Optimal. Leaf size=210 \[ -\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.49, antiderivative size = 210, 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} \[ -\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \]
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 {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^4+20 d^3 e x-35 d^2 e^2 x^2+40 d e^3 x^3-32 e^4 x^4}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^4-60 d^3 e x+120 d^2 e^2 x^2-180 d e^3 x^3+144 e^4 x^4}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^4+60 d^3 e x-135 d^2 e^2 x^2+240 d e^3 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {\int \frac {-180 d^5 e+435 d^4 e^2 x-720 d^3 e^3 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^8}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\int \frac {-870 d^6 e^2+1620 d^5 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{10}}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {\left (18 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {\left (9 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {(18 e) \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 )}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 118, normalized size = 0.56 \[ -\frac {-270 e^3 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (5 d^5-15 d^4 e x+70 d^3 e^2 x^2+674 d^2 e^3 x^3+1002 d e^4 x^4+424 e^5 x^5\right )}{x^3 (d+e x)^3}+270 e^3 \log (x)}{15 d^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 213, normalized size = 1.01 \[ -\frac {324 \, e^{6} x^{6} + 972 \, d e^{5} x^{5} + 972 \, d^{2} e^{4} x^{4} + 324 \, d^{3} e^{3} x^{3} + 270 \, {\left (e^{6} x^{6} + 3 \, d e^{5} x^{5} + 3 \, d^{2} e^{4} x^{4} + d^{3} e^{3} x^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (424 \, e^{5} x^{5} + 1002 \, d e^{4} x^{4} + 674 \, d^{2} e^{3} x^{3} + 70 \, d^{3} e^{2} x^{2} - 15 \, d^{4} e x + 5 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{6} e^{3} x^{6} + 3 \, d^{7} e^{2} x^{5} + 3 \, d^{8} e x^{4} + d^{9} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 1, normalized size = 0.00 \[ +\infty \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 412, normalized size = 1.96 \[ \frac {18 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{5}}+\frac {10 e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, d^{6}}-\frac {10 e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, d^{6}}-\frac {10 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{4} x}{d^{8}}-\frac {18 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}{d^{7}}+\frac {10 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{3}}{d^{7}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}}}{5 \left (x +\frac {d}{e}\right )^{4} d^{5} e}-\frac {7 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}}}{5 \left (x +\frac {d}{e}\right )^{3} d^{6}}-\frac {10 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e}{\left (x +\frac {d}{e}\right )^{2} d^{7}}-\frac {10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}{d^{8} x}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{d^{7} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{6} x^{3}} \]
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 \[ \int \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x^{4}}\,{d x} \]
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 \[ \int \frac {\sqrt {d^2-e^2\,x^2}}{x^4\,{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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