Optimal. Leaf size=143 \[ -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2} \]
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Rubi [A] time = 0.14, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}-\frac {3 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \]
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^5 (d+e x)} \, dx &=\int \frac {d-e x}{x^5 \sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}-\frac {\int \frac {4 d^2 e-3 d e^2 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{4 d^2}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}+\frac {\int \frac {9 d^3 e^2-8 d^2 e^3 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{12 d^4}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\int \frac {16 d^4 e^3-9 d^3 e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{24 d^6}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}+\frac {\left (3 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}+\frac {\left (3 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {\left (3 e^2\right ) \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 )}{8 d^3}\\ &=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 95, normalized size = 0.66 \[ \frac {-9 e^4 x^4 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (-6 d^3+8 d^2 e x-9 d e^2 x^2+16 e^3 x^3\right )+9 e^4 x^4 \log (x)}{24 d^4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 86, normalized size = 0.60 \[ \frac {9 \, e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{3} x^{3} - 9 \, d e^{2} x^{2} + 8 \, d^{2} e x - 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, d^{4} x^{4}} \]
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 = 304, normalized size = 2.13 \[ -\frac {3 e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}\, d^{3}}-\frac {e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, d^{4}}+\frac {e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, d^{4}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{5} x}{d^{6}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{4}}{8 d^{5}}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{4}}{d^{5}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}{d^{6} x}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}{8 d^{5} x^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{3 d^{4} x^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 d^{3} x^{4}} \]
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 \[ \int \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{5}}\,{d x} \]
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 \[ \int \frac {\sqrt {d^2-e^2\,x^2}}{x^5\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{5} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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