Optimal. Leaf size=138 \[ \frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
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Rubi [A] time = 0.0572427, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {669, 641, 217, 203} \[ \frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]
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)^7}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{5} \int \frac{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{7}{3} \int \frac{(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}-7 \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-(7 d) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{2 (d+e x)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-(7 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)^6}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{14 (d+e x)^4}{15 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac{14 (d+e x)^2}{3 e \sqrt{d^2-e^2 x^2}}+\frac{7 \sqrt{d^2-e^2 x^2}}{e}-\frac{7 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.240212, size = 119, normalized size = 0.86 \[ \frac{(d+e x) \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (-381 d^2 e x+167 d^3+277 d e^2 x^2-15 e^3 x^3\right )-105 (d-e x)^3 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{15 e (d-e x)^2 \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.106, size = 253, normalized size = 1.8 \begin{align*} -{{e}^{5}{x}^{6} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+27\,{\frac{{e}^{3}{d}^{2}{x}^{4}}{ \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{73\,e{d}^{4}{x}^{2}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{167\,{d}^{6}}{15\,e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,d{e}^{4}{x}^{5}}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{7\,d{e}^{2}{x}^{3}}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{176\,dx}{15}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-7\,{\frac{d}{\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }+{\frac{35\,{d}^{3}{e}^{2}{x}^{3}}{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{61\,{d}^{5}x}{10} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{71\,{d}^{3}x}{30} \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 = 2.28444, size = 443, normalized size = 3.21 \begin{align*} \frac{7}{15} \, d e^{6} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{7}{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{27 \, d^{2} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{35 \, d^{3} e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{73 \, d^{4} e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} - \frac{61 \, d^{5} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{167 \, d^{6}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{127 \, d^{3} x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{22 \, d x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}}} - \frac{7 \, 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 = 2.28796, size = 378, normalized size = 2.74 \begin{align*} \frac{167 \, d e^{3} x^{3} - 501 \, d^{2} e^{2} x^{2} + 501 \, d^{3} e x - 167 \, d^{4} + 210 \,{\left (d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{3} x^{3} - 277 \, d e^{2} x^{2} + 381 \, d^{2} e x - 167 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} 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 )^{7}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34006, size = 144, normalized size = 1.04 \begin{align*} -7 \, d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (167 \, d^{6} e^{\left (-1\right )} +{\left (120 \, d^{5} -{\left (365 \, d^{4} e +{\left (160 \, d^{3} e^{2} -{\left (405 \, d^{2} e^{3} -{\left (15 \, x e^{5} - 232 \, d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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