Optimal. Leaf size=82 \[ \frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 3, 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}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{15 d^3 \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) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{5 d 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}{5 d}\\ &=\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d 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}{15 d^3}\\ &=\frac {4 x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{15 d^5 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 82, normalized size = 1.00 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-3 d^4+12 d^3 e x+12 d^2 e^2 x^2-8 d e^3 x^3-8 e^4 x^4\right )}{15 d^5 e (d-e x)^2 (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. \(163\) vs.
\(2(70)=140\).
time = 0.07, size = 164, normalized size = 2.00
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (8 e^{4} x^{4}+8 d \,e^{3} x^{3}-12 d^{2} x^{2} e^{2}-12 d^{3} e x +3 d^{4}\right )}{15 d^{5} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(70\) |
trager | \(-\frac {\left (8 e^{4} x^{4}+8 d \,e^{3} x^{3}-12 d^{2} x^{2} e^{2}-12 d^{3} e x +3 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{5} \left (e x +d \right )^{3} \left (-e x +d \right )^{2} e}\) | \(79\) |
default | \(\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{e}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 81, normalized size = 0.99 \begin {gather*} -\frac {1}{5 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d x e^{2} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e\right )}} + \frac {4 \, x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} + \frac {8 \, x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5}} \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. 158 vs.
\(2 (67) = 134\).
time = 2.22, size = 158, normalized size = 1.93 \begin {gather*} -\frac {3 \, x^{5} e^{5} + 3 \, d x^{4} e^{4} - 6 \, d^{2} x^{3} e^{3} - 6 \, d^{3} x^{2} e^{2} + 3 \, d^{4} x e + 3 \, d^{5} + {\left (8 \, x^{4} e^{4} + 8 \, d x^{3} e^{3} - 12 \, d^{2} x^{2} e^{2} - 12 \, d^{3} x e + 3 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{5} x^{5} e^{6} + d^{6} x^{4} e^{5} - 2 \, d^{7} x^{3} e^{4} - 2 \, d^{8} x^{2} e^{3} + d^{9} x e^{2} + d^{10} 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 )}\, 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 = 2.76, size = 78, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}\,\left (3\,d^4-12\,d^3\,e\,x-12\,d^2\,e^2\,x^2+8\,d\,e^3\,x^3+8\,e^4\,x^4\right )}{15\,d^5\,e\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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