt {d+e x}}{d x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \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) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{d}+\frac {(2 b e 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 {b n \sqrt {d+e x}}{d x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \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 e 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 e 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 {b n \sqrt {d+e x}}{d x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \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 e 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 e 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 {b n \sqrt {d+e x}}{d x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{d^{3/2}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{d x}+\frac {e \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 e 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 {b e 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 = 0.26, size = 392, normalized size = 1.73 \[ -\frac {4 a \sqrt {d} \sqrt {d+e x}+2 a e x \log \left (\sqrt {d}-\sqrt {d+e x}\right )-2 a e x \log \left (\sqrt {d+e x}+\sqrt {d}\right )+2 b e x \log \left (c x^n\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )+4 b \sqrt {d} \sqrt {d+e x} \log \left (c x^n\right )-2 b e x \log \left (c x^n\right ) \log \left (\sqrt {d+e x}+\sqrt {d}\right )-2 b e n x \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+2 b e n x \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right )+4 b \sqrt {d} n \sqrt {d+e x}-b e n x \log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )+b e n x \log ^2\left (\sqrt {d+e x}+\sqrt {d}\right )-2 b e n x \log \left (\frac {1}{2} \left (\frac {\sqrt {d+e x}}{\sqrt {d}}+1\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 b e n x \log \left (\sqrt {d+e x}+\sqrt {d}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+4 b e n x \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 d^{3/2} x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} b \log \left (c x^{n}\right ) + \sqrt {e x + d} a}{e x^{3} + d x^{2}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x + d} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\sqrt {e x +d}\, x^{2}}\, dx \]
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 \[ -\frac {1}{2} \, a {\left (\frac {2 \, \sqrt {e x + d} e}{{\left (e x + d\right )} d - d^{2}} + \frac {e \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{d^{\frac {3}{2}}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{\sqrt {e x + d} x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\sqrt {d+e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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