Optimal. Leaf size=152 \[ \frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.13, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {857, 823, 835, 807, 266, 63, 208} \[ \frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 835
Rule 857
Rubi steps
\begin {align*} \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-5 d e^2+4 e^3 x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^3 e^4+8 d^2 e^5 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^6 e^4}\\ &=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {\int \frac {-16 d^4 e^5+15 d^3 e^6 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^8 e^4}\\ &=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {\left (5 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^5}\\ &=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {\left (5 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^5}\\ &=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 \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^5}\\ &=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 115, normalized size = 0.76 \[ \frac {-15 e^2 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (3 d^4-3 d^3 e x-23 d^2 e^2 x^2+d e^3 x^3+16 e^4 x^4\right )}{x^2 (e x-d) (d+e x)^2}+15 e^2 \log (x)}{6 d^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 201, normalized size = 1.32 \[ \frac {14 \, e^{5} x^{5} + 14 \, d e^{4} x^{4} - 14 \, d^{2} e^{3} x^{3} - 14 \, d^{3} e^{2} x^{2} + 15 \, {\left (e^{5} x^{5} + d e^{4} x^{4} - d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{4} x^{4} + d e^{3} x^{3} - 23 \, d^{2} e^{2} x^{2} - 3 \, d^{3} e x + 3 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, {\left (d^{6} e^{3} x^{5} + d^{7} e^{2} x^{4} - d^{8} e x^{3} - d^{9} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 216, normalized size = 1.42 \[ -\frac {5 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}\, d^{5}}-\frac {2 e^{3} x}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}}-\frac {2 e^{3} x}{3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{6}}+\frac {e}{3 \left (x +\frac {d}{e}\right ) \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{4}}+\frac {5 e^{2}}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5}}+\frac {e}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} x}-\frac {1}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} x^{2}} \]
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 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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