Optimal. Leaf size=142 \[ -\frac {b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e n \sqrt {1-\frac {e^2 x^2}{d^2}} \sin ^{-1}\left (\frac {e x}{d}\right )}{d \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}} \]
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Rubi [A]
time = 0.26, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2387, 2373,
283, 222} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {ArcSin}\left (\frac {e x}{d}\right )}{d \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 283
Rule 2373
Rule 2387
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \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^2 \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 )}{d^2 x \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^2} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )}{d^2 x \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 )}{d^2 x \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 {1}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {b e n \sqrt {1-\frac {e^2 x^2}{d^2}} \sin ^{-1}\left (\frac {e x}{d}\right )}{d \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 )}{d^2 x \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 70, normalized size = 0.49 \begin {gather*} -\frac {b e n x \tan ^{-1}\left (\frac {e x}{\sqrt {d-e x} \sqrt {d+e x}}\right )+\sqrt {d-e x} \sqrt {d+e x} \left (a+b n+b \log \left (c x^n\right )\right )}{d^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{2} \sqrt {-e x +d}\, \sqrt {e x +d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 87, normalized size = 0.61 \begin {gather*} -\frac {{\left (\arcsin \left (\frac {x e}{d}\right ) e + \frac {\sqrt {-x^{2} e^{2} + d^{2}}}{x}\right )} b n}{d^{2}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} b \log \left (c x^{n}\right )}{d^{2} x} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} a}{d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 77, normalized size = 0.54 \begin {gather*} \frac {2 \, b n x \arctan \left (\frac {{\left (\sqrt {x e + d} \sqrt {-x e + d} - d\right )} e^{\left (-1\right )}}{x}\right ) e - {\left (b n \log \left (x\right ) + b n + b \log \left (c\right ) + a\right )} \sqrt {x e + d} \sqrt {-x e + d}}{d^{2} x} \end {gather*}
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 \begin {gather*} \int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \sqrt {d - e x} \sqrt {d + e x}}\, dx \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 = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\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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