Optimal. Leaf size=154 \[ \frac{6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 d^7 x}+\frac{8 d-5 e x}{5 d^6 x \sqrt{d^2-e^2 x^2}}+\frac{1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^7} \]
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Rubi [A] time = 0.135612, antiderivative size = 154, 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, 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 857
Rule 823
Rule 807
Rule 266
Rule 63
Rule 208
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*}
Mathematica [A] time = 0.133733, size = 122, normalized size = 0.79 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (-52 d^3 e^2 x^2-87 d^2 e^3 x^3+38 d^4 e x+15 d^5+33 d e^4 x^4+48 e^5 x^5\right )}{x (d-e x)^2 (d+e x)^3}-15 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 e \log (x)}{15 d^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 268, normalized size = 1.7 \begin{align*} -{\frac{e}{3\,{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{e}{{d}^{6}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+{\frac{e}{{d}^{6}}\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\,{d}^{3}} \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\,{e}^{2}x}{15\,{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{e}^{2}x}{15\,{d}^{7}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{1}{{d}^{3}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{4\,{e}^{2}x}{3\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{e}^{2}x}{3\,{d}^{7}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94252, size = 549, normalized size = 3.56 \begin{align*} -\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 )}} \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^{2} \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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