Optimal. Leaf size=113 \[ \frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 113, 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 = {815, 844, 217, 203, 266, 63, 208} \[ \frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 815
Rule 844
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx &=\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int \frac {\left (-4 d^3 e^2-3 d^2 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{4 e^2}\\ &=\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int \frac {8 d^5 e^4+3 d^4 e^5 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 e^4}\\ &=\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+d^5 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{8} \left (3 d^4 e\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{2} d^5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{8} \left (3 d^4 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 )\\ &=\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d^5 \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}\\ &=\frac {1}{8} d^2 (8 d+3 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{12} (4 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {3}{8} d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-d^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 124, normalized size = 1.10 \[ d^4 \left (-\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+\frac {3 d^3 \sqrt {d^2-e^2 x^2} \sin ^{-1}\left (\frac {e x}{d}\right )}{8 \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {1}{24} \sqrt {d^2-e^2 x^2} \left (32 d^3+15 d^2 e x-8 d e^2 x^2-6 e^3 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 107, normalized size = 0.95 \[ -\frac {3}{4} \, d^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + d^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - \frac {1}{24} \, {\left (6 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} - 15 \, d^{2} e x - 32 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 99, normalized size = 0.88 \[ \frac {3}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) \mathrm {sgn}\relax (d) - d^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {1}{24} \, {\left (32 \, d^{3} + {\left (15 \, d^{2} e - 2 \, {\left (3 \, x e^{3} + 4 \, d e^{2}\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 151, normalized size = 1.34 \[ -\frac {d^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}+\frac {3 d^{4} e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e x}{8}+\sqrt {-e^{2} x^{2}+d^{2}}\, d^{3}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e x}{4}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 124, normalized size = 1.10 \[ \frac {3}{8} \, d^{4} \arcsin \left (\frac {e x}{d}\right ) - d^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {3}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} + \frac {1}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e x + \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.90, size = 107, normalized size = 0.95 \[ \frac {d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{3}-d^4\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )+d^3\,\sqrt {d^2-e^2\,x^2}+\frac {e\,x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.41, size = 469, normalized size = 4.15 \[ d^{3} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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