Optimal. Leaf size=254 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}+\frac {19 e^{11} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7} \]
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Rubi [A] time = 0.33, 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 = {1807, 835, 807, 266, 47, 63, 208} \[ -\frac {19 e^9 \sqrt {d^2-e^2 x^2}}{256 d^2 x^2}+\frac {19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac {19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac {74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac {19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac {37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}+\frac {19 e^{11} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 1807
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [C] time = 0.06, size = 112, normalized size = 0.44 \[ -\frac {\left (d^2-e^2 x^2\right )^{7/2} \left (63 d^{11}+259 d^9 e^2 x^2+74 d^7 e^4 x^4+99 e^{11} x^{11} \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )+297 e^{11} x^{11} \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )\right )}{693 d^{10} x^{11}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 164, normalized size = 0.65 \[ -\frac {65835 \, e^{11} x^{11} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (94720 \, e^{10} x^{10} + 65835 \, d e^{9} x^{9} + 47360 \, d^{2} e^{8} x^{8} - 251790 \, d^{3} e^{7} x^{7} - 629760 \, d^{4} e^{6} x^{6} - 201432 \, d^{5} e^{5} x^{5} + 657920 \, d^{6} e^{4} x^{4} + 587664 \, d^{7} e^{3} x^{3} - 89600 \, d^{8} e^{2} x^{2} - 266112 \, d^{9} e x - 80640 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{887040 \, d^{3} x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 746, normalized size = 2.94 \[ \frac {x^{11} {\left (\frac {4158 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{22}}{x} + \frac {8470 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{20}}{x^{2}} - \frac {3465 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{18}}{x^{3}} - \frac {40590 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{16}}{x^{4}} - \frac {57750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{14}}{x^{5}} + \frac {6930 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{12}}{x^{6}} + \frac {138600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{10}}{x^{7}} + \frac {244860 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{8}}{x^{8}} + \frac {152460 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{6}}{x^{9}} - \frac {568260 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} e^{4}}{x^{10}} + 630 \, e^{24}\right )} e^{9}}{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 {{\left (\frac {568260 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{30} e^{152}}{x} - \frac {152460 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{30} e^{150}}{x^{2}} - \frac {244860 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{30} e^{148}}{x^{3}} - \frac {138600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{30} e^{146}}{x^{4}} - \frac {6930 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{30} e^{144}}{x^{5}} + \frac {57750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{30} e^{142}}{x^{6}} + \frac {40590 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{30} e^{140}}{x^{7}} + \frac {3465 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{30} e^{138}}{x^{8}} - \frac {8470 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} d^{30} e^{136}}{x^{9}} - \frac {4158 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} d^{30} e^{134}}{x^{10}} - \frac {630 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{11} d^{30} e^{132}}{x^{11}}\right )} e^{\left (-143\right )}}{14192640 \, d^{33}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 303, normalized size = 1.19 \[ \frac {19 e^{11} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{256 \sqrt {d^{2}}\, d^{2}}-\frac {19 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{11}}{256 d^{4}}-\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{11}}{768 d^{6}}-\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{11}}{1280 d^{8}}-\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{9}}{1280 d^{8} x^{2}}+\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{7}}{1920 d^{6} x^{4}}-\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{480 d^{4} x^{6}}-\frac {74 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{693 d^{3} x^{7}}-\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{80 d^{2} x^{8}}-\frac {37 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{99 d \,x^{9}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{10 x^{10}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{11 x^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 297, normalized size = 1.17 \[ \frac {19 \, e^{11} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{256 \, d^{3}} - \frac {19 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{11}}{256 \, d^{4}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{11}}{768 \, d^{6}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{11}}{1280 \, d^{8}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{9}}{1280 \, d^{8} x^{2}} + \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{7}}{1920 \, d^{6} x^{4}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{480 \, d^{4} x^{6}} - \frac {74 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{693 \, d^{3} x^{7}} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{80 \, d^{2} x^{8}} - \frac {37 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{99 \, d x^{9}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{10 \, x^{10}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{11 \, x^{11}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{12}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 74.52, size = 2397, normalized size = 9.44 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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