Optimal. Leaf size=114 \[ \frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3} \]
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Rubi [A] time = 0.11, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {850, 835, 807, 266, 63, 208} \[ -\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 850
Rubi steps
\begin {align*} \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx &=\int \frac {d-e x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {\int \frac {3 d^2 e-2 d e^2 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}+\frac {\int \frac {4 d^3 e^2-3 d^2 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {e^3 \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {e^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e \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 )}{2 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 84, normalized size = 0.74 \[ \frac {\left (-2 d^2+3 d e x-4 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}+3 e^3 x^3 \log \left (\sqrt {d^2-e^2 x^2}+d\right )-3 e^3 x^3 \log (x)}{6 d^3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 75, normalized size = 0.66 \[ -\frac {3 \, e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (4 \, e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, d^{3} x^{3}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 280, normalized size = 2.46 \[ \frac {e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}\, d^{2}}+\frac {e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, d^{3}}-\frac {e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, d^{3}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{4} x}{d^{5}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}{2 d^{4}}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{3}}{d^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}{d^{5} x}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{2 d^{4} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{3} x^{3}} \]
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 \[ \int \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{4}}\,{d x} \]
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 \[ \int \frac {\sqrt {d^2-e^2\,x^2}}{x^4\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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