Optimal. Leaf size=182 \[ \frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}-\frac{13 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^6} \]
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Rubi [A] time = 0.361906, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1805, 1807, 807, 266, 63, 208} \[ \frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}-\frac{13 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^6} \]
Antiderivative was successfully verified.
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Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 d^3-15 d^2 e x-20 d e^2 x^2-16 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{15 d^3+45 d^2 e x+75 d e^2 x^2+62 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-15 d^3-45 d^2 e x-90 d e^2 x^2}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{\int \frac{90 d^4 e+195 d^3 e^2 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^8}\\ &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}+\frac{\left (13 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^5}\\ &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}+\frac{\left (13 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^5}\\ &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 \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^5}\\ &=\frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^6}\\ \end{align*}
Mathematica [C] time = 0.0758142, size = 119, normalized size = 0.65 \[ \frac{e \left (9 d^5 e x \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right )+3 d^5 e x \, _2F_1\left (-\frac{5}{2},2;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right )+285 d^4 e^2 x^2-380 d^2 e^4 x^4-45 d^6+152 e^6 x^6\right )}{15 d^6 x \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 222, normalized size = 1.2 \begin{align*}{\frac{19\,{e}^{3}x}{5\,{d}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{76\,{e}^{3}x}{15\,{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{152\,{e}^{3}x}{15\,{d}^{6}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+{\frac{13\,{e}^{2}}{10\,d} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{13\,{e}^{2}}{6\,{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{13\,{e}^{2}}{2\,{d}^{5}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{13\,{e}^{2}}{2\,{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{d}{2\,{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-3\,{\frac{e}{x \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{5/2}}} \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 = 1.73885, size = 424, normalized size = 2.33 \begin{align*} \frac{254 \, e^{5} x^{5} - 762 \, d e^{4} x^{4} + 762 \, d^{2} e^{3} x^{3} - 254 \, d^{3} e^{2} x^{2} + 195 \,{\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 (304 \, e^{4} x^{4} - 717 \, d e^{3} x^{3} + 479 \, d^{2} e^{2} x^{2} - 45 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{6} e^{3} x^{5} - 3 \, d^{7} e^{2} x^{4} + 3 \, d^{8} e x^{3} - d^{9} 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{\left (d + e x\right )^{3}}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22188, size = 350, normalized size = 1.92 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{107 \, x e^{7}}{d^{6}} + \frac{90 \, e^{6}}{d^{5}}\right )} - \frac{245 \, e^{5}}{d^{4}}\right )} x - \frac{205 \, e^{4}}{d^{3}}\right )} x + \frac{150 \, e^{3}}{d^{2}}\right )} x + \frac{127 \, e^{2}}{d}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{13 \, e^{2} \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{2 \, d^{6}} + \frac{x^{2}{\left (\frac{12 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + e^{6}\right )}}{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6}} - \frac{{\left (\frac{12 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{6} e^{8}}{x} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6} e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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