Optimal. Leaf size=95 \[ -\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {869, 792, 197}
\begin {gather*} -\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 792
Rule 869
Rubi steps
\begin {align*} \int \frac {x^2}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x (2 d+2 e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d e}\\ &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e^2}\\ &=-\frac {x^2}{5 d e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d+e x)}{15 d e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{15 d^3 e^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 82, normalized size = 0.86 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (2 d^4+2 d^3 e x-3 d^2 e^2 x^2+2 d e^3 x^3+2 e^4 x^4\right )}{15 d^3 e^3 (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. \(233\) vs.
\(2(83)=166\).
time = 0.06, size = 234, normalized size = 2.46
method | result | size |
gosper | \(\frac {\left (-e x +d \right ) \left (2 e^{4} x^{4}+2 d \,e^{3} x^{3}-3 d^{2} x^{2} e^{2}+2 d^{3} e x +2 d^{4}\right )}{15 d^{3} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\) | \(70\) |
trager | \(\frac {\left (2 e^{4} x^{4}+2 d \,e^{3} x^{3}-3 d^{2} x^{2} e^{2}+2 d^{3} e x +2 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} e^{3} \left (e x +d \right )^{3} \left (-e x +d \right )^{2}}\) | \(79\) |
default | \(\frac {1}{3 e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2}}+\frac {d^{2} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{3}}\) | \(234\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 100, normalized size = 1.05 \begin {gather*} -\frac {d}{5 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{4} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{3}\right )}} - \frac {x e^{\left (-2\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {e^{\left (-3\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, x e^{\left (-2\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 = 2.01, size = 158, normalized size = 1.66 \begin {gather*} \frac {2 \, x^{5} e^{5} + 2 \, d x^{4} e^{4} - 4 \, d^{2} x^{3} e^{3} - 4 \, d^{3} x^{2} e^{2} + 2 \, d^{4} x e + 2 \, d^{5} + {\left (2 \, x^{4} e^{4} + 2 \, d x^{3} e^{3} - 3 \, d^{2} x^{2} e^{2} + 2 \, d^{3} x e + 2 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{5} e^{8} + d^{4} x^{4} e^{7} - 2 \, d^{5} x^{3} e^{6} - 2 \, d^{6} x^{2} e^{5} + d^{7} x e^{4} + d^{8} 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 {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 = 2.79, size = 78, normalized size = 0.82 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^4+2\,d^3\,e\,x-3\,d^2\,e^2\,x^2+2\,d\,e^3\,x^3+2\,e^4\,x^4\right )}{15\,d^3\,e^3\,{\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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