Optimal. Leaf size=489 \[ -\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {Li}_2\left (-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}}+1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}} \]
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Rubi [A] time = 0.72, 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 = {2342, 266, 51, 63, 208, 2350, 47, 5984, 5918, 2402, 2315} \[ -\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {PolyLog}\left (2,-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}}+1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {e^2 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )}{4 d^2 x^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e^2 n \sqrt {1-\frac {e^2 x^2}{d^2}} \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{2 d^2 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 266
Rule 2315
Rule 2342
Rule 2350
Rule 2402
Rule 5918
Rule 5984
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.88, size = 255, normalized size = 0.52 \[ \frac {\frac {b n \left (e^2 x^2-d^2\right ) \left (2 d^3 \, _3F_2\left (\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {5}{2},\frac {5}{2};\frac {d^2}{e^2 x^2}\right )+9 e^2 x^2 (2 \log (x)+1) \left (d \sqrt {1-\frac {d^2}{e^2 x^2}}-e x \sin ^{-1}\left (\frac {d}{e x}\right )\right )\right )}{e^2 x^4 \sqrt {1-\frac {d^2}{e^2 x^2}} \sqrt {d-e x} \sqrt {d+e x}}-18 e^2 \log \left (\sqrt {d-e x} \sqrt {d+e x}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\frac {18 d \sqrt {d-e x} \sqrt {d+e x} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x^2}+18 e^2 \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{36 d^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {-e x + d} b \log \left (c x^{n}\right ) + \sqrt {e x + d} \sqrt {-e x + d} a}{e^{2} x^{5} - d^{2} x^{3}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x + d} \sqrt {-e x + d} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\sqrt {-e x +d}\, \sqrt {e x +d}\, x^{3}}\, dx \]
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 \[ -\frac {1}{2} \, a {\left (\frac {e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{3}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{d^{2} x^{2}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{\sqrt {e x + d} \sqrt {-e x + d} x^{3}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \]
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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