Optimal. Leaf size=216 \[ \frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1821, 827,
858, 223, 209, 272, 65, 214} \begin {gather*} \frac {13}{2} d^3 e^5 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}+\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {25}{8} d^3 e^5 \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
Rule 1821
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^6} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-15 d^4 e-13 d^3 e^2 x-5 d^2 e^3 x^2\right )}{x^5} \, dx}{5 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (52 d^5 e^2-25 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx}{20 d^4}\\ &=-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {\left (150 d^6 e^3+312 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx}{72 d^4}\\ &=\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {\int \frac {\left (-1248 d^7 e^4+600 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x^2} \, dx}{192 d^4}\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {-1200 d^8 e^5-2496 d^7 e^6 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{384 d^4}\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{8} \left (25 d^4 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{2} \left (13 d^3 e^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {1}{16} \left (25 d^4 e^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{2} \left (13 d^3 e^6\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 {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {1}{8} \left (25 d^4 e^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 {d^2 e^4 (52 d+25 e x) \sqrt {d^2-e^2 x^2}}{8 x}+\frac {d e^3 (25 d-52 e x) \left (d^2-e^2 x^2\right )^{3/2}}{24 x^2}-\frac {e^2 (52 d+25 e x) \left (d^2-e^2 x^2\right )^{5/2}}{60 x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{5 x^5}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {13}{2} d^3 e^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {25}{8} d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.69, size = 189, normalized size = 0.88 \begin {gather*} \frac {1}{120} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-24 d^7-90 d^6 e x-32 d^5 e^2 x^2+345 d^4 e^3 x^3+656 d^3 e^4 x^4+80 d^2 e^5 x^5+180 d e^6 x^6+40 e^7 x^7\right )}{x^5}+750 d^3 e^5 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )+780 d^3 e^4 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )\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. \(659\) vs.
\(2(188)=376\).
time = 0.07, size = 660, normalized size = 3.06
method | result | size |
risch | \(-\frac {d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (-656 e^{4} x^{4}-345 d \,e^{3} x^{3}+32 d^{2} x^{2} e^{2}+90 d^{3} e x +24 d^{4}\right )}{120 x^{5}}+\frac {e^{7} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}+\frac {2 e^{5} d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3}+\frac {3 e^{6} d x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {13 e^{6} d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {25 e^{5} d^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\) | \(210\) |
default | \(d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\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}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \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 )}{2 d^{2}}\right )}{4 d^{2}}\right )+e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \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 )}{2 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\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}}\right )}{3 d^{2}}\right )\) | \(660\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 259, normalized size = 1.20 \begin {gather*} \frac {13}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{5} - \frac {25}{8} \, d^{3} e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {13}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} d x e^{6} + \frac {25}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e^{5} + \frac {13 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{6}}{3 \, d} + \frac {25}{24} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{8 \, d^{2}} + \frac {52 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{15 \, d x} + \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{8 \, d^{2} x^{2}} - \frac {13 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{15 \, d x^{3}} - \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{4 \, x^{4}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.64, size = 168, normalized size = 0.78 \begin {gather*} -\frac {1560 \, d^{3} x^{5} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e^{5} - 375 \, d^{3} x^{5} e^{5} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - 80 \, d^{3} x^{5} e^{5} - {\left (40 \, x^{7} e^{7} + 180 \, d x^{6} e^{6} + 80 \, d^{2} x^{5} e^{5} + 656 \, d^{3} x^{4} e^{4} + 345 \, d^{4} x^{3} e^{3} - 32 \, d^{5} x^{2} e^{2} - 90 \, d^{6} x e - 24 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{120 \, x^{5}} \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 = 8.13, size = 1178, normalized size = 5.45 \begin {gather*} d^{7} \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) + 3 d^{6} e \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + d^{5} e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) - 5 d^{4} e^{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 ) - 5 d^{3} e^{4} \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^{5} \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 ) + 3 d e^{6} \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^{7} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 425 vs.
\(2 (181) = 362\).
time = 1.63, size = 425, normalized size = 1.97 \begin {gather*} \frac {13}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{5} \mathrm {sgn}\left (d\right ) - \frac {25}{8} \, d^{3} e^{5} \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 {43 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{3}}{16 \, x} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e}{8 \, x^{2}} - \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-1\right )}}{96 \, x^{3}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-3\right )}}{64 \, x^{4}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{3} e^{\left (-5\right )}}{160 \, x^{5}} + \frac {{\left (6 \, d^{3} e^{5} + \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{3}}{x} + \frac {50 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} e}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} e^{\left (-1\right )}}{x^{3}} - \frac {2580 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} e^{\left (-3\right )}}{x^{4}}\right )} x^{5} e^{10}}{960 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5}} + \frac {1}{6} \, {\left (4 \, d^{2} e^{5} + {\left (2 \, x e^{7} + 9 \, d e^{6}\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \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 = -0.00 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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