Optimal. Leaf size=119 \[ \frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6} \]
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Rubi [A] time = 0.10675, antiderivative size = 119, normalized size of antiderivative = 1., 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, 12, 266, 63, 208} \[ \frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6} \]
Antiderivative was successfully verified.
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Rule 857
Rule 823
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-5 d e^2+4 e^3 x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-15 d^3 e^4+8 d^2 e^5 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\int -\frac{15 d^5 e^6}{x \sqrt{d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{d^5}\\ &=\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-8 e x}{15 d^6 \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^5}\\ &=\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-8 e x}{15 d^6 \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^5 e^2}\\ &=\frac{5 d-4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{15 d-8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}\\ \end{align*}
Mathematica [A] time = 0.095976, size = 106, normalized size = 0.89 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-27 d^2 e^2 x^2+8 d^3 e x+23 d^4-7 d e^3 x^3+8 e^4 x^4\right )}{(d-e x)^2 (d+e x)^3}-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 \log (x)}{15 d^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 196, normalized size = 1.7 \begin{align*}{\frac{1}{3\,{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{d}^{5}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{5}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{1}{5\,e{d}^{2}} \left ({\frac{d}{e}}+x \right ) ^{-1} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,ex}{15\,{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,ex}{15\,{d}^{6}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.70179, size = 493, normalized size = 4.14 \begin{align*} \frac{23 \, e^{5} x^{5} + 23 \, d e^{4} x^{4} - 46 \, d^{2} e^{3} x^{3} - 46 \, d^{3} e^{2} x^{2} + 23 \, d^{4} e x + 23 \, d^{5} + 15 \,{\left (e^{5} x^{5} + d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} - 2 \, d^{3} e^{2} x^{2} + d^{4} e x + d^{5}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (8 \, e^{4} x^{4} - 7 \, d e^{3} x^{3} - 27 \, d^{2} e^{2} x^{2} + 8 \, d^{3} e x + 23 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{6} e^{5} x^{5} + d^{7} e^{4} x^{4} - 2 \, d^{8} e^{3} x^{3} - 2 \, d^{9} e^{2} x^{2} + d^{10} e x + d^{11}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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