Optimal. Leaf size=196 \[ -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}+\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2} \]
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Rubi [A] time = 0.52, antiderivative size = 196, 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 = {852, 1805, 1807, 807, 266, 63, 208} \begin {gather*} -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 1807
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x^6 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-d^4+4 d^3 e x-7 d^2 e^2 x^2+8 d e^3 x^3-8 e^4 x^4+\frac {8 e^5 x^5}{d}}{x^6 \sqrt {d^2-e^2 x^2}} \, dx}{d^2}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {\int \frac {-20 d^5 e+39 d^4 e^2 x-40 d^3 e^3 x^2+40 d^2 e^4 x^3-40 d e^5 x^4}{x^5 \sqrt {d^2-e^2 x^2}} \, dx}{5 d^4}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {\int \frac {-156 d^6 e^2+220 d^5 e^3 x-160 d^4 e^4 x^2+160 d^3 e^5 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{20 d^6}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {\int \frac {-660 d^7 e^3+792 d^6 e^4 x-480 d^5 e^5 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{60 d^8}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {\int \frac {-1584 d^8 e^4+1620 d^7 e^5 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{120 d^{10}}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {\left (27 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^3}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {\left (27 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^3}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {\left (27 e^3\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 )}{2 d^3}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 118, normalized size = 0.60 \begin {gather*} -\frac {-135 e^5 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (2 d^5-8 d^4 e x+16 d^3 e^2 x^2-29 d^2 e^3 x^3+77 d e^4 x^4+212 e^5 x^5\right )}{x^5 (d+e x)}+135 e^5 \log (x)}{10 d^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.06, size = 131, normalized size = 0.67 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-2 d^5+8 d^4 e x-16 d^3 e^2 x^2+29 d^2 e^3 x^3-77 d e^4 x^4-212 e^5 x^5\right )}{10 d^4 x^5 (d+e x)}-\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 147, normalized size = 0.75 \begin {gather*} -\frac {80 \, e^{6} x^{6} + 80 \, d e^{5} x^{5} + 135 \, {\left (e^{6} x^{6} + d e^{5} x^{5}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (212 \, e^{5} x^{5} + 77 \, d e^{4} x^{4} - 29 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} - 8 \, d^{4} e x + 2 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{10 \, {\left (d^{4} e x^{6} + d^{5} x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 628, normalized size = 3.20 \begin {gather*} \frac {27 e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}\, d^{3}}+\frac {333 e^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}-\frac {333 e^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}-\frac {333 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x}{8 d^{6}}+\frac {333 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{6} x}{8 d^{6}}-\frac {27 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{5}}{2 d^{5}}-\frac {111 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6} x}{4 d^{8}}+\frac {111 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{6} x}{4 d^{8}}-\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}{2 d^{7}}-\frac {111 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6} x}{5 d^{10}}-\frac {27 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}{10 d^{9}}+\frac {111 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{5}}{5 d^{9}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e}{\left (x +\frac {d}{e}\right )^{4} d^{7}}+\frac {3 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{2}}{\left (x +\frac {d}{e}\right )^{3} d^{8}}+\frac {11 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{3}}{\left (x +\frac {d}{e}\right )^{2} d^{9}}-\frac {111 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{5 d^{10} x}+\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{2 d^{9} x^{2}}-\frac {16 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{5 d^{8} x^{3}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{d^{7} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{6}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^6\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{x^{6} \left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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