Optimal. Leaf size=88 \[ \frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
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Rubi [A] time = 0.08, 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 = {857, 823, 12, 266, 63, 208} \[ \frac {3 d-2 e x}{3 d^4 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 857
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 {\operatorname {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 {\operatorname {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 = 0.10, size = 83, normalized size = 0.94 \[ \frac {\frac {\sqrt {d^2-e^2 x^2} \left (4 d^2+d e x-2 e^2 x^2\right )}{(d-e x) (d+e x)^2}-3 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+3 \log (x)}{3 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 155, normalized size = 1.76 \[ \frac {4 \, e^{3} x^{3} + 4 \, d e^{2} x^{2} - 4 \, d^{2} e x - 4 \, d^{3} + 3 \, {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (2 \, e^{2} x^{2} - d e x - 4 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2} - d^{6} e x - d^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 142, normalized size = 1.61 \[ -\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{3}}-\frac {2 e x}{3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{4}}+\frac {1}{3 \left (x +\frac {d}{e}\right ) \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{2} e}+\frac {1}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{3}} \]
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 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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