Optimal. Leaf size=110 \[ \frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 837,
12, 272, 65, 214} \begin {gather*} -\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 837
Rule 866
Rule 1819
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^4+12 d^3 e x+5 d^2 e^2 x^2}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^4-24 d^3 e x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {15 d^6 e^2}{x \sqrt {d^2-e^2 x^2}} \, dx}{15 d^8 e^2}\\ &=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^2}\\ &=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^2}\\ &=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}-\frac {\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^2 e^2}\\ &=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 88, normalized size = 0.80 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (13 d^2+19 d e x+8 e^2 x^2\right )}{(d+e x)^3}+10 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{5 d^3} \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. \(414\) vs.
\(2(96)=192\).
time = 0.07, size = 415, normalized size = 3.77
method | result | size |
default | \(-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}}{d^{4}}+\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{3} d^{3} \left (x +\frac {d}{e}\right )^{3}}-\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}}{e^{3} d}-\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}}{e \,d^{3}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}}{d^{4}}\) | \(415\) |
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 = 1.87, size = 148, normalized size = 1.35 \begin {gather*} \frac {13 \, x^{3} e^{3} + 39 \, d x^{2} e^{2} + 39 \, d^{2} x e + 13 \, d^{3} + 5 \, {\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (8 \, x^{2} e^{2} + 19 \, d x e + 13 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (d^{3} x^{3} e^{3} + 3 \, d^{4} x^{2} e^{2} + 3 \, d^{5} x e + d^{6}\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 \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. 195 vs.
\(2 (96) = 192\).
time = 1.65, size = 195, normalized size = 1.77 \begin {gather*} -\frac {\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^{3}} - \frac {2 \, {\left (\frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-4\right )}}{x^{2}} + \frac {55 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-6\right )}}{x^{3}} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-8\right )}}{x^{4}} + 13\right )}}{5 \, d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \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 {\sqrt {d^2-e^2\,x^2}}{x\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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