Optimal. Leaf size=81 \[ \frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
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Rubi [A] time = 0.0231172, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {669, 653, 217, 203} \[ \frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 669
Rule 653
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\int \frac{(d+e x)^2}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=\frac{2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d+e x)}{e \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.120354, size = 100, normalized size = 1.23 \[ -\frac{(d+e x) \left (4 d (d-2 e x) \sqrt{1-\frac{e^2 x^2}{d^2}}-3 (d-e x)^2 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{3 d e (d-e x) \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 132, normalized size = 1.6 \begin{align*}{\frac{{e}^{2}{x}^{3}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,x}{3}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+4\,{\frac{de{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-{\frac{4\,{d}^{3}}{3\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{d}^{2}x}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.68638, size = 207, normalized size = 2.56 \begin{align*} \frac{1}{3} \, e^{4} x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{4 \, d e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{7 \, d^{2} x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{4 \, d^{3}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{5 \, x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}}} + \frac{\arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03055, size = 236, normalized size = 2.91 \begin{align*} -\frac{2 \,{\left (2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} + 3 \,{\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 2 \, \sqrt{-e^{2} x^{2} + d^{2}}{\left (2 \, e x - d\right )}\right )}}{3 \,{\left (e^{3} x^{2} - 2 \, d e^{2} x + d^{2} e\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 )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41552, size = 89, normalized size = 1.1 \begin{align*} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{4 \,{\left (d^{3} e^{\left (-1\right )} -{\left (2 \, x e^{2} + 3 \, d e\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{3 \,{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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