Optimal. Leaf size=209 \[ -\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1821, 827,
829, 858, 223, 209, 272, 65, 214} \begin {gather*} \frac {45}{8} d^4 e^4 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}+\frac {45}{8} d^4 e^4 \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 1821
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-12 d^4 e-9 d^3 e^2 x-4 d^2 e^3 x^2\right )}{x^4} \, dx}{4 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {\int \frac {\left (27 d^5 e^2-36 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx}{12 d^4}\\ &=-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {\left (144 d^6 e^3+216 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx}{192 d^4}\\ &=\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {5 \int \frac {\left (-432 d^7 e^4+864 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{384 d^4}\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {864 d^9 e^6-864 d^8 e^7 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{768 d^4 e^2}\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{8} \left (45 d^5 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{8} \left (45 d^4 e^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{16} \left (45 d^5 e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{8} \left (45 d^4 e^5\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 {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{8} \left (45 d^5 e^2\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 {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 187, normalized size = 0.89 \begin {gather*} \frac {1}{8} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^7-8 d^6 e x-3 d^5 e^2 x^2+48 d^4 e^3 x^3-48 d^3 e^4 x^4+3 d^2 e^5 x^5+8 d e^6 x^6+2 e^7 x^7\right )}{x^4}-90 d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )-45 d^4 e \left (-e^2\right )^{3/2} \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. \(597\) vs.
\(2(183)=366\).
time = 0.07, size = 598, normalized size = 2.86
method | result | size |
risch | \(-\frac {d^{4} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (-48 e^{3} x^{3}+3 d \,e^{2} x^{2}+8 d^{2} e x +2 d^{3}\right )}{8 x^{4}}+\frac {e^{7} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4}+\frac {3 e^{5} d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}{8}+\frac {45 e^{5} d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+e^{6} d \,x^{2} \sqrt {-e^{2} x^{2}+d^{2}}-6 e^{4} d^{3} \sqrt {-e^{2} x^{2}+d^{2}}+\frac {45 e^{4} d^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\) | \(223\) |
default | \(d^{3} \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 )+3 d \,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 )+e^{3} \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} e \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 )\) | \(598\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 233, normalized size = 1.11 \begin {gather*} \frac {45}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{4} + \frac {45}{8} \, d^{4} e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {45}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2} x e^{5} - \frac {45}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3} e^{4} + \frac {15}{4} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{5} - \frac {15}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} - \frac {9 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{8 \, d} + \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{x} - \frac {9 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{8 \, d x^{2}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{x^{3}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.48, size = 168, normalized size = 0.80 \begin {gather*} -\frac {90 \, d^{4} x^{4} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e^{4} + 45 \, d^{4} x^{4} e^{4} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + 48 \, d^{4} x^{4} e^{4} - {\left (2 \, x^{7} e^{7} + 8 \, d x^{6} e^{6} + 3 \, d^{2} x^{5} e^{5} - 48 \, d^{3} x^{4} e^{4} + 48 \, d^{4} x^{3} e^{3} - 3 \, d^{5} x^{2} e^{2} - 8 \, d^{6} x e - 2 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{8 \, x^{4}} \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 = 9.47, size = 1028, normalized size = 4.92 \begin {gather*} d^{7} \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 ) + 3 d^{6} e \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 ) + d^{5} e^{2} \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^{4} e^{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 ) - 5 d^{3} e^{4} \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^{5} \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 ) + 3 d e^{6} \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^{7} \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 ) \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. 365 vs.
\(2 (176) = 352\).
time = 1.38, size = 365, normalized size = 1.75 \begin {gather*} \frac {45}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{4} \mathrm {sgn}\left (d\right ) + \frac {45}{8} \, d^{4} e^{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 {23 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{2}}{8 \, x} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{\left (-2\right )}}{8 \, x^{3}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{\left (-4\right )}}{64 \, x^{4}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4}}{8 \, x^{2}} + \frac {{\left (d^{4} e^{4} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{2}}{x} - \frac {184 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{\left (-2\right )}}{x^{3}} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4}}{x^{2}}\right )} x^{4} e^{8}}{64 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}} - \frac {1}{8} \, {\left (48 \, d^{3} e^{4} - {\left (3 \, d^{2} e^{5} + 2 \, {\left (x e^{7} + 4 \, d e^{6}\right )} x\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^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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