Optimal. Leaf size=169 \[ -\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{4 d^4} \]
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Rubi [A] time = 0.195176, antiderivative size = 169, 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 = {1807, 835, 807, 266, 63, 208} \[ -\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{4 d^4} \]
Antiderivative was successfully verified.
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Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{x^6 \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{\int \frac{-10 d^3 e-9 d^2 e^2 x}{x^5 \sqrt{d^2-e^2 x^2}} \, dx}{5 d^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}+\frac{\int \frac{36 d^4 e^2+30 d^3 e^3 x}{x^4 \sqrt{d^2-e^2 x^2}} \, dx}{20 d^4}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{\int \frac{-90 d^5 e^3-72 d^4 e^4 x}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{60 d^6}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}+\frac{\int \frac{144 d^6 e^4+90 d^5 e^5 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{120 d^8}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{\left (3 e^5\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{4 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{\left (3 e^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{8 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{\left (3 e^3\right ) \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 )}{4 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}-\frac{e \sqrt{d^2-e^2 x^2}}{2 d x^4}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{3 e^3 \sqrt{d^2-e^2 x^2}}{4 d^3 x^2}-\frac{6 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{4 d^4}\\ \end{align*}
Mathematica [C] time = 0.0436116, size = 79, normalized size = 0.47 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (10 e^5 x^5 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1-\frac{e^2 x^2}{d^2}\right )+3 d^3 e^2 x^2+d^5+6 d e^4 x^4\right )}{5 d^5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 164, normalized size = 1. \begin{align*} -{\frac{3\,{e}^{2}}{5\,{d}^{2}{x}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{6\,{e}^{4}}{5\,{d}^{4}x}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{1}{5\,{x}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{e}{2\,d{x}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,{e}^{3}}{4\,{d}^{3}{x}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,{e}^{5}}{4\,{d}^{3}}\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}}}}} \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 = 1.86618, size = 207, normalized size = 1.22 \begin{align*} \frac{15 \, e^{5} x^{5} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (24 \, e^{4} x^{4} + 15 \, d e^{3} x^{3} + 12 \, d^{2} e^{2} x^{2} + 10 \, d^{3} e x + 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{20 \, d^{4} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 8.22559, size = 520, normalized size = 3.08 \begin{align*} d^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{5 d^{2} x^{4}} - \frac{4 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{15 d^{4} x^{2}} - \frac{8 e^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{15 d^{6}} & \text{for}\: \frac{\left |{d^{2}}\right |}{\left |{e^{2}}\right | \left |{x^{2}}\right |} > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{5 d^{2} x^{4}} - \frac{4 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{15 d^{4} x^{2}} - \frac{8 i e^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{15 d^{6}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{1}{4 e x^{5} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{e}{8 d^{2} x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{3 e^{3}}{8 d^{4} x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - \frac{3 e^{4} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{for}\: \frac{\left |{d^{2}}\right |}{\left |{e^{2}}\right | \left |{x^{2}}\right |} > 1 \\\frac{i}{4 e x^{5} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{i e}{8 d^{2} x^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} - \frac{3 i e^{3}}{8 d^{4} x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + \frac{3 i e^{4} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{8 d^{5}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac{2 e^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text{for}\: \frac{\left |{d^{2}}\right |}{\left |{e^{2}}\right | \left |{x^{2}}\right |} > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac{2 i e^{3} \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16473, size = 493, normalized size = 2.92 \begin{align*} \frac{x^{5}{\left (\frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{10}}{x} + \frac{15 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{8}}{x^{2}} + \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{6}}{x^{3}} + \frac{110 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{4}}{x^{4}} + e^{12}\right )} e^{3}}{160 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4}} - \frac{3 \, e^{5} \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 )}{4 \, d^{4}} - \frac{{\left (\frac{110 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{16} e^{38}}{x} + \frac{40 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{16} e^{36}}{x^{2}} + \frac{15 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{16} e^{34}}{x^{3}} + \frac{5 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{16} e^{32}}{x^{4}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{16} e^{30}}{x^{5}}\right )} e^{\left (-35\right )}}{160 \, d^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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