Optimal. Leaf size=146 \[ -\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (30 d-41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {852, 1805, 807, 266, 63, 208} \[ -\frac {e (30 d-41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac {(d-e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2+10 d e x-8 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^2-30 d e x+26 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (30 d-41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^2+30 d e x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (30 d-41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {(2 e) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^5}\\ &=-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (30 d-41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {e \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d^5}\\ &=-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (30 d-41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {2 \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 e}\\ &=-\frac {2 e (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (10 d-13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (30 d-41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 112, normalized size = 0.77 \[ \frac {30 e \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (15 d^4+76 d^3 e x+32 d^2 e^2 x^2-82 d e^3 x^3-56 e^4 x^4\right )}{x (e x-d) (d+e x)^3}-30 e \log (x)}{15 d^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 194, normalized size = 1.33 \[ -\frac {46 \, e^{5} x^{5} + 92 \, d e^{4} x^{4} - 92 \, d^{3} e^{2} x^{2} - 46 \, d^{4} e x + 30 \, {\left (e^{5} x^{5} + 2 \, d e^{4} x^{4} - 2 \, d^{3} e^{2} x^{2} - d^{4} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (56 \, e^{4} x^{4} + 82 \, d e^{3} x^{3} - 32 \, d^{2} e^{2} x^{2} - 76 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{6} e^{4} x^{5} + 2 \, d^{7} e^{3} x^{4} - 2 \, d^{9} e x^{2} - d^{10} x\right )}} \]
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 = 234, normalized size = 1.60 \[ \frac {2 e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{5}}+\frac {2 e^{2} x}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}}+\frac {26 e^{2} x}{15 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{6}}-\frac {1}{5 \left (x +\frac {d}{e}\right )^{2} \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3} e}-\frac {13}{15 \left (x +\frac {d}{e}\right ) \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{4}}-\frac {2 e}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{5}}-\frac {1}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} x} \]
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 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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