Optimal. Leaf size=223 \[ \frac {19 d^9 x \sqrt {d^2-e^2 x^2}}{256 e^2}+\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {19 d^{11} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^3} \]
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Rubi [A]
time = 0.19, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1823, 847, 794,
201, 223, 209} \begin {gather*} \frac {19 d^{11} \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^3}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}+\frac {19 d^9 x \sqrt {d^2-e^2 x^2}}{256 e^2}+\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rule 1823
Rubi steps
\begin {align*} \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^2 \left (d^2-e^2 x^2\right )^{5/2} \left (-11 d^3 e^2-37 d^2 e^3 x-33 d e^4 x^2\right ) \, dx}{11 e^2}\\ &=-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^2 \left (209 d^3 e^4+370 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{110 e^4}\\ &=-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x \left (-740 d^4 e^5-1881 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{990 e^6}\\ &=-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {\left (19 d^5\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e^2}\\ &=\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {\left (19 d^7\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{96 e^2}\\ &=\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {\left (19 d^9\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{128 e^2}\\ &=\frac {19 d^9 x \sqrt {d^2-e^2 x^2}}{256 e^2}+\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {\left (19 d^{11}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{256 e^2}\\ &=\frac {19 d^9 x \sqrt {d^2-e^2 x^2}}{256 e^2}+\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {\left (19 d^{11}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2}\\ &=\frac {19 d^9 x \sqrt {d^2-e^2 x^2}}{256 e^2}+\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {19 d^{11} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 177, normalized size = 0.79 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (-94720 d^{10}-65835 d^9 e x-47360 d^8 e^2 x^2+251790 d^7 e^3 x^3+629760 d^6 e^4 x^4+201432 d^5 e^5 x^5-657920 d^4 e^6 x^6-587664 d^3 e^7 x^7+89600 d^2 e^8 x^8+266112 d e^9 x^9+80640 e^{10} x^{10}\right )+65835 d^{11} \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{887040 e^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. \(423\) vs.
\(2(191)=382\).
time = 0.07, size = 424, normalized size = 1.90
method | result | size |
risch | \(-\frac {\left (-80640 e^{10} x^{10}-266112 d \,e^{9} x^{9}-89600 d^{2} e^{8} x^{8}+587664 d^{3} e^{7} x^{7}+657920 d^{4} e^{6} x^{6}-201432 d^{5} e^{5} x^{5}-629760 d^{6} e^{4} x^{4}-251790 d^{7} e^{3} x^{3}+47360 d^{8} e^{2} x^{2}+65835 d^{9} e x +94720 d^{10}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{887040 e^{3}}+\frac {19 d^{11} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 e^{2} \sqrt {e^{2}}}\) | \(163\) |
default | \(e^{3} \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )+3 e^{2} d \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )+3 e \,d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )+d^{3} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )\) | \(424\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 182, normalized size = 0.82 \begin {gather*} \frac {19}{256} \, d^{11} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} + \frac {19}{256} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{9} x e^{\left (-2\right )} + \frac {19}{384} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} x e^{\left (-2\right )} + \frac {19}{480} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x e^{\left (-2\right )} - \frac {1}{11} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x^{4} e - \frac {37}{99} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{2} e^{\left (-1\right )} - \frac {19}{80} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x e^{\left (-2\right )} - \frac {74}{693} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} e^{\left (-3\right )} - \frac {3}{10} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 149, normalized size = 0.67 \begin {gather*} -\frac {1}{887040} \, {\left (131670 \, d^{11} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (80640 \, x^{10} e^{10} + 266112 \, d x^{9} e^{9} + 89600 \, d^{2} x^{8} e^{8} - 587664 \, d^{3} x^{7} e^{7} - 657920 \, d^{4} x^{6} e^{6} + 201432 \, d^{5} x^{5} e^{5} + 629760 \, d^{6} x^{4} e^{4} + 251790 \, d^{7} x^{3} e^{3} - 47360 \, d^{8} x^{2} e^{2} - 65835 \, d^{9} x e - 94720 \, d^{10}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 138.75, size = 1681, normalized size = 7.54 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 139, normalized size = 0.62 \begin {gather*} \frac {19}{256} \, d^{11} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{887040} \, {\left (94720 \, d^{10} e^{\left (-3\right )} + {\left (65835 \, d^{9} e^{\left (-2\right )} + 2 \, {\left (23680 \, d^{8} e^{\left (-1\right )} - {\left (125895 \, d^{7} + 4 \, {\left (78720 \, d^{6} e + {\left (25179 \, d^{5} e^{2} - 2 \, {\left (41120 \, d^{4} e^{3} + 7 \, {\left (5247 \, d^{3} e^{4} - 8 \, {\left (100 \, d^{2} e^{5} + 9 \, {\left (10 \, x e^{7} + 33 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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