Optimal. Leaf size=210 \[ -\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \]
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Rubi [A]
time = 0.31, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819,
1821, 821, 272, 65, 214} \begin {gather*} -\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 1821
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^4+20 d^3 e x-35 d^2 e^2 x^2+40 d e^3 x^3-32 e^4 x^4}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^4-60 d^3 e x+120 d^2 e^2 x^2-180 d e^3 x^3+144 e^4 x^4}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^4+60 d^3 e x-135 d^2 e^2 x^2+240 d e^3 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {\int \frac {-180 d^5 e+435 d^4 e^2 x-720 d^3 e^3 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^8}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\int \frac {-870 d^6 e^2+1620 d^5 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{10}}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {\left (18 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {\left (9 e^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {(18 e) \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 )}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 127, normalized size = 0.60 \begin {gather*} -\frac {\frac {\sqrt {d^2-e^2 x^2} \left (5 d^5-15 d^4 e x+70 d^3 e^2 x^2+674 d^2 e^3 x^3+1002 d e^4 x^4+424 e^5 x^5\right )}{x^3 (d+e x)^3}+540 e^3 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{15 d^6} \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. \(619\) vs.
\(2(186)=372\).
time = 0.09, size = 620, normalized size = 2.95
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (29 e^{2} x^{2}-6 d e x +d^{2}\right )}{3 d^{6} x^{3}}-\frac {93 e^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d^{6} \left (x +\frac {d}{e}\right )}+\frac {18 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{5} \sqrt {d^{2}}}-\frac {13 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d^{5} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d^{4} \left (x +\frac {d}{e}\right )^{3}}\) | \(216\) |
default | \(\frac {20 e^{3} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{7}}-\frac {4 \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 d^{6} \left (x +\frac {d}{e}\right )^{3}}-\frac {4 e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{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 )}{2 d^{2}}\right )}{d^{5}}+\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}}{d^{4}}+\frac {10 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{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 )}{d^{2}}\right )}{d^{6}}+\frac {10 e^{2} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}\right )}{d^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{6} x^{3}}-\frac {20 e^{3} \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 )}{d^{7}}\) | \(620\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.42, size = 199, normalized size = 0.95 \begin {gather*} -\frac {324 \, x^{6} e^{6} + 972 \, d x^{5} e^{5} + 972 \, d^{2} x^{4} e^{4} + 324 \, d^{3} x^{3} e^{3} + 270 \, {\left (x^{6} e^{6} + 3 \, d x^{5} e^{5} + 3 \, d^{2} x^{4} e^{4} + d^{3} x^{3} e^{3}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (424 \, x^{5} e^{5} + 1002 \, d x^{4} e^{4} + 674 \, d^{2} x^{3} e^{3} + 70 \, d^{3} x^{2} e^{2} - 15 \, d^{4} x e + 5 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{6} x^{6} e^{3} + 3 \, d^{7} x^{5} e^{2} + 3 \, d^{8} x^{4} e + d^{9} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )^{4}}\, dx \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. 420 vs.
\(2 (179) = 358\).
time = 1.46, size = 420, normalized size = 2.00 \begin {gather*} \frac {18 \, e^{3} \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 )}{d^{6}} - \frac {x^{3} {\left (\frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e}{x} - \frac {335 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-1\right )}}{x^{2}} - \frac {7559 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-3\right )}}{x^{3}} - \frac {25195 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-5\right )}}{x^{4}} - \frac {36035 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-7\right )}}{x^{5}} - \frac {24225 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-9\right )}}{x^{6}} - \frac {6585 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-11\right )}}{x^{7}} - 5 \, e^{3}\right )} e^{6}}{120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{6} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} - \frac {\frac {117 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{12} e}{x} - \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{12} e^{\left (-1\right )}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{12} e^{\left (-3\right )}}{x^{3}}}{24 \, d^{18}} \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 {\sqrt {d^2-e^2\,x^2}}{x^4\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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