Optimal. Leaf size=58 \[ \frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {793, 191} \begin {gather*} \frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 793
Rubi steps
\begin {align*} \int \frac {x}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 e}\\ &=\frac {x}{3 d^2 e \sqrt {d^2-e^2 x^2}}+\frac {1}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (d^2+d e x+e^2 x^2\right )}{3 d^2 e^2 (d-e x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 56, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (d^2+d e x+e^2 x^2\right )}{3 d^2 e^2 (d-e x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 101, normalized size = 1.74 \begin {gather*} \frac {e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3} - {\left (e^{2} x^{2} + d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{2} e^{5} x^{3} + d^{3} e^{4} x^{2} - d^{4} e^{3} x - d^{5} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.76 \begin {gather*} \frac {\left (-e x +d \right ) \left (e^{2} x^{2}+d e x +d^{2}\right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 67, normalized size = 1.16 \begin {gather*} \frac {1}{3 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{3} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{2}\right )}} + \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.71, size = 52, normalized size = 0.90 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2+d\,e\,x+e^2\,x^2\right )}{3\,d^2\,e^2\,{\left (d+e\,x\right )}^2\,\left (d-e\,x\right )} \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 {x}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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