Optimal. Leaf size=118 \[ \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {852, 1805, 823, 12, 266, 63, 208} \[ \frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 852
Rule 1805
Rubi steps
\begin {align*} \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac {(d-e x)^2}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2+8 d e x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-15 d^4 e^2+16 d^3 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^2}\\ &=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {15 d^6 e^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^4}\\ &=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4}\\ &=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\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^4 e^2}\\ &=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 95, normalized size = 0.81 \[ \frac {-15 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (26 d^3+22 d^2 e x-17 d e^2 x^2-16 e^3 x^3\right )}{(d-e x) (d+e x)^3}+15 \log (x)}{15 d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 168, normalized size = 1.42 \[ \frac {26 \, e^{4} x^{4} + 52 \, d e^{3} x^{3} - 52 \, d^{3} e x - 26 \, d^{4} + 15 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{3} - 2 \, d^{3} e x - d^{4}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{3} x^{3} + 17 \, d e^{2} x^{2} - 22 \, d^{2} e x - 26 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{5} e^{4} x^{4} + 2 \, d^{6} e^{3} x^{3} - 2 \, d^{8} e x - d^{9}\right )}} \]
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 = 0.01, size = 187, normalized size = 1.58 \[ -\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{4}}-\frac {16 e x}{15 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{5}}+\frac {1}{5 \left (x +\frac {d}{e}\right )^{2} \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{2} e^{2}}+\frac {8}{15 \left (x +\frac {d}{e}\right ) \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3} e}+\frac {1}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{4}} \]
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 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} x}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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