Optimal. Leaf size=132 \[ \frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2}+\frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{7 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e} \]
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Rubi [A] time = 0.0508933, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {655, 671, 641, 195, 217, 203} \[ \frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2}+\frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{7 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e} \]
Antiderivative was successfully verified.
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Rule 655
Rule 671
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx &=\int (d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{1}{6} (7 d) \int (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{1}{6} \left (7 d^2\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{1}{8} \left (7 d^4\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2}+\frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{1}{16} \left (7 d^6\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2}+\frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{1}{16} \left (7 d^6\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{7}{16} d^4 x \sqrt{d^2-e^2 x^2}+\frac{7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac{7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac{(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac{7 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e}\\ \end{align*}
Mathematica [A] time = 0.0778846, size = 102, normalized size = 0.77 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-192 d^3 e^2 x^2+10 d^2 e^3 x^3+135 d^4 e x+96 d^5+96 d e^4 x^4-40 e^5 x^5\right )+105 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{240 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 228, normalized size = 1.7 \begin{align*}{\frac{1}{5\,{e}^{3}d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{9}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-2}}+{\frac{1}{5\,de} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{7\,x}{30} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{d}^{2}x}{24} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{d}^{4}x}{16}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{7\,{d}^{6}}{16}\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 [C] time = 1.58455, size = 188, normalized size = 1.42 \begin{align*} -\frac{7 i \, d^{6} \arcsin \left (\frac{e x}{d} + 2\right )}{16 \, e} + \frac{7}{16} \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4} x + \frac{7 \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5}}{8 \, e} + \frac{7}{24} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2} x + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}{6 \,{\left (e^{2} x + d e\right )}} + \frac{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d}{30 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17988, size = 231, normalized size = 1.75 \begin{align*} -\frac{210 \, d^{6} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (40 \, e^{5} x^{5} - 96 \, d e^{4} x^{4} - 10 \, d^{2} e^{3} x^{3} + 192 \, d^{3} e^{2} x^{2} - 135 \, d^{4} e x - 96 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 15.2106, size = 498, normalized size = 3.77 \begin{align*} d^{4} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) - 2 d^{3} e \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) + 2 d e^{3} \left (\begin{cases} - \frac{2 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\\frac{x^{4} \sqrt{d^{2}}}{4} & \text{otherwise} \end{cases}\right ) - e^{4} \left (\begin{cases} - \frac{i d^{6} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{16 e^{5}} + \frac{i d^{5} x}{16 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d^{3} x^{3}}{48 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{5 i d x^{5}}{24 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{7}}{6 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{6} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{16 e^{5}} - \frac{d^{5} x}{16 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d^{3} x^{3}}{48 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{5 d x^{5}}{24 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{7}}{6 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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