Optimal. Leaf size=143 \[ \frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac{e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^2} \]
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Rubi [A] time = 0.120188, antiderivative size = 143, 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 = {850, 835, 807, 266, 47, 63, 208} \[ \frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac{e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^2} \]
Antiderivative was successfully verified.
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Rule 850
Rule 835
Rule 807
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx &=\int \frac{(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac{\int \frac{\left (6 d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{6 d^2}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac{e^2 \int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{6 d}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{12 d}\\ &=-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{e^4 \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d}\\ &=\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac{e^6 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{32 d}\\ &=\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{e^4 \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 )}{16 d}\\ &=\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^2}\\ \end{align*}
Mathematica [A] time = 0.200641, size = 117, normalized size = 0.82 \[ \frac{\sqrt{d^2-e^2 x^2} \left (70 d^3 e^2 x^2-96 d^2 e^3 x^3+48 d^4 e x-40 d^5-15 d e^4 x^4+48 e^5 x^5\right )-15 e^6 x^6 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 e^6 x^6 \log (x)}{240 d^2 x^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.124, size = 521, normalized size = 3.6 \begin{align*} -{\frac{{e}^{6}}{16\,d}\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}{x}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{6}}{80\,{d}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{6}}{48\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{6}}{16\,{d}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{1}{6\,{d}^{3}{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{6}}{5\,{d}^{7}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{3}}{5\,{d}^{6}{x}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{5}}{5\,{d}^{8}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{7}x}{5\,{d}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{7}x}{4\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{7}x}{8\,{d}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{3\,{e}^{7}}{8\,{d}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{5\,{e}^{2}}{24\,{d}^{5}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{e}^{4}}{16\,{d}^{7}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{7}x}{4\,{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{3\,{e}^{7}x}{8\,{d}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{3\,{e}^{7}}{8\,{d}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72457, size = 232, normalized size = 1.62 \begin{align*} \frac{15 \, e^{6} x^{6} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (48 \, e^{5} x^{5} - 15 \, d e^{4} x^{4} - 96 \, d^{2} e^{3} x^{3} + 70 \, d^{3} e^{2} x^{2} + 48 \, d^{4} e x - 40 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \, d^{2} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 20.1386, size = 930, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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