Optimal. Leaf size=252 \[ -\frac{2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 b e^2 n \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 d^4 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 b e^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{3 d^3 \sqrt{d-e x} \sqrt{d+e x}} \]
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Rubi [A] time = 0.475077, antiderivative size = 252, normalized size of antiderivative = 1., 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 = {2342, 271, 264, 2350, 12, 451, 277, 216} \[ -\frac{2 e^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^4 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^2 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 b e^2 n \left (d^2-e^2 x^2\right )}{3 d^4 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{b n \left (d^2-e^2 x^2\right )^2}{9 d^4 x^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{2 b e^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{3 d^3 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 2342
Rule 271
Rule 264
Rule 2350
Rule 12
Rule 451
Rule 277
Rule 216
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*}
Mathematica [A] time = 0.295317, size = 116, normalized size = 0.46 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (3 a \left (d^2+2 e^2 x^2\right )+3 b \left (d^2+2 e^2 x^2\right ) \log \left (c x^n\right )+b n \left (d^2+5 e^2 x^2\right )\right )+6 b e^3 n x^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d-e x} \sqrt{d+e x}}\right )}{9 d^4 x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.67, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{4}}{\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51681, size = 311, normalized size = 1.23 \begin{align*} \frac{12 \, b e^{3} n x^{3} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right ) -{\left (b d^{2} n + 3 \, a d^{2} +{\left (5 \, b e^{2} n + 6 \, a e^{2}\right )} x^{2} + 3 \,{\left (2 \, b e^{2} x^{2} + b d^{2}\right )} \log \left (c\right ) + 3 \,{\left (2 \, b e^{2} n x^{2} + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt{e x + d} \sqrt{-e x + d}}{9 \, d^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{\sqrt{e x + d} \sqrt{-e x + d} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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