Optimal. Leaf size=88 \[ \frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
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Rubi [A]
time = 0.05, antiderivative size = 88, 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, 837, 12,
272, 65, 214} \begin {gather*} \frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 837
Rule 871
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-3 d e^2+2 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {3 d^3 e^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{3 d^6 e^4}\\ &=\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^3}\\ &=\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \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^3}\\ &=\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \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^3 e^2}\\ &=\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 95, normalized size = 1.08 \begin {gather*} \frac {\frac {\left (4 d^2+d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{(d-e x) (d+e x)^2}+6 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{3 d^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. \(170\) vs.
\(2(78)=156\).
time = 0.06, size = 171, normalized size = 1.94
method | result | size |
default | \(-\frac {-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d}\) | \(171\) |
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 = 4.03, size = 150, normalized size = 1.70 \begin {gather*} \frac {4 \, x^{3} e^{3} + 4 \, d x^{2} e^{2} - 4 \, d^{2} x e - 4 \, d^{3} + 3 \, {\left (x^{3} e^{3} + d x^{2} e^{2} - d^{2} x e - d^{3}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (2 \, x^{2} e^{2} - d x e - 4 \, d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d^{4} x^{3} e^{3} + d^{5} x^{2} e^{2} - d^{6} x e - d^{7}\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 \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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