Optimal. Leaf size=251 \[ -\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \]
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Rubi [A] time = 0.52, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {2342, 266, 43, 2350, 12, 446, 80, 50, 63, 208} \[ -\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 50
Rule 63
Rule 80
Rule 208
Rule 266
Rule 446
Rule 2342
Rule 2350
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {d^2 \left (-2 d^2-e^2 x^2\right ) \sqrt {1-\frac {e^2 x^2}{d^2}}}{3 e^4 x} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {\left (-2 d^2-e^2 x^2\right ) \sqrt {1-\frac {e^2 x^2}{d^2}}}{x} \, dx}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b d^2 n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \operatorname {Subst}\left (\int \frac {\left (-2 d^2-e^2 x\right ) \sqrt {1-\frac {e^2 x}{d^2}}}{x} \, dx,x,x^2\right )}{6 e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^4 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} \, dx,x,x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b d^4 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 )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (2 b d^6 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 )}{3 e^6 \sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {2 b d^2 n \left (d^2-e^2 x^2\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \left (d^2-e^2 x^2\right )^2}{9 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {2 b d^4 n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {d^2 \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (d^2-e^2 x^2\right )^2 \left (a+b \log \left (c x^n\right )\right )}{3 e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 163, normalized size = 0.65 \[ -\frac {\sqrt {d-e x} \sqrt {d+e x} \left (d^2 \left (6 a+6 b \log \left (c x^n\right )-6 b n \log (x)-5 b n\right )+e^2 x^2 \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)-b n\right )\right )+6 b d^3 n \log \left (\sqrt {d-e x} \sqrt {d+e x}+d\right )-6 b d^3 n \log (x)+3 b n \log (x) \sqrt {d-e x} \sqrt {d+e x} \left (2 d^2+e^2 x^2\right )}{9 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 125, normalized size = 0.50 \[ \frac {6 \, b d^{3} n \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) + {\left (5 \, b d^{2} n - 6 \, a d^{2} + {\left (b e^{2} n - 3 \, a e^{2}\right )} x^{2} - 3 \, {\left (b e^{2} x^{2} + 2 \, b d^{2}\right )} \log \relax (c) - 3 \, {\left (b e^{2} n x^{2} + 2 \, b d^{2} n\right )} \log \relax (x)\right )} \sqrt {e x + d} \sqrt {-e x + d}}{9 \, e^{4}} \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{\sqrt {e x + d} \sqrt {-e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{3}}{\sqrt {-e x +d}\, \sqrt {e x +d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.54, size = 199, normalized size = 0.79 \[ -\frac {1}{9} \, b n {\left (\frac {3 \, d^{3} \log \left (d + \sqrt {-e^{2} x^{2} + d^{2}}\right )}{e^{4}} - \frac {3 \, d^{3} \log \left (-d + \sqrt {-e^{2} x^{2} + d^{2}}\right )}{e^{4}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} - {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{e^{4}}\right )} - \frac {1}{3} \, b {\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} x^{2}}{e^{2}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{e^{4}}\right )} \log \left (c x^{n}\right ) - \frac {1}{3} \, a {\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} x^{2}}{e^{2}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{e^{4}}\right )} \]
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 \[ \int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )}{\sqrt {d - e x} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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