Optimal. Leaf size=115 \[ -\frac {1}{2} d e (2 d+3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 115, 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 = {850, 813, 815, 844, 217, 203, 266, 63, 208} \[ -\frac {1}{2} d e (2 d+3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 813
Rule 815
Rule 844
Rule 850
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)} \, dx &=\int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {1}{2} \int \frac {\left (2 d^2 e+6 d e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx\\ &=-\frac {1}{2} d e (2 d+3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}+\frac {\int \frac {-4 d^4 e^3-6 d^3 e^4 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{4 e^2}\\ &=-\frac {1}{2} d e (2 d+3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\left (d^4 e\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (3 d^3 e^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {1}{2} d e (2 d+3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {1}{2} \left (d^4 e\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{2} \left (3 d^3 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {1}{2} d e (2 d+3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {d^4 \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 {1}{2} d e (2 d+3 e x) \sqrt {d^2-e^2 x^2}-\frac {(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 114, normalized size = 0.99 \[ -d^3 e \log (x)+d^3 e \log \left (\sqrt {d^2-e^2 x^2}+d\right )-\frac {3}{2} d^3 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\sqrt {d^2-e^2 x^2} \left (-\frac {d^3}{x}-\frac {4 d^2 e}{3}-\frac {1}{2} d e^2 x+\frac {e^3 x^2}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 123, normalized size = 1.07 \[ \frac {18 \, d^{3} e x \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 6 \, d^{3} e x \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 8 \, d^{3} e x + {\left (2 \, e^{3} x^{3} - 3 \, d e^{2} x^{2} - 8 \, d^{2} e x - 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, x} \]
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 = 380, normalized size = 3.30 \[ \frac {d^{4} e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}+\frac {3 d^{3} e^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}}-\frac {15 d^{3} e^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}-\frac {15 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{2} x}{8}+\frac {3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d \,e^{2} x}{8}-\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e -\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2} x}{4 d}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{2} x}{4 d}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{3}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2} x}{d^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}{5 d^{2}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e}{5 d^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 131, normalized size = 1.14 \[ -\frac {3}{2} \, d^{3} e \arcsin \left (\frac {e x}{d}\right ) + d^{3} e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2} x - \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e - \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^2\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.20, size = 386, normalized size = 3.36 \[ d^{3} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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