Optimal. Leaf size=225 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}+\frac {33 e^{10} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2}-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7} \]
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Rubi [A] time = 0.30, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}+\frac {33 e^{10} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2} \]
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^{11}} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-30 d^4 e-33 d^3 e^2 x-10 d^2 e^3 x^2\right )}{x^{10}} \, dx}{10 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}+\frac {\int \frac {\left (297 d^5 e^2+150 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx}{90 d^4}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {\int \frac {\left (-1200 d^6 e^3-297 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{720 d^6}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^4\right ) \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{80 d}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^4\right ) \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )}{160 d}\\ &=-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac {\left (11 e^6\right ) \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{64 d}\\ &=\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^8\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{256 d}\\ &=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac {\left (33 e^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{512 d}\\ &=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^8\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}\\ &=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {33 e^{10} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 102, normalized size = 0.45 \[ -\frac {e \left (d^2-e^2 x^2\right )^{7/2} \left (7 d^9+5 d^7 e^2 x^2+9 e^9 x^9 \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )+3 e^9 x^9 \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )\right )}{21 d^9 x^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 153, normalized size = 0.68 \[ -\frac {3465 \, e^{10} x^{10} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (6400 \, e^{9} x^{9} + 3465 \, d e^{8} x^{8} - 10240 \, d^{2} e^{7} x^{7} - 24570 \, d^{3} e^{6} x^{6} - 7680 \, d^{4} e^{5} x^{5} + 23352 \, d^{5} e^{4} x^{4} + 20480 \, d^{6} e^{3} x^{3} - 3024 \, d^{7} e^{2} x^{2} - 8960 \, d^{8} e x - 2688 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{26880 \, d^{2} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 683, normalized size = 3.04 \[ \frac {x^{10} {\left (\frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{20}}{x} + \frac {525 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{18}}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{16}}{x^{3}} - \frac {3570 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{14}}{x^{4}} - \frac {3360 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{12}}{x^{5}} + \frac {5880 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{10}}{x^{6}} + \frac {16800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{8}}{x^{7}} + \frac {10500 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{6}}{x^{8}} - \frac {31920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{4}}{x^{9}} + 42 \, e^{22}\right )} e^{8}}{430080 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} d^{2}} + \frac {33 \, e^{10} \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^{2}} + \frac {{\left (\frac {31920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{18} e^{128}}{x} - \frac {10500 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{18} e^{126}}{x^{2}} - \frac {16800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{18} e^{124}}{x^{3}} - \frac {5880 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{18} e^{122}}{x^{4}} + \frac {3360 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{18} e^{120}}{x^{5}} + \frac {3570 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{18} e^{118}}{x^{6}} + \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{18} e^{116}}{x^{7}} - \frac {525 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{18} e^{114}}{x^{8}} - \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} d^{18} e^{112}}{x^{9}} - \frac {42 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} d^{18} e^{110}}{x^{10}}\right )} e^{\left (-120\right )}}{430080 \, d^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 278, normalized size = 1.24 \[ \frac {33 e^{10} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{256 \sqrt {d^{2}}\, d}-\frac {33 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{10}}{256 d^{3}}-\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{10}}{256 d^{5}}-\frac {33 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{10}}{1280 d^{7}}-\frac {33 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{8}}{1280 d^{7} x^{2}}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{640 d^{5} x^{4}}-\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{160 d^{3} x^{6}}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{21 d^{2} x^{7}}-\frac {33 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{80 d \,x^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{3 x^{9}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{10 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 272, normalized size = 1.21 \[ \frac {33 \, e^{10} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{256 \, d^{2}} - \frac {33 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{10}}{256 \, d^{3}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{10}}{256 \, d^{5}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{10}}{1280 \, d^{7}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{8}}{1280 \, d^{7} x^{2}} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{640 \, d^{5} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{160 \, d^{3} x^{6}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{21 \, d^{2} x^{7}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{80 \, d x^{8}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{3 \, x^{9}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{10 \, x^{10}} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{11}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 49.87, size = 2159, normalized size = 9.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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