Optimal. Leaf size=121 \[ -\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} d^2 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d^2 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {827, 858, 223,
209, 272, 65, 214} \begin {gather*} -\frac {3}{2} d^2 e^2 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac {3}{2} d^2 e^2 \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 858
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx &=-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{8} \int \frac {\left (-4 d^2 e+4 d e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx\\ &=-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac {3}{16} \int \frac {-8 d^3 e^2-8 d^2 e^3 x}{x \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {1}{2} \left (3 d^3 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{2} \left (3 d^2 e^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {1}{4} \left (3 d^3 e^2\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^2 e^3\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 {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} d^2 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} \left (3 d^3\right ) \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 )\\ &=-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} d^2 e^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d^2 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 141, normalized size = 1.17 \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2} \left (d^3+2 d^2 e x+2 d e^2 x^2+e^3 x^3\right )}{2 x^2}-3 d^2 e^2 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {3}{2} d^2 e \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. \(218\) vs.
\(2(105)=210\).
time = 0.08, size = 219, normalized size = 1.81
method | result | size |
risch | \(-\frac {d^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (2 e x +d \right )}{2 x^{2}}-\frac {e^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{2}-\frac {3 e^{3} d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-e^{2} d \sqrt {-e^{2} x^{2}+d^{2}}+\frac {3 e^{2} d^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}\) | \(150\) |
default | \(d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )+e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )\) | \(219\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 151, normalized size = 1.25 \begin {gather*} -\frac {3}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{2} + \frac {3}{2} \, d^{2} e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} x e^{3} - \frac {3}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} d e^{2} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{2 \, d} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e}{x} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}{2 \, d x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.30, size = 125, normalized size = 1.03 \begin {gather*} \frac {6 \, d^{2} x^{2} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e^{2} - 3 \, d^{2} x^{2} e^{2} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - 2 \, d^{2} x^{2} e^{2} - {\left (x^{3} e^{3} + 2 \, d x^{2} e^{2} + 2 \, d^{2} x e + d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{2 \, x^{2}} \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 = 4.26, size = 461, normalized size = 3.81 \begin {gather*} d^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) + d^{2} e \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 e^{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 ) - e^{3} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs.
\(2 (104) = 208\).
time = 1.60, size = 211, normalized size = 1.74 \begin {gather*} -\frac {3}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{2} \mathrm {sgn}\left (d\right ) + \frac {3}{2} \, d^{2} e^{2} \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 {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} e^{\left (-2\right )}}{8 \, x^{2}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2}}{2 \, x} + \frac {{\left (d^{2} e^{2} + \frac {4 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2}}{x}\right )} x^{2} e^{4}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}} - \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} {\left (x e^{3} + 2 \, d e^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.74, size = 120, normalized size = 0.99 \begin {gather*} \frac {3\,d^2\,e^2\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )}{2}-\frac {d^3\,\sqrt {d^2-e^2\,x^2}}{2\,x^2}-d\,e^2\,\sqrt {d^2-e^2\,x^2}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ \frac {e^2\,x^2}{d^2}\right )}{x\,{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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