Optimal. Leaf size=120 \[ \frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}+\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 120, 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, 807, 266, 63, 208} \[ \frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 857
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-4 d e^2+3 e^3 x}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-8 d^3 e^4+3 d^2 e^5 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^6 e^4}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}-\frac {e \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}-\frac {e \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\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^4 e}\\ &=\frac {4 d-3 e x}{3 d^4 x \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{3 d^5 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 101, normalized size = 0.84 \[ \frac {3 e \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (3 d^3+7 d^2 e x-5 d e^2 x^2-8 e^3 x^3\right )}{x (e x-d) (d+e x)^2}-3 e \log (x)}{3 d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 181, normalized size = 1.51 \[ -\frac {4 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 4 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x + 3 \, {\left (e^{4} x^{4} + d e^{3} x^{3} - d^{2} e^{2} x^{2} - d^{3} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (8 \, e^{3} x^{3} + 5 \, d e^{2} x^{2} - 7 \, d^{2} e x - 3 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{5} e^{3} x^{4} + d^{6} e^{2} x^{3} - d^{7} e x^{2} - d^{8} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 1, normalized size = 0.01 \[ +\infty \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 188, normalized size = 1.57 \[ \frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{4}}+\frac {2 e^{2} x}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{5}}+\frac {2 e^{2} x}{3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{5}}-\frac {1}{3 \left (x +\frac {d}{e}\right ) \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3}}-\frac {e}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{4}}-\frac {1}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} x} \]
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 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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