Optimal. Leaf size=81 \[ \frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {857, 807, 266, 63, 208} \begin {gather*} -\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 857
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x) \sqrt {d^2-e^2 x^2}} \, dx &=\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {\int \frac {-2 d e^2+e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {e \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^2}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac {e \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e 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^2 e}\\ &=-\frac {2 \sqrt {d^2-e^2 x^2}}{d^3 x}+\frac {\sqrt {d^2-e^2 x^2}}{d^2 x (d+e x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 62, normalized size = 0.77 \begin {gather*} \frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {(d+2 e x) \sqrt {d^2-e^2 x^2}}{x (d+e x)}}{d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 82, normalized size = 1.01 \begin {gather*} \frac {(-d-2 e x) \sqrt {d^2-e^2 x^2}}{d^3 x (d+e x)}-\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 88, normalized size = 1.09 \begin {gather*} -\frac {e^{2} x^{2} + d e x + {\left (e^{2} x^{2} + d e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (2 \, e x + d\right )}}{d^{3} e x^{2} + d^{4} x} \end {gather*}
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 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 108, normalized size = 1.33 \begin {gather*} \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^{2}}-\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}{\left (x +\frac {d}{e}\right ) d^{3}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{3} x} \end {gather*}
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*} \int \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} {\left (e x + d\right )} x^{2}}\,{d x} \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^2\,\sqrt {d^2-e^2\,x^2}\,\left (d+e\,x\right )} \,d x \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^{2} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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