Optimal. Leaf size=143 \[ -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 849, 821,
272, 65, 214} \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 864
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx &=\int \frac {d-e x}{x^5 \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}-\frac {\int \frac {4 d^2 e-3 d e^2 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{4 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}+\frac {\int \frac {9 d^3 e^2-8 d^2 e^3 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{12 d^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\int \frac {16 d^4 e^3-9 d^3 e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{24 d^6}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}+\frac {\left (3 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}+\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {\left (3 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 )}{8 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 101, normalized size = 0.71 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3+8 d^2 e x-9 d e^2 x^2+16 e^3 x^3\right )+18 e^4 x^4 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{24 d^4 x^4} \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. \(476\) vs.
\(2(123)=246\).
time = 0.07, size = 477, normalized size = 3.34
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-16 e^{3} x^{3}+9 d \,e^{2} x^{2}-8 d^{2} e x +6 d^{3}\right )}{24 d^{4} x^{4}}-\frac {3 e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 d^{3} \sqrt {d^{2}}}\) | \(99\) |
default | \(-\frac {e^{4} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{5}}+\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 d^{2} x^{4}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{4 d^{2}}}{d}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{d^{3}}-\frac {e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )}{d^{4}}+\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{4} x^{3}}+\frac {e^{4} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{5}}\) | \(477\) |
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 = 3.71, size = 82, normalized size = 0.57 \begin {gather*} \frac {9 \, x^{4} e^{4} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (16 \, x^{3} e^{3} - 9 \, d x^{2} e^{2} + 8 \, d^{2} x e - 6 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{24 \, d^{4} x^{4}} \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^{5} \left (d + e x\right )}\, 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. 299 vs.
\(2 (116) = 232\).
time = 1.43, size = 299, normalized size = 2.09 \begin {gather*} -\frac {x^{4} {\left (\frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{2}}{x} + \frac {72 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-2\right )}}{x^{3}} - \frac {24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2}}{x^{2}} - 3 \, e^{4}\right )} e^{8}}{192 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4}} - \frac {3 \, 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 )}{8 \, d^{4}} + \frac {\frac {72 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{12} e^{2}}{x} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{12} e^{\left (-2\right )}}{x^{3}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{12} e^{\left (-4\right )}}{x^{4}} - \frac {24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{12}}{x^{2}}}{192 \, d^{16}} \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^5\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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