Optimal. Leaf size=108 \[ \frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+\frac{5 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
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Rubi [A] time = 0.0398871, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {669, 641, 217, 203} \[ \frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+\frac{5 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 669
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{5}{3} \int \frac{(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+5 \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+(5 d) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+(5 d) \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)^4}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac{10 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-\frac{5 \sqrt{d^2-e^2 x^2}}{e}+\frac{5 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.157079, size = 109, normalized size = 1.01 \[ \frac{(d+e x) \left (\left (23 d^2-34 d e x+3 e^2 x^2\right ) \sqrt{1-\frac{e^2 x^2}{d^2}}-15 (d-e x)^2 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{3 e (e x-d) \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 160, normalized size = 1.5 \begin{align*} -{{e}^{3}{x}^{4} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+14\,{\frac{e{d}^{2}{x}^{2}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}}-{\frac{23\,{d}^{4}}{3\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,d{e}^{2}{x}^{3}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{23\,dx}{3}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+5\,{\frac{d}{\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }+{\frac{11\,{d}^{3}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 [A] time = 1.87467, size = 244, normalized size = 2.26 \begin{align*} \frac{5}{3} \, d 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{e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{14 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{11 \, d^{3} x}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{23 \, d^{4}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e} - \frac{13 \, d x}{3 \, \sqrt{-e^{2} x^{2} + d^{2}}} + \frac{5 \, d \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 = 2.1472, size = 277, normalized size = 2.56 \begin{align*} -\frac{23 \, d e^{2} x^{2} - 46 \, d^{2} e x + 23 \, d^{3} + 30 \,{\left (d e^{2} x^{2} - 2 \, d^{2} e x + d^{3}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (3 \, e^{2} x^{2} - 34 \, d e x + 23 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{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 )^{5}}{\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 = 1.34395, size = 116, normalized size = 1.07 \begin{align*} 5 \, d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (23 \, d^{4} e^{\left (-1\right )} +{\left (12 \, d^{3} -{\left (42 \, d^{2} e -{\left (3 \, x e^{3} - 28 \, d e^{2}\right )} x\right )} x\right )} x\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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