Optimal. Leaf size=91 \[ \frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}}+\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {807, 673, 197}
\begin {gather*} -\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 673
Rule 807
Rubi steps
\begin {align*} \int \frac {x}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {2 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e}\\ &=\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac {4 x}{15 d^3 e \sqrt {d^2-e^2 x^2}}+\frac {1}{5 e^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}-\frac {2}{15 d e^2 (d+e x) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 69, normalized size = 0.76 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (d^3+2 d^2 e x+8 d e^2 x^2+4 e^3 x^3\right )}{15 d^3 e^2 (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. \(261\) vs.
\(2(79)=158\).
time = 0.07, size = 262, normalized size = 2.88
method | result | size |
gosper | \(\frac {\left (-e x +d \right ) \left (4 e^{3} x^{3}+8 d \,e^{2} x^{2}+2 d^{2} e x +d^{3}\right )}{15 \left (e x +d \right ) d^{3} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(64\) |
trager | \(\frac {\left (4 e^{3} x^{3}+8 d \,e^{2} x^{2}+2 d^{2} e x +d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} \left (e x +d \right )^{3} e^{2} \left (-e x +d \right )}\) | \(66\) |
default | \(\frac {-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{e^{2}}-\frac {d \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right )^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}+\frac {3 e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{3}}\) | \(262\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 126, normalized size = 1.38 \begin {gather*} \frac {1}{5 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} x^{2} e^{4} + 2 \, \sqrt {-x^{2} e^{2} + d^{2}} d x e^{3} + \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e^{2}\right )}} - \frac {2}{15 \, {\left (\sqrt {-x^{2} e^{2} + d^{2}} d x e^{3} + \sqrt {-x^{2} e^{2} + d^{2}} d^{2} e^{2}\right )}} + \frac {4 \, x e^{\left (-1\right )}}{15 \, \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 [A]
time = 2.35, size = 109, normalized size = 1.20 \begin {gather*} \frac {x^{4} e^{4} + 2 \, d x^{3} e^{3} - 2 \, d^{3} x e - d^{4} - {\left (4 \, x^{3} e^{3} + 8 \, d x^{2} e^{2} + 2 \, d^{2} x e + d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{4} e^{6} + 2 \, d^{4} x^{3} e^{5} - 2 \, d^{6} x e^{3} - d^{7} e^{2}\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 )^{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 = 1.03, size = 160, normalized size = 1.76 \begin {gather*} -\frac {1}{120} \, {\left (-\frac {32 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{3}} - \frac {15}{d^{3} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {3 \, d^{12} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 5 \, d^{12} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} - 15 \, d^{12} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}}{d^{15} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.88, size = 65, normalized size = 0.71 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (d^3+2\,d^2\,e\,x+8\,d\,e^2\,x^2+4\,e^3\,x^3\right )}{15\,d^3\,e^2\,{\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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