Optimal. Leaf size=94 \[ -\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{x}-e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+4 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {852, 1805, 1807, 844, 217, 203, 266, 63, 208} \[ -\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{x}-e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+4 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 844
Rule 852
Rule 1805
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-d^4+4 d^3 e x+d^2 e^2 x^2}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{d^2}\\ &=-\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{x}+\frac {\int \frac {-4 d^5 e-d^4 e^2 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4}\\ &=-\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{x}-(4 d e) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-e^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{x}-(2 d e) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e^2 \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{x}-e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {(4 d) \operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e}\\ &=-\frac {8 e (d-e x)}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{x}-e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+4 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.19, size = 84, normalized size = 0.89 \[ \sqrt {d^2-e^2 x^2} \left (-\frac {8 e}{d+e x}-\frac {1}{x}\right )+4 e \log \left (\sqrt {d^2-e^2 x^2}+d\right )-e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-4 e \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 127, normalized size = 1.35 \[ -\frac {8 \, e^{2} x^{2} + 8 \, d e x - 2 \, {\left (e^{2} x^{2} + d e x\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 4 \, {\left (e^{2} x^{2} + d e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (9 \, e x + d\right )}}{e x^{2} + d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 515, normalized size = 5.48 \[ \frac {4 d e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}+\frac {7 e^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}}-\frac {15 e^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}-\frac {15 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x}{8 d^{2}}+\frac {7 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{2} x}{8 d^{2}}-\frac {4 \sqrt {-e^{2} x^{2}+d^{2}}\, e}{d}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2} x}{4 d^{4}}+\frac {7 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{2} x}{12 d^{4}}-\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{3 d^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2} x}{d^{6}}-\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}{5 d^{5}}+\frac {7 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e}{15 d^{5}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{4} d^{3} e^{3}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{3} d^{4} e^{2}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{3 \left (x +\frac {d}{e}\right )^{2} d^{5} e}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{6} x} \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^2\,{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{x^{2} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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