Optimal. Leaf size=489 \[ -\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {Li}_2\left (-\frac {1+\sqrt {1-\frac {e^2 x^2}{d^2}}}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}} \]
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Rubi [A]
time = 0.49, antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2387, 272,
44, 65, 214, 2392, 43, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {PolyLog}\left (2,-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}}+1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 272
Rule 2352
Rule 2387
Rule 2392
Rule 2449
Rule 6055
Rule 6131
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^3 \sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {a+b \log \left (c x^n\right )}{x^3 \sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \left (-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}}}{2 x^3}-\frac {e^2 \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 d^2 x}\right ) \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {\sqrt {1-\frac {e^2 x^2}{d^2}}}{x^3} \, dx}{2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{x} \, dx}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {e^2 x}{d^2}}}{x^2} \, dx,x,x^2\right )}{4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\sqrt {1-\frac {e^2 x}{d^2}}\right )}{x} \, dx,x,x^2\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {e^2 x}{d^2}}} \, dx,x,x^2\right )}{8 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}(x)}{-1+x^2} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {d^2 x^2}{e^2}} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}(x)}{1-x} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {Li}_2\left (1-\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.59, size = 255, normalized size = 0.52 \begin {gather*} \frac {\frac {b n \left (-d^2+e^2 x^2\right ) \left (2 d^3 \, _3F_2\left (\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};\frac {d^2}{e^2 x^2}\right )+9 e^2 x^2 \left (d \sqrt {1-\frac {d^2}{e^2 x^2}}-e x \sin ^{-1}\left (\frac {d}{e x}\right )\right ) (1+2 \log (x))\right )}{e^2 \sqrt {1-\frac {d^2}{e^2 x^2}} x^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {18 d \sqrt {d-e x} \sqrt {d+e x} \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{x^2}+18 e^2 \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-18 e^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+\sqrt {d-e x} \sqrt {d+e x}\right )}{36 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{3} \sqrt {-e x +d}\, \sqrt {e x +d}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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