Optimal. Leaf size=113 \[ -\frac {4 (-e)^{5/2} \sqrt {d+e x^2}}{45 d^3 x}-\frac {2 (-e)^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{30 d x^5} \]
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Rubi [A] time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5151, 271, 264} \[ -\frac {4 (-e)^{5/2} \sqrt {d+e x^2}}{45 d^3 x}-\frac {2 (-e)^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{30 d x^5}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 5151
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^7} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}+\frac {1}{6} \sqrt {-e} \int \frac {1}{x^6 \sqrt {d+e x^2}} \, dx\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{30 d x^5}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}+\frac {\left (2 (-e)^{3/2}\right ) \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{15 d}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{30 d x^5}-\frac {2 (-e)^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}+\frac {\left (4 (-e)^{5/2}\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{45 d^2}\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{30 d x^5}-\frac {2 (-e)^{3/2} \sqrt {d+e x^2}}{45 d^2 x^3}-\frac {4 (-e)^{5/2} \sqrt {d+e x^2}}{45 d^3 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{6 x^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 78, normalized size = 0.69 \[ \frac {\sqrt {-e} x \sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )-15 d^3 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{90 d^3 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 68, normalized size = 0.60 \[ -\frac {15 \, d^{3} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (8 \, e^{2} x^{5} - 4 \, d e x^{3} + 3 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{90 \, d^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 282, normalized size = 2.50 \[ \frac {x^{5} {\left (\frac {25 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{2} e^{\left (-1\right )}}{x^{2}} + \frac {150 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{4} e^{\left (-5\right )}}{x^{4}} + 3 \, e^{3}\right )} e^{10}}{2880 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{5} d^{3}} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{6 \, x^{6}} - \frac {{\left (\frac {150 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )} d^{12} e^{16}}{x} + \frac {25 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{3} d^{12} e^{12}}{x^{3}} + \frac {3 \, {\left (\sqrt {-x^{2} e^{2} - d e} e - \sqrt {-d e} e\right )}^{5} d^{12} e^{8}}{x^{5}}\right )} e^{\left (-15\right )}}{2880 \, d^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 117, normalized size = 1.04 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{6 x^{6}}+\frac {\sqrt {-e}\, e \sqrt {e \,x^{2}+d}}{18 d^{2} x^{3}}-\frac {\sqrt {-e}\, e^{2} \sqrt {e \,x^{2}+d}}{9 d^{3} x}-\frac {\sqrt {-e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{30 d^{2} x^{5}}+\frac {\sqrt {-e}\, e \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{45 d^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 109, normalized size = 0.96 \[ -\frac {{\left (2 \, e^{2} x^{4} + d e x^{2} - d^{2}\right )} \sqrt {-e} e}{18 \, \sqrt {e x^{2} + d} d^{3} x^{3}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{6 \, x^{6}} + \frac {{\left (2 \, e^{2} x^{4} - d e x^{2} - 3 \, d^{2}\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{90 \, d^{3} x^{5}} \]
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 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.50, size = 352, normalized size = 3.12 \[ - \frac {d^{4} e^{\frac {9}{2}} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{30 d^{5} e^{4} x^{4} + 60 d^{4} e^{5} x^{6} + 30 d^{3} e^{6} x^{8}} - \frac {d^{3} e^{\frac {11}{2}} x^{2} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{45 d^{5} e^{4} x^{4} + 90 d^{4} e^{5} x^{6} + 45 d^{3} e^{6} x^{8}} - \frac {d^{2} e^{\frac {13}{2}} x^{4} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{30 d^{5} e^{4} x^{4} + 60 d^{4} e^{5} x^{6} + 30 d^{3} e^{6} x^{8}} - \frac {2 d e^{\frac {15}{2}} x^{6} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{15 d^{5} e^{4} x^{4} + 30 d^{4} e^{5} x^{6} + 15 d^{3} e^{6} x^{8}} - \frac {4 e^{\frac {17}{2}} x^{8} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{45 d^{5} e^{4} x^{4} + 90 d^{4} e^{5} x^{6} + 45 d^{3} e^{6} x^{8}} - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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