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 = 0.07, 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 = {864, 827, 829,
858, 223, 209, 272, 65, 214} \begin {gather*} -\frac {3}{2} d^3 e \text {ArcTan}\left (\frac {e 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}+d^3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 827
Rule 829
Rule 858
Rule 864
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 ) \text {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 ) \text {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 \text {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 = 0.35, size = 143, normalized size = 1.24 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3-8 d^2 e x-3 d e^2 x^2+2 e^3 x^3\right )}{6 x}-2 d^3 e \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {3}{2} d^3 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(429\) vs.
\(2(101)=202\).
time = 0.08, size = 430, normalized size = 3.74
method | result | size |
risch | \(-\frac {d^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{x}+\frac {e^{3} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}-\frac {4 e \,d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}-\frac {e^{2} d x \sqrt {-e^{2} x^{2}+d^{2}}}{2}-\frac {3 e^{2} d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}+\frac {e \,d^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\) | \(164\) |
default | \(\frac {e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{d^{2}}+\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}}{d}-\frac {e \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{d^{2}}\) | \(430\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 130, normalized size = 1.13 \begin {gather*} -\frac {3}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e + d^{3} e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} d x e^{2} - \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e - \frac {\sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{x} - \frac {1}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.25, size = 121, normalized size = 1.05 \begin {gather*} \frac {18 \, d^{3} x \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e - 6 \, d^{3} x e \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - 8 \, d^{3} x e + {\left (2 \, x^{3} e^{3} - 3 \, d x^{2} e^{2} - 8 \, d^{2} x e - 6 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.18, size = 386, normalized size = 3.36 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.16, size = 156, normalized size = 1.36 \begin {gather*} -\frac {3}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e \mathrm {sgn}\left (d\right ) + d^{3} e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {d^{3} x e^{3}}{2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-1\right )}}{2 \, x} - \frac {1}{6} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (8 \, d^{2} e - {\left (2 \, x e^{3} - 3 \, d e^{2}\right )} x\right )} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^2\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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