Optimal. Leaf size=193 \[ \frac {3}{16} d^5 e (16 d-5 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {15}{16} d^7 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-3 d^7 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 193, 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, 1823,
829, 858, 223, 209, 272, 65, 214} \begin {gather*} -\frac {15}{16} d^7 e \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}}{x}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-3 d^7 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {3}{16} d^5 e (16 d-5 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 829
Rule 858
Rule 1821
Rule 1823
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^2} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-3 d^4 e+3 d^3 e^2 x-d^2 e^3 x^2\right )}{x} \, dx}{d^2}\\ &=-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {\int \frac {\left (21 d^4 e^3-21 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x} \, dx}{7 d^2 e^2}\\ &=\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {\int \frac {\left (-126 d^6 e^5+105 d^5 e^6 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{42 d^2 e^4}\\ &=\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {\int \frac {\left (504 d^8 e^7-315 d^7 e^8 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{168 d^2 e^6}\\ &=\frac {3}{16} d^5 e (16 d-5 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {\int \frac {-1008 d^{10} e^9+315 d^9 e^{10} x}{x \sqrt {d^2-e^2 x^2}} \, dx}{336 d^2 e^8}\\ &=\frac {3}{16} d^5 e (16 d-5 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}+\left (3 d^8 e\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{16} \left (15 d^7 e^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {3}{16} d^5 e (16 d-5 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}+\frac {1}{2} \left (3 d^8 e\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{16} \left (15 d^7 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 {3}{16} d^5 e (16 d-5 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {15}{16} d^7 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {\left (3 d^8\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 )}{e}\\ &=\frac {3}{16} d^5 e (16 d-5 e x) \sqrt {d^2-e^2 x^2}+\frac {1}{8} d^3 e (8 d-5 e x) \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{10} d e (6 d-5 e x) \left (d^2-e^2 x^2\right )^{5/2}-\frac {1}{7} e \left (d^2-e^2 x^2\right )^{7/2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{x}-\frac {15}{16} d^7 e \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-3 d^7 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 187, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-560 d^7+2496 d^6 e x+525 d^5 e^2 x^2-992 d^4 e^3 x^3-770 d^3 e^4 x^4+96 d^2 e^5 x^5+280 d e^6 x^6+80 e^7 x^7\right )}{560 x}+6 d^7 e \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {15}{16} d^7 \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. \(356\) vs.
\(2(169)=338\).
time = 0.10, size = 357, normalized size = 1.85
method | result | size |
risch | \(-\frac {d^{7} \sqrt {-e^{2} x^{2}+d^{2}}}{x}+\frac {e^{7} x^{6} \sqrt {-e^{2} x^{2}+d^{2}}}{7}+\frac {6 e^{5} d^{2} x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{35}-\frac {62 e^{3} d^{4} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{35}+\frac {156 e \,d^{6} \sqrt {-e^{2} x^{2}+d^{2}}}{35}+\frac {e^{6} d \,x^{5} \sqrt {-e^{2} x^{2}+d^{2}}}{2}-\frac {11 e^{4} d^{3} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{8}+\frac {15 e^{2} d^{5} x \sqrt {-e^{2} x^{2}+d^{2}}}{16}-\frac {15 e^{2} d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}}-\frac {3 e \,d^{8} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\) | \(265\) |
default | \(-\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7}+3 d \,e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )+3 d^{2} e \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )\) | \(357\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 212, normalized size = 1.10 \begin {gather*} -\frac {15}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e - 3 \, d^{7} 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 {15}{16} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5} x e^{2} + 3 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6} e - \frac {5}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} e + \frac {1}{2} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x e^{2} + \frac {3}{5} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{x} - \frac {1}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.32, size = 161, normalized size = 0.83 \begin {gather*} \frac {1050 \, d^{7} x \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e + 1680 \, d^{7} x e \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + 2496 \, d^{7} x e + {\left (80 \, x^{7} e^{7} + 280 \, d x^{6} e^{6} + 96 \, d^{2} x^{5} e^{5} - 770 \, d^{3} x^{4} e^{4} - 992 \, d^{4} x^{3} e^{3} + 525 \, d^{5} x^{2} e^{2} + 2496 \, d^{6} x e - 560 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{560 \, 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 = 11.82, size = 1057, normalized size = 5.48 \begin {gather*} d^{7} \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 ) + 3 d^{6} 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^{5} 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 ) - 5 d^{4} 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 ) - 5 d^{3} e^{4} \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 ) + d^{2} e^{5} \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) + 3 d e^{6} \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{7} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 199, normalized size = 1.03 \begin {gather*} -\frac {15}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e \mathrm {sgn}\left (d\right ) - 3 \, d^{7} 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^{7} 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^{7} e^{\left (-1\right )}}{2 \, x} + \frac {1}{560} \, {\left (2496 \, d^{6} e + {\left (525 \, d^{5} e^{2} - 2 \, {\left (496 \, d^{4} e^{3} + {\left (385 \, d^{3} e^{4} - 4 \, {\left (12 \, d^{2} e^{5} + 5 \, {\left (2 \, x e^{7} + 7 \, d e^{6}\right )} x\right )} x\right )} x\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 = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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