Optimal. Leaf size=58 \[ \frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 197}
\begin {gather*} \frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d}\\ &=\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 60, normalized size = 1.03 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-d^2+2 d e x+2 e^2 x^2\right )}{3 d^3 e (d-e x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs.
\(2(50)=100\).
time = 0.06, size = 104, normalized size = 1.79
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-2 e^{2} x^{2}-2 d e x +d^{2}\right )}{3 d^{3} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(46\) |
trager | \(-\frac {\left (-2 e^{2} x^{2}-2 d e x +d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{3} \left (e x +d \right )^{2} e \left (-e x +d \right )}\) | \(55\) |
default | \(\frac {-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{e}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 62, normalized size = 1.07 \begin {gather*} -\frac {1}{3 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} d x e^{2} + \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e\right )}} + \frac {2 \, x}{3 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (48) = 96\).
time = 4.10, size = 98, normalized size = 1.69 \begin {gather*} -\frac {x^{3} e^{3} + d x^{2} e^{2} - d^{2} x e - d^{3} + {\left (2 \, x^{2} e^{2} + 2 \, d x e - d^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (d^{3} x^{3} e^{4} + d^{4} x^{2} e^{3} - d^{5} x e^{2} - d^{6} e\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 {1}{\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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.71, size = 56, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (-d^2+2\,d\,e\,x+2\,e^2\,x^2\right )}{3\,d^3\,e\,{\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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