Optimal. Leaf size=187 \[ -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {55 e^9 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d}-\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6} \]
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Rubi [A] time = 0.26, antiderivative size = 187, 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 = {1807, 807, 266, 47, 63, 208} \[ -\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}+\frac {55 e^9 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-27 d^4 e-29 d^3 e^2 x-9 d^2 e^3 x^2\right )}{x^9} \, dx}{9 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}+\frac {\int \frac {\left (232 d^5 e^2+99 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{72 d^4}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{8} \left (11 e^3\right ) \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{16} \left (11 e^3\right ) \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac {1}{96} \left (55 e^5\right ) \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{128} \left (55 e^7\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}-\frac {1}{256} \left (55 e^9\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )\\ &=-\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {1}{128} \left (55 e^7\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 )\\ &=-\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {55 e^9 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 218, normalized size = 1.17 \[ \frac {-112 d^{10}-16 d^8 e^2 x^2-168 d^7 e^3 x^3+1184 d^6 e^4 x^4+714 d^5 e^5 x^5-2336 d^4 e^6 x^6-1239 d^3 e^7 x^7+1744 d^2 e^8 x^8+315 d e^9 x^9 \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )+693 d e^9 x^9-464 e^{10} x^{10}}{1008 d x^9 \sqrt {d^2-e^2 x^2}}-\frac {3 e^9 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )}{7 d^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 142, normalized size = 0.76 \[ -\frac {3465 \, e^{9} x^{9} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (3712 \, e^{8} x^{8} - 4599 \, d e^{7} x^{7} - 10240 \, d^{2} e^{6} x^{6} - 3066 \, d^{3} e^{5} x^{5} + 8448 \, d^{4} e^{4} x^{4} + 7224 \, d^{5} e^{3} x^{3} - 1024 \, d^{6} e^{2} x^{2} - 3024 \, d^{7} e x - 896 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8064 \, d x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 620, normalized size = 3.32 \[ \frac {x^{9} {\left (\frac {189 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{18}}{x} + \frac {324 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{16}}{x^{2}} - \frac {672 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{14}}{x^{3}} - \frac {3024 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{12}}{x^{4}} - \frac {1512 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{10}}{x^{5}} + \frac {9744 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{8}}{x^{6}} + \frac {18144 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{6}}{x^{7}} - \frac {16632 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{4}}{x^{8}} + 28 \, e^{20}\right )} e^{7}}{129024 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} d} + \frac {55 \, e^{9} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{128 \, d} + \frac {{\left (\frac {16632 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{8} e^{106}}{x} - \frac {18144 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8} e^{104}}{x^{2}} - \frac {9744 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{8} e^{102}}{x^{3}} + \frac {1512 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{8} e^{100}}{x^{4}} + \frac {3024 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{8} e^{98}}{x^{5}} + \frac {672 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{8} e^{96}}{x^{6}} - \frac {324 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{8} e^{94}}{x^{7}} - \frac {189 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{8} e^{92}}{x^{8}} - \frac {28 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} d^{8} e^{90}}{x^{9}}\right )} e^{\left (-99\right )}}{129024 \, d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 250, normalized size = 1.34 \[ \frac {55 e^{9} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}-\frac {55 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{9}}{128 d^{2}}-\frac {55 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{9}}{384 d^{4}}-\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{9}}{128 d^{6}}-\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{7}}{128 d^{6} x^{2}}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{192 d^{4} x^{4}}-\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{48 d^{2} x^{6}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{63 d \,x^{7}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{8 x^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{9 x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 247, normalized size = 1.32 \[ \frac {55 \, e^{9} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{128 \, d} - \frac {55 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{9}}{128 \, d^{2}} - \frac {55 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9}}{384 \, d^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{9}}{128 \, d^{6}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{7}}{128 \, d^{6} x^{2}} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{192 \, d^{4} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{48 \, d^{2} x^{6}} - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{63 \, d x^{7}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{8 \, x^{8}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{9 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{10}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 36.71, size = 1889, normalized size = 10.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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