Optimal. Leaf size=183 \[ \frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}-\frac{19 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^5} \]
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Rubi [A] time = 0.391562, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {852, 1805, 1807, 807, 266, 63, 208} \[ \frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}-\frac{19 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^5} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx &=\int \frac{(d-e x)^4}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 d^4+20 d^3 e x-35 d^2 e^2 x^2+32 d e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{15 d^4-60 d^3 e x+120 d^2 e^2 x^2-104 d e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-15 d^4+60 d^3 e x-135 d^2 e^2 x^2}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}+\frac{\int \frac{-120 d^5 e+285 d^4 e^2 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^8}\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{\left (19 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^4}\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{\left (19 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^4}\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{19 \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^4}\\ &=\frac{8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (135 d-164 e x)}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^4 x^2}+\frac{4 e \sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{19 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^5}\\ \end{align*}
Mathematica [A] time = 0.242391, size = 107, normalized size = 0.58 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (713 d^2 e^2 x^2+75 d^3 e x-15 d^4+1059 d e^3 x^3+448 e^4 x^4\right )}{x^2 (d+e x)^3}-285 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+285 e^2 \log (x)}{30 d^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 389, normalized size = 2.1 \begin{align*}{\frac{19\,{e}^{2}}{2\,{d}^{6}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{19\,{e}^{2}}{2\,{d}^{4}}\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}}}}}-4\,{\frac{{e}^{2}}{{d}^{6}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-4\,{\frac{{e}^{3}}{{d}^{5}\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ) }+{\frac{1}{5\,{d}^{4}{e}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}+{\frac{16}{15\,e{d}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}+6\,{\frac{1}{{d}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{3/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{1}{2\,{d}^{6}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+4\,{\frac{e \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{3/2}}{{d}^{7}x}}+4\,{\frac{{e}^{3}x\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}{{d}^{7}}}+4\,{\frac{{e}^{3}}{{d}^{5}\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}} \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{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65852, size = 428, normalized size = 2.34 \begin{align*} \frac{398 \, e^{5} x^{5} + 1194 \, d e^{4} x^{4} + 1194 \, d^{2} e^{3} x^{3} + 398 \, d^{3} e^{2} x^{2} + 285 \,{\left (e^{5} x^{5} + 3 \, d e^{4} x^{4} + 3 \, d^{2} e^{3} x^{3} + d^{3} e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (448 \, e^{4} x^{4} + 1059 \, d e^{3} x^{3} + 713 \, d^{2} e^{2} x^{2} + 75 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{5} e^{3} x^{5} + 3 \, d^{6} e^{2} x^{4} + 3 \, d^{7} e x^{3} + d^{8} 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{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{3} \left (d + e x\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19897, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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