Optimal. Leaf size=182 \[ \frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{9 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^7} \]
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Rubi [A] time = 0.358016, 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} \[ \frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{9 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^7} \]
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)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 d^2-10 d e x-10 e^2 x^2-\frac{8 e^3 x^3}{d}}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{15 d^2+30 d e x+45 e^2 x^2+\frac{36 e^3 x^3}{d}}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-15 d^2-30 d e x-60 e^2 x^2}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{\int \frac{60 d^3 e+135 d^2 e^2 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^8}\\ &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}+\frac{\left (9 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^6}\\ &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^6}\\ &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{9 \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^6}\\ &=\frac{2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d+11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{9 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^7}\\ \end{align*}
Mathematica [C] time = 0.0612163, size = 117, normalized size = 0.64 \[ \frac{e \left (d^5 e x \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right )+d^5 e x \, _2F_1\left (-\frac{5}{2},2;-\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right )+60 d^4 e^2 x^2-80 d^2 e^4 x^4-10 d^6+32 e^6 x^6\right )}{5 d^7 x \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 224, normalized size = 1.2 \begin{align*}{\frac{9\,{e}^{2}}{10\,{d}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{3\,{e}^{2}}{2\,{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{9\,{e}^{2}}{2\,{d}^{6}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{9\,{e}^{2}}{2\,{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}{2\,{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-2\,{\frac{e}{dx \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{12\,{e}^{3}x}{5\,{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{16\,{e}^{3}x}{5\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{32\,{e}^{3}x}{5\,{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(-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 = 2.0328, size = 440, normalized size = 2.42 \begin{align*} \frac{54 \, e^{6} x^{6} - 108 \, d e^{5} x^{5} + 108 \, d^{3} e^{3} x^{3} - 54 \, d^{4} e^{2} x^{2} + 45 \,{\left (e^{6} x^{6} - 2 \, d e^{5} x^{5} + 2 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (64 \, e^{5} x^{5} - 83 \, d e^{4} x^{4} - 58 \, d^{2} e^{3} x^{3} + 94 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x - 5 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\left (d^{7} e^{4} x^{6} - 2 \, d^{8} e^{3} x^{5} + 2 \, d^{10} e x^{3} - d^{11} 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 )^{2}}{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 = 1.16959, size = 351, normalized size = 1.93 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left (2 \,{\left (x{\left (\frac{11 \, x e^{7}}{d^{7}} + \frac{10 \, e^{6}}{d^{6}}\right )} - \frac{25 \, e^{5}}{d^{5}}\right )} x - \frac{45 \, e^{4}}{d^{4}}\right )} x + \frac{30 \, e^{3}}{d^{3}}\right )} x + \frac{27 \, e^{2}}{d^{2}}\right )}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{9 \, 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^{7}} + \frac{x^{2}{\left (\frac{8 \,{\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^{7}} - \frac{{\left (\frac{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{7} e^{8}}{x} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{7} e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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