Optimal. Leaf size=142 \[ \frac{7}{8} d^3 x \sqrt{d^2-e^2 x^2}+\frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{7 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e} \]
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Rubi [A] time = 0.0594916, antiderivative size = 142, 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} \[ \frac{7}{8} d^3 x \sqrt{d^2-e^2 x^2}+\frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{7 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e} \]
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)^3} \, dx &=\int (d-e x)^3 \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{1}{5} (7 d) \int (d-e x)^2 \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{1}{4} \left (7 d^2\right ) \int (d-e x) \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{1}{4} \left (7 d^3\right ) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=\frac{7}{8} d^3 x \sqrt{d^2-e^2 x^2}+\frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{1}{8} \left (7 d^5\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{7}{8} d^3 x \sqrt{d^2-e^2 x^2}+\frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{1}{8} \left (7 d^5\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}{8} d^3 x \sqrt{d^2-e^2 x^2}+\frac{7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac{7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac{(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{7 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}\\ \end{align*}
Mathematica [A] time = 0.074503, size = 91, normalized size = 0.64 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-112 d^2 e^2 x^2+15 d^3 e x+136 d^4+90 d e^3 x^3-24 e^4 x^4\right )+105 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{120 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 274, normalized size = 1.9 \begin{align*}{\frac{1}{3\,{e}^{4}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 ) ^{-3}}+{\frac{2}{5\,{e}^{3}{d}^{2}} \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{2}{5\,e{d}^{2}} \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}{15\,d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{7\,dx}{12} \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}^{3}x}{8}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{7\,{d}^{5}}{8}\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 = 1.58163, size = 216, normalized size = 1.52 \begin{align*} -\frac{7 i \, d^{5} \arcsin \left (\frac{e x}{d} + 2\right )}{8 \, e} + \frac{7}{8} \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x + \frac{7 \, \sqrt{e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{4 \, e} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}{5 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d}{20 \,{\left (e^{2} x + d e\right )}} + \frac{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{2}}{12 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28994, size = 208, normalized size = 1.46 \begin{align*} -\frac{210 \, d^{5} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (24 \, e^{4} x^{4} - 90 \, d e^{3} x^{3} + 112 \, d^{2} e^{2} x^{2} - 15 \, d^{3} e x - 136 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{120 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 21.5633, size = 442, normalized size = 3.11 \begin{align*} d^{3} \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 ) - 3 d^{2} 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 ) + 3 d e^{2} \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 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^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) - 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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