Optimal. Leaf size=186 \[ \frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8} \]
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Rubi [A] time = 0.168233, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8} \]
Antiderivative was successfully verified.
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Rule 857
Rule 823
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-7 d e^2+6 e^3 x}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-35 d^3 e^4+24 d^2 e^5 x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-105 d^5 e^6+48 d^4 e^7 x}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{\int \frac{-96 d^6 e^7+105 d^5 e^8 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^{12} e^6}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}+\frac{\left (7 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^7}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}+\frac{\left (7 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^7}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \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^7}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8}\\ \end{align*}
Mathematica [A] time = 0.143633, size = 137, normalized size = 0.74 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (176 d^4 e^2 x^2-4 d^3 e^3 x^3-249 d^2 e^4 x^4+15 d^5 e x-15 d^6-9 d e^5 x^5+96 e^6 x^6\right )}{x^2 (d-e x)^2 (d+e x)^3}-105 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+105 e^2 \log (x)}{30 d^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 298, normalized size = 1.6 \begin{align*}{\frac{7\,{e}^{2}}{6\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{e}^{2}}{2\,{d}^{7}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{7\,{e}^{2}}{2\,{d}^{7}}\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{e}{5\,{d}^{4}} \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}^{3}x}{15\,{d}^{6}} \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}^{3}x}{15\,{d}^{8}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{1}{2\,{d}^{3}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{e}{{d}^{4}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{e}^{3}x}{3\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{e}^{3}x}{3\,{d}^{8}}{\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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17023, size = 595, normalized size = 3.2 \begin{align*} \frac{116 \, e^{7} x^{7} + 116 \, d e^{6} x^{6} - 232 \, d^{2} e^{5} x^{5} - 232 \, d^{3} e^{4} x^{4} + 116 \, d^{4} e^{3} x^{3} + 116 \, d^{5} e^{2} x^{2} + 105 \,{\left (e^{7} x^{7} + d e^{6} x^{6} - 2 \, d^{2} e^{5} x^{5} - 2 \, d^{3} e^{4} x^{4} + d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (96 \, e^{6} x^{6} - 9 \, d e^{5} x^{5} - 249 \, d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} + 176 \, d^{4} e^{2} x^{2} + 15 \, d^{5} e x - 15 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{8} e^{5} x^{7} + d^{9} e^{4} x^{6} - 2 \, d^{10} e^{3} x^{5} - 2 \, d^{11} e^{2} x^{4} + d^{12} e x^{3} + d^{13} x^{2}\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^{3} \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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