Optimal. Leaf size=89 \[ \frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}}-\frac {d}{5 e^3 (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {7}{15 e^3 (d+e x) \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {866, 1649, 792,
197} \begin {gather*} \frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}}-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 (d-e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 792
Rule 866
Rule 1649
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac {x^2 (d-e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {\left (\frac {2 d^2}{e^2}-\frac {5 d x}{e}\right ) (d-e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 (d-e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 e^2}\\ &=-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 (d-e x)}{15 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 70, normalized size = 0.79 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (4 d^3+8 d^2 e x+2 d e^2 x^2+e^3 x^3\right )}{15 d^2 e^3 (d-e x) (d+e x)^3} \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. \(286\) vs.
\(2(77)=154\).
time = 0.08, size = 287, normalized size = 3.22
method | result | size |
gosper | \(\frac {\left (-e x +d \right ) \left (e^{3} x^{3}+2 d \,e^{2} x^{2}+8 d^{2} e x +4 d^{3}\right )}{15 \left (e x +d \right ) d^{2} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(65\) |
trager | \(\frac {\left (e^{3} x^{3}+2 d \,e^{2} x^{2}+8 d^{2} e x +4 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} e^{3} \left (e x +d \right )^{3} \left (-e x +d \right )}\) | \(67\) |
default | \(\frac {x}{d^{2} e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {2 d \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{3}}+\frac {d^{2} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right )^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}+\frac {3 e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{4}}\) | \(287\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 124, normalized size = 1.39 \begin {gather*} -\frac {d}{5 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} x^{2} e^{5} + 2 \, \sqrt {-x^{2} e^{2} + d^{2}} d x e^{4} + \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e^{3}\right )}} + \frac {x e^{\left (-2\right )}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} + \frac {7}{15 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} x e^{4} + \sqrt {-x^{2} e^{2} + d^{2}} d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.80, size = 111, normalized size = 1.25 \begin {gather*} \frac {4 \, x^{4} e^{4} + 8 \, d x^{3} e^{3} - 8 \, d^{3} x e - 4 \, d^{4} - {\left (x^{3} e^{3} + 2 \, d x^{2} e^{2} + 8 \, d^{2} x e + 4 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{2} x^{4} e^{7} + 2 \, d^{3} x^{3} e^{6} - 2 \, d^{5} x e^{4} - d^{6} e^{3}\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^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.05, size = 171, normalized size = 1.92 \begin {gather*} -\frac {1}{120} \, {\left (-\frac {8 i \, e^{\left (-2\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2}} - \frac {15 \, e^{\left (-2\right )}}{d^{2} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {{\left (3 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} - 5 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} - 15 \, d^{8} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-10\right )}}{d^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.90, size = 66, normalized size = 0.74 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (4\,d^3+8\,d^2\,e\,x+2\,d\,e^2\,x^2+e^3\,x^3\right )}{15\,d^2\,e^3\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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