Optimal. Leaf size=154 \[ \frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^7}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}+\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {857, 823, 807, 266, 63, 208} \[ \frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^7} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 857
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-6 d e^2+5 e^3 x}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-24 d^3 e^4+15 d^2 e^5 x}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-48 d^5 e^6+15 d^4 e^7 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}-\frac {e \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^6}\\ &=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}-\frac {e \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^6}\\ &=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\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^6 e}\\ &=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^7}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 122, normalized size = 0.79 \[ -\frac {-15 e \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (15 d^5+38 d^4 e x-52 d^3 e^2 x^2-87 d^2 e^3 x^3+33 d e^4 x^4+48 e^5 x^5\right )}{x (d-e x)^2 (d+e x)^3}+15 e \log (x)}{15 d^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 265, normalized size = 1.72 \[ -\frac {23 \, e^{6} x^{6} + 23 \, d e^{5} x^{5} - 46 \, d^{2} e^{4} x^{4} - 46 \, d^{3} e^{3} x^{3} + 23 \, d^{4} e^{2} x^{2} + 23 \, d^{5} e x + 15 \, {\left (e^{6} x^{6} + d e^{5} x^{5} - 2 \, d^{2} e^{4} x^{4} - 2 \, d^{3} e^{3} x^{3} + d^{4} e^{2} x^{2} + d^{5} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (48 \, e^{5} x^{5} + 33 \, d e^{4} x^{4} - 87 \, d^{2} e^{3} x^{3} - 52 \, d^{3} e^{2} x^{2} + 38 \, d^{4} e x + 15 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{7} e^{5} x^{6} + d^{8} e^{4} x^{5} - 2 \, d^{9} e^{3} x^{4} - 2 \, d^{10} e^{2} x^{3} + d^{11} e x^{2} + d^{12} x\right )}} \]
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 = 268, normalized size = 1.74 \[ \frac {4 e^{2} x}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5}}+\frac {4 e^{2} x}{15 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{5}}-\frac {1}{5 \left (x +\frac {d}{e}\right ) \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{3}}-\frac {e}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{4}}-\frac {1}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3} x}+\frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{6}}+\frac {8 e^{2} x}{3 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7}}+\frac {8 e^{2} x}{15 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{7}}-\frac {e}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}} \]
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 \[ \int \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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