Optimal. Leaf size=115 \[ \frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197}
\begin {gather*} -\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d}\\ &=-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d^2}\\ &=\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{21 d^4}\\ &=\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 93, normalized size = 0.81 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-6 d^5+9 d^4 e x+24 d^3 e^2 x^2+4 d^2 e^3 x^3-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \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. \(215\) vs.
\(2(99)=198\).
time = 0.48, size = 216, normalized size = 1.88
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (8 e^{5} x^{5}+16 d \,e^{4} x^{4}-4 d^{2} e^{3} x^{3}-24 d^{3} e^{2} x^{2}-9 d^{4} e x +6 d^{5}\right )}{21 \left (e x +d \right ) d^{6} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(88\) |
trager | \(-\frac {\left (8 e^{5} x^{5}+16 d \,e^{4} x^{4}-4 d^{2} e^{3} x^{3}-24 d^{3} e^{2} x^{2}-9 d^{4} e x +6 d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{21 d^{6} \left (e x +d \right )^{4} \left (-e x +d \right )^{2} e}\) | \(90\) |
default | \(\frac {-\frac {1}{7 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {5 e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{7 d}}{e^{2}}\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 148, normalized size = 1.29 \begin {gather*} -\frac {1}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x^{2} e^{3} + 2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} - \frac {1}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} + \frac {4 \, x}{21 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {8 \, x}{21 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 190, normalized size = 1.65 \begin {gather*} -\frac {6 \, x^{6} e^{6} + 12 \, d x^{5} e^{5} - 6 \, d^{2} x^{4} e^{4} - 24 \, d^{3} x^{3} e^{3} - 6 \, d^{4} x^{2} e^{2} + 12 \, d^{5} x e + 6 \, d^{6} + {\left (8 \, x^{5} e^{5} + 16 \, d x^{4} e^{4} - 4 \, d^{2} x^{3} e^{3} - 24 \, d^{3} x^{2} e^{2} - 9 \, d^{4} x e + 6 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{21 \, {\left (d^{6} x^{6} e^{7} + 2 \, d^{7} x^{5} e^{6} - d^{8} x^{4} e^{5} - 4 \, d^{9} x^{3} e^{4} - d^{10} x^{2} e^{3} + 2 \, d^{11} x e^{2} + d^{12} 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 {5}{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 = 1.55, size = 220, normalized size = 1.91 \begin {gather*} \frac {1}{672} \, {\left ({\left (\frac {14 \, {\left (\frac {15 \, d}{x e + d} - 7\right )} e^{\left (-5\right )}}{d^{6} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (3 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 21 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 70 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 210 \, d^{36} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6}\right )} e^{\left (-35\right )}}{d^{42} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{7}}\right )} e^{5} + \frac {256 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{6}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 139, normalized size = 1.21 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {11\,x}{42\,d^4}-\frac {5}{28\,d^3\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{28\,d^3\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{14\,d^4\,e\,{\left (d+e\,x\right )}^3}+\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{21\,d^6\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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