Optimal. Leaf size=118 \[ \frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{35 d^4 e^3 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {864, 833, 792,
198, 197} \begin {gather*} \frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{35 d^4 e^3 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 792
Rule 833
Rule 864
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\int \frac {x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {\int \frac {x \left (2 d^3-3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d^2 e^2}\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 e^3}\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{35 d^2 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}{35 d^2 e^3}\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{35 d^4 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 104, normalized size = 0.88 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-2 d^6-2 d^5 e x+5 d^4 e^2 x^2+5 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5-2 e^6 x^6\right )}{35 d^4 e^4 (d-e x)^3 (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. \(421\) vs.
\(2(102)=204\).
time = 0.07, size = 422, normalized size = 3.58
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (2 e^{6} x^{6}+2 d \,e^{5} x^{5}-5 d^{2} e^{4} x^{4}-5 d^{3} e^{3} x^{3}-5 d^{4} e^{2} x^{2}+2 e \,d^{5} x +2 d^{6}\right )}{35 d^{4} e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(92\) |
trager | \(-\frac {\left (2 e^{6} x^{6}+2 d \,e^{5} x^{5}-5 d^{2} e^{4} x^{4}-5 d^{3} e^{3} x^{3}-5 d^{4} e^{2} x^{2}+2 e \,d^{5} x +2 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{35 d^{4} e^{4} \left (e x +d \right )^{4} \left (-e x +d \right )^{3}}\) | \(101\) |
default | \(\frac {\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}}{e}-\frac {d}{5 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{e^{3}}-\frac {d^{3} \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{10 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{e^{4}}\) | \(422\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 121, normalized size = 1.03 \begin {gather*} \frac {d^{2}}{7 \, {\left ({\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x e^{5} + {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{4}\right )}} + \frac {8 \, x e^{\left (-3\right )}}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {d e^{\left (-4\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {x e^{\left (-3\right )}}{35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} - \frac {2 \, x e^{\left (-3\right )}}{35 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4}} \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. 221 vs.
\(2 (97) = 194\).
time = 2.32, size = 221, normalized size = 1.87 \begin {gather*} -\frac {2 \, x^{7} e^{7} + 2 \, d x^{6} e^{6} - 6 \, d^{2} x^{5} e^{5} - 6 \, d^{3} x^{4} e^{4} + 6 \, d^{4} x^{3} e^{3} + 6 \, d^{5} x^{2} e^{2} - 2 \, d^{6} x e - 2 \, d^{7} - {\left (2 \, x^{6} e^{6} + 2 \, d x^{5} e^{5} - 5 \, d^{2} x^{4} e^{4} - 5 \, d^{3} x^{3} e^{3} - 5 \, d^{4} x^{2} e^{2} + 2 \, d^{5} x e + 2 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{35 \, {\left (d^{4} x^{7} e^{11} + d^{5} x^{6} e^{10} - 3 \, d^{6} x^{5} e^{9} - 3 \, d^{7} x^{4} e^{8} + 3 \, d^{8} x^{3} e^{7} + 3 \, d^{9} x^{2} e^{6} - d^{10} x e^{5} - d^{11} e^{4}\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^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{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 = 2.95, size = 161, normalized size = 1.36 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}}{56\,d\,e^4\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{56\,d\,e^4}+\frac {x}{35\,d^2\,e^3}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {2\,d}{35\,e^4}-\frac {11\,x}{70\,e^3}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {2\,x\,\sqrt {d^2-e^2\,x^2}}{35\,d^4\,e^3\,\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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