Optimal. Leaf size=201 \[ -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4} \]
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Rubi [A] time = 0.16, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {835, 807, 266, 47, 63, 208} \[ \frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rubi steps
\begin {align*} \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {\int \frac {\left (-8 d^2 e-3 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx}{8 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}+\frac {\int \frac {\left (21 d^3 e^2+16 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {\int \frac {\left (-96 d^4 e^3-21 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{336 d^6}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {e^4 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{16 d^3}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {e^4 \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{32 d^3}\\ &=-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {\left (3 e^6\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{128 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {\left (3 e^8\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{256 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {\left (3 e^6\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 )}{128 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 73, normalized size = 0.36 \[ -\frac {e \left (d^2-e^2 x^2\right )^{5/2} \left (5 d^7+2 d^5 e^2 x^2+7 e^7 x^7 \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};1-\frac {e^2 x^2}{d^2}\right )\right )}{35 d^9 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 131, normalized size = 0.65 \[ \frac {105 \, e^{8} x^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (256 \, e^{7} x^{7} + 105 \, d e^{6} x^{6} + 128 \, d^{2} e^{5} x^{5} + 70 \, d^{3} e^{4} x^{4} - 1024 \, d^{4} e^{3} x^{3} - 840 \, d^{5} e^{2} x^{2} + 640 \, d^{6} e x + 560 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4480 \, d^{4} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 431, normalized size = 2.14 \[ \frac {x^{8} {\left (\frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{16}}{x} - \frac {112 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{12}}{x^{3}} - \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{10}}{x^{4}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{8}}{x^{5}} + \frac {1680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{4}}{x^{7}} + 35 \, e^{18}\right )} e^{6}}{71680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{4}} - \frac {3 \, e^{8} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{128 \, d^{4}} - \frac {{\left (\frac {1680 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{28} e^{86}}{x} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{28} e^{82}}{x^{3}} - \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{28} e^{80}}{x^{4}} - \frac {112 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{28} e^{78}}{x^{5}} + \frac {80 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{28} e^{74}}{x^{7}} + \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{28} e^{72}}{x^{8}}\right )} e^{\left (-80\right )}}{71680 \, d^{32}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 236, normalized size = 1.17 \[ -\frac {3 e^{8} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}\, d^{3}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8}}{128 d^{5}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8}}{128 d^{7}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}{128 d^{7} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}{64 d^{5} x^{4}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}{35 d^{4} x^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}{16 d^{3} x^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}{7 d^{2} x^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 d \,x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 230, normalized size = 1.14 \[ -\frac {3 \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{128 \, d^{4}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}}{128 \, d^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{128 \, d^{7}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}{128 \, d^{7} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{64 \, d^{5} x^{4}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{35 \, d^{4} x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{16 \, d^{3} x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{7 \, d^{2} x^{7}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{8 \, d x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.04, size = 212, normalized size = 1.05 \[ \frac {3\,d^3\,\sqrt {d^2-e^2\,x^2}}{128\,x^8}-\frac {11\,d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{128\,x^8}-\frac {11\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{128\,d\,x^8}+\frac {3\,{\left (d^2-e^2\,x^2\right )}^{7/2}}{128\,d^3\,x^8}+\frac {8\,e^3\,\sqrt {d^2-e^2\,x^2}}{35\,x^5}-\frac {e^5\,\sqrt {d^2-e^2\,x^2}}{35\,d^2\,x^3}-\frac {2\,e^7\,\sqrt {d^2-e^2\,x^2}}{35\,d^4\,x}-\frac {d^2\,e\,\sqrt {d^2-e^2\,x^2}}{7\,x^7}+\frac {e^8\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,3{}\mathrm {i}}{128\,d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 22.55, size = 1159, normalized size = 5.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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