Optimal. Leaf size=252 \[ -\frac {2 b e^2 n \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b e^3 n \sqrt {1-\frac {e^2 x^2}{d^2}} \sin ^{-1}\left (\frac {e x}{d}\right )}{3 d^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}} \]
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Rubi [A]
time = 0.31, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2387, 277, 270,
2392, 12, 462, 283, 222} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b e^3 n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {ArcSin}\left (\frac {e x}{d}\right )}{3 d^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b e^2 n \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 270
Rule 277
Rule 283
Rule 462
Rule 2387
Rule 2392
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^4 \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^4 \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 )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 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 {\left (-d^2-2 e^2 x^2\right ) \sqrt {1-\frac {e^2 x^2}{d^2}}}{3 d^2 x^4} \, 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 )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 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 {\left (-d^2-2 e^2 x^2\right ) \sqrt {1-\frac {e^2 x^2}{d^2}}}{x^4} \, dx}{3 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 d^4 x^3 \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 )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (2 b e^2 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}{3 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {2 b e^2 n \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 d^4 x^3 \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 )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (2 b e^4 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {1}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{3 d^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {2 b e^2 n \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 d^4 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b e^3 n \sqrt {1-\frac {e^2 x^2}{d^2}} \sin ^{-1}\left (\frac {e x}{d}\right )}{3 d^3 \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 )}{3 d^2 x^3 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 x \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 116, normalized size = 0.46 \begin {gather*} -\frac {6 b e^3 n x^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d-e x} \sqrt {d+e x}}\right )+\sqrt {d-e x} \sqrt {d+e x} \left (3 a \left (d^2+2 e^2 x^2\right )+b n \left (d^2+5 e^2 x^2\right )+3 b \left (d^2+2 e^2 x^2\right ) \log \left (c x^n\right )\right )}{9 d^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x^{4} \sqrt {-e x +d}\, \sqrt {e x +d}}\, dx\]
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 [A]
time = 0.38, size = 131, normalized size = 0.52 \begin {gather*} \frac {12 \, b n x^{3} \arctan \left (\frac {{\left (\sqrt {x e + d} \sqrt {-x e + d} - d\right )} e^{\left (-1\right )}}{x}\right ) e^{3} - {\left (b d^{2} n + {\left (5 \, b n + 6 \, a\right )} x^{2} e^{2} + 3 \, a d^{2} + 3 \, {\left (2 \, b x^{2} e^{2} + b d^{2}\right )} \log \left (c\right ) + 3 \, {\left (2 \, b n x^{2} e^{2} + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {x e + d} \sqrt {-x e + d}}{9 \, d^{4} x^{3}} \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^{4} \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 = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,\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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