Optimal. Leaf size=183 \[ \frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^7 x}-\frac {9 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^7}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.37, 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} \[ \frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^7 x}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}-\frac {9 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^7} \]
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 {1}{x^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac {(d-e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2+10 d e x-10 e^2 x^2+\frac {8 e^3 x^3}{d}}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^2-30 d e x+45 e^2 x^2-\frac {36 e^3 x^3}{d}}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^2+30 d e x-60 e^2 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}+\frac {\int \frac {-60 d^3 e+135 d^2 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^8}\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^7 x}+\frac {\left (9 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^6}\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^7 x}+\frac {\left (9 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^6}\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^7 x}-\frac {9 \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^6}\\ &=\frac {2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (10 d-11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^7 x}-\frac {9 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^7}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 127, normalized size = 0.69 \[ \frac {-45 e^2 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (5 d^5-10 d^4 e x-94 d^3 e^2 x^2-58 d^2 e^3 x^3+83 d e^4 x^4+64 e^5 x^5\right )}{x^2 (e x-d) (d+e x)^3}+45 e^2 \log (x)}{10 d^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 215, normalized size = 1.17 \[ \frac {54 \, e^{6} x^{6} + 108 \, d e^{5} x^{5} - 108 \, d^{3} e^{3} x^{3} - 54 \, d^{4} e^{2} x^{2} + 45 \, {\left (e^{6} x^{6} + 2 \, d e^{5} x^{5} - 2 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (64 \, e^{5} x^{5} + 83 \, d e^{4} x^{4} - 58 \, d^{2} e^{3} x^{3} - 94 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x + 5 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{10 \, {\left (d^{7} e^{4} x^{6} + 2 \, d^{8} e^{3} x^{5} - 2 \, d^{10} e x^{3} - d^{11} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 259, normalized size = 1.42 \[ -\frac {9 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}\, d^{6}}-\frac {4 e^{3} x}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{7}}-\frac {12 e^{3} x}{5 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{7}}+\frac {1}{5 \left (x +\frac {d}{e}\right )^{2} \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{4}}+\frac {6 e}{5 \left (x +\frac {d}{e}\right ) \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{5}}+\frac {9 e^{2}}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}}+\frac {2 e}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} x}-\frac {1}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} x^{2}} \]
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 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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