Optimal. Leaf size=81 \[ -\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {871, 821, 272,
65, 214} \begin {gather*} \frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {e \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 65
Rule 214
Rule 272
Rule 821
Rule 871
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {\int \frac {-2 d e^2+e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {e \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^2}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {e \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}+\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}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 77, normalized size = 0.95 \begin {gather*} -\frac {\frac {(d+2 e x) \sqrt {d^2-e^2 x^2}}{x (d+e x)}+2 e \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 108, normalized size = 1.33
method | result | size |
default | \(-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{3} \left (x +\frac {d}{e}\right )}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{3} x}+\frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}\) | \(108\) |
risch | \(-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{3} \left (x +\frac {d}{e}\right )}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{3} x}+\frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}\) | \(108\) |
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 = 1.43, size = 88, normalized size = 1.09 \begin {gather*} -\frac {x^{2} e^{2} + d x e + {\left (x^{2} e^{2} + d x e\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + \sqrt {-x^{2} e^{2} + d^{2}} {\left (2 \, x e + d\right )}}{d^{3} x^{2} e + d^{4} x} \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^{2} \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. 164 vs.
\(2 (74) = 148\).
time = 0.79, size = 164, normalized size = 2.02 \begin {gather*} \frac {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 )}{d^{3}} + \frac {x {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{x} + e\right )} e^{2}}{2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{3} x} \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^2\,\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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