Optimal. Leaf size=183 \[ \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {19 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^5}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}-\frac {19 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^5} \]
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^3 (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {8 e^2 (d-e x)}{5 d \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+32 d e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \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^2-104 d e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \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}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {\int \frac {-120 d^5 e+285 d^4 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^8}\\ &=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {\left (19 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^4}\\ &=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {\left (19 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^4}\\ &=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {19 \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^4}\\ &=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {19 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^5}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 107, normalized size = 0.58 \[ \frac {-285 e^2 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (-15 d^4+75 d^3 e x+713 d^2 e^2 x^2+1059 d e^3 x^3+448 e^4 x^4\right )}{x^2 (d+e x)^3}+285 e^2 \log (x)}{30 d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 202, normalized size = 1.10 \[ \frac {398 \, e^{5} x^{5} + 1194 \, d e^{4} x^{4} + 1194 \, d^{2} e^{3} x^{3} + 398 \, d^{3} e^{2} x^{2} + 285 \, {\left (e^{5} x^{5} + 3 \, d e^{4} x^{4} + 3 \, d^{2} e^{3} x^{3} + d^{3} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (448 \, e^{4} x^{4} + 1059 \, d e^{3} x^{3} + 713 \, d^{2} e^{2} x^{2} + 75 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{5} e^{3} x^{5} + 3 \, d^{6} e^{2} x^{4} + 3 \, d^{7} e x^{3} + d^{8} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 1, normalized size = 0.01 \[ +\infty \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 389, normalized size = 2.13 \[ -\frac {19 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}\, d^{4}}-\frac {4 e^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, d^{5}}+\frac {4 e^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, d^{5}}+\frac {4 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{3} x}{d^{7}}+\frac {19 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2}}{2 d^{6}}-\frac {4 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{2}}{d^{6}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}}}{5 \left (x +\frac {d}{e}\right )^{4} d^{4} e^{2}}+\frac {16 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}}}{15 \left (x +\frac {d}{e}\right )^{3} d^{5} e}+\frac {6 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}}}{\left (x +\frac {d}{e}\right )^{2} d^{6}}+\frac {4 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{d^{7} x}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{6} x^{2}} \]
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 \[ \int \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {d^2-e^2\,x^2}}{x^3\,{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{3} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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