Optimal. Leaf size=113 \[ -\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \]
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Rubi [A]
time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {871, 849, 821,
272, 65, 214} \begin {gather*} \frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 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 871
Rubi steps
\begin {align*} \int \frac {1}{x^3 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {\int \frac {-3 d e^2+2 e^3 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {\int \frac {-4 d^2 e^3+3 d e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{2 d^4 e^2}\\ &=-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {\left (3 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^3}\\ &=-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^3}\\ &=-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {3 \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 )}{2 d^3}\\ &=-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^4 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 93, normalized size = 0.82 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (-d^2+d e x+4 e^2 x^2\right )}{x^2 (d+e x)}+6 e^2 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 183, normalized size = 1.62
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-2 e x +d \right )}{2 d^{4} x^{2}}+\frac {e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{4} \left (x +\frac {d}{e}\right )}-\frac {3 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{3} \sqrt {d^{2}}}\) | \(117\) |
default | \(\frac {e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{4} \left (x +\frac {d}{e}\right )}+\frac {-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}}{d}+\frac {e \sqrt {-e^{2} x^{2}+d^{2}}}{d^{4} x}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{3} \sqrt {d^{2}}}\) | \(183\) |
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 = 1.45, size = 108, normalized size = 0.96 \begin {gather*} \frac {2 \, x^{3} e^{3} + 2 \, d x^{2} e^{2} + 3 \, {\left (x^{3} e^{3} + d x^{2} e^{2}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (4 \, x^{2} e^{2} + d x e - d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{2 \, {\left (d^{4} x^{3} e + d^{5} x^{2}\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 {1}{x^{3} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \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. 233 vs.
\(2 (98) = 196\).
time = 1.26, size = 233, normalized size = 2.06 \begin {gather*} -\frac {3 \, e^{2} \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 )}{2 \, d^{4}} - \frac {x^{2} {\left (\frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{\left (-2\right )}}{x^{2}} + \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}}{x} - e^{2}\right )} e^{4}}{8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} - \frac {\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{\left (-2\right )}}{x^{2}} - \frac {4 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4}}{x}}{8 \, d^{8}} \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 {1}{x^3\,\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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