Optimal. Leaf size=254 \[ -\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {19 e^{11} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1821, 849,
821, 272, 43, 65, 214} \begin {gather*} -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}+\frac {19 e^{11} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 1821
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{12}} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-33 d^4 e-37 d^3 e^2 x-11 d^2 e^3 x^2\right )}{x^{11}} \, dx}{11 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}+\frac {\int \frac {\left (370 d^5 e^2+209 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx}{110 d^4}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {\int \frac {\left (-1881 d^6 e^3-740 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx}{990 d^6}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}+\frac {\int \frac {\left (5920 d^7 e^4+1881 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{7920 d^8}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^5\right ) \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{80 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^5\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )}{160 d^2}\\ &=-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac {\left (19 e^7\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{192 d^2}\\ &=\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^9\right ) \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{256 d^2}\\ &=-\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac {\left (19 e^{11}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{512 d^2}\\ &=-\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {\left (19 e^9\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{256 d^2}\\ &=-\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac {19 e^{11} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.84, size = 175, normalized size = 0.69 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (-80640 d^{10}-266112 d^9 e x-89600 d^8 e^2 x^2+587664 d^7 e^3 x^3+657920 d^6 e^4 x^4-201432 d^5 e^5 x^5-629760 d^4 e^6 x^6-251790 d^3 e^7 x^7+47360 d^2 e^8 x^8+65835 d e^9 x^9+94720 e^{10} x^{10}\right )}{x^{11}}-131670 e^{11} \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{887040 d^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. \(625\) vs.
\(2(218)=436\).
time = 0.18, size = 626, normalized size = 2.46
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-94720 e^{10} x^{10}-65835 d \,e^{9} x^{9}-47360 d^{2} e^{8} x^{8}+251790 d^{3} e^{7} x^{7}+629760 d^{4} e^{6} x^{6}+201432 d^{5} e^{5} x^{5}-657920 d^{6} e^{4} x^{4}-587664 d^{7} e^{3} x^{3}+89600 d^{8} e^{2} x^{2}+266112 d^{9} e x +80640 d^{10}\right )}{887040 x^{11} d^{3}}+\frac {19 e^{11} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{256 d^{2} \sqrt {d^{2}}}\) | \(176\) |
default | \(d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 d^{2} x^{11}}+\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 d^{2} x^{9}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 d^{4} x^{7}}\right )}{11 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 d^{2} x^{10}}+\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )}{10 d^{2}}\right )+e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 d^{2} x^{9}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 d^{4} x^{7}}\right )\) | \(626\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 276, normalized size = 1.09 \begin {gather*} \frac {19 \, e^{11} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{256 \, d^{3}} - \frac {19 \, \sqrt {-x^{2} e^{2} + d^{2}} e^{11}}{256 \, d^{4}} - \frac {19 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{11}}{768 \, d^{6}} - \frac {19 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{11}}{1280 \, d^{8}} - \frac {19 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{9}}{1280 \, d^{8} x^{2}} + \frac {19 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{7}}{1920 \, d^{6} x^{4}} - \frac {19 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{480 \, d^{4} x^{6}} - \frac {74 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{693 \, d^{3} x^{7}} - \frac {19 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{80 \, d^{2} x^{8}} - \frac {37 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{99 \, d x^{9}} - \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{10 \, x^{10}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{11 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.64, size = 153, normalized size = 0.60 \begin {gather*} -\frac {65835 \, x^{11} e^{11} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (94720 \, x^{10} e^{10} + 65835 \, d x^{9} e^{9} + 47360 \, d^{2} x^{8} e^{8} - 251790 \, d^{3} x^{7} e^{7} - 629760 \, d^{4} x^{6} e^{6} - 201432 \, d^{5} x^{5} e^{5} + 657920 \, d^{6} x^{4} e^{4} + 587664 \, d^{7} x^{3} e^{3} - 89600 \, d^{8} x^{2} e^{2} - 266112 \, d^{9} x e - 80640 \, d^{10}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{887040 \, d^{3} x^{11}} \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 = 154.53, size = 2397, normalized size = 9.44 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 744 vs.
\(2 (203) = 406\).
time = 0.57, size = 744, normalized size = 2.93 \begin {gather*} \frac {x^{11} {\left (\frac {4158 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{9}}{x} + \frac {8470 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{7}}{x^{2}} - \frac {3465 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{5}}{x^{3}} - \frac {40590 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{3}}{x^{4}} - \frac {57750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e}{x^{5}} + \frac {6930 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-1\right )}}{x^{6}} + \frac {138600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-3\right )}}{x^{7}} + \frac {244860 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-5\right )}}{x^{8}} + \frac {152460 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-7\right )}}{x^{9}} - \frac {568260 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{\left (-9\right )}}{x^{10}} + 630 \, e^{11}\right )} e^{22}}{14192640 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{11} d^{3}} + \frac {19 \, e^{11} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{256 \, d^{3}} + \frac {\frac {568260 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{30} e^{9}}{x} - \frac {152460 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{30} e^{7}}{x^{2}} - \frac {244860 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{30} e^{5}}{x^{3}} - \frac {138600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{30} e^{3}}{x^{4}} - \frac {6930 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{30} e}{x^{5}} + \frac {57750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{30} e^{\left (-1\right )}}{x^{6}} + \frac {40590 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{30} e^{\left (-3\right )}}{x^{7}} + \frac {3465 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{30} e^{\left (-5\right )}}{x^{8}} - \frac {8470 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} d^{30} e^{\left (-7\right )}}{x^{9}} - \frac {4158 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} d^{30} e^{\left (-9\right )}}{x^{10}} - \frac {630 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{11} d^{30} e^{\left (-11\right )}}{x^{11}}}{14192640 \, d^{33}} \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 \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{12}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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