Optimal. Leaf size=96 \[ d (d-e x) \sqrt {d^2-e^2 x^2}-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^3 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-d^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 96, 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, 1809, 815, 844, 217, 203, 266, 63, 208} \[ d (d-e x) \sqrt {d^2-e^2 x^2}-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^3 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-d^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 815
Rule 844
Rule 852
Rule 1809
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x} \, dx\\ &=-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int \frac {\left (-3 d^2 e^2+6 d e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{3 e^2}\\ &=d (d-e x) \sqrt {d^2-e^2 x^2}-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int \frac {6 d^4 e^4-6 d^3 e^5 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{6 e^4}\\ &=d (d-e x) \sqrt {d^2-e^2 x^2}-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}+d^4 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\left (d^3 e\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=d (d-e x) \sqrt {d^2-e^2 x^2}-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{2} d^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\left (d^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=d (d-e x) \sqrt {d^2-e^2 x^2}-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}-d^3 \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^2}\\ &=d (d-e x) \sqrt {d^2-e^2 x^2}-\frac {1}{3} \left (d^2-e^2 x^2\right )^{3/2}-d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 96, normalized size = 1.00 \[ d^3 \log (x)+\sqrt {d^2-e^2 x^2} \left (\frac {2 d^2}{3}-d e x+\frac {e^2 x^2}{3}\right )-d^3 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+d^3 \left (-\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 95, normalized size = 0.99 \[ 2 \, d^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \frac {1}{3} \, {\left (e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 290, normalized size = 3.02 \[ -\frac {d^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}-\frac {d^{3} e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}}-\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d e x +\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}-\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e x}{3 d}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 d^{2}}-\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}}}{15 d^{2}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{3 \left (x +\frac {d}{e}\right )^{2} d^{2} e^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 103, normalized size = 1.07 \[ -d^{3} \arcsin \left (\frac {e x}{d}\right ) - d^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \sqrt {-e^{2} x^{2} + d^{2}} d e x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} - \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x\,{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 14.83, size = 267, normalized size = 2.78 \[ d^{2} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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