Optimal. Leaf size=132 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
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Rubi [A] time = 0.0508519, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {663, 665, 195, 217, 203} \[ -\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 663
Rule 665
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)^5} \, dx &=-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-7 \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\\ &=-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-35 \int \frac{\left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx\\ &=-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-(35 d) \int \sqrt{d^2-e^2 x^2} \, dx\\ &=-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{1}{2} \left (35 d^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{1}{2} \left (35 d^3\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{35}{2} d x \sqrt{d^2-e^2 x^2}-\frac{35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac{14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac{2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0993727, size = 85, normalized size = 0.64 \[ \frac{1}{6} \sqrt{d^2-e^2 x^2} \left (-\frac{96 d^3}{e (d+e x)}-\frac{70 d^2}{e}+15 d x-2 e x^2\right )-\frac{35 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 364, normalized size = 2.8 \begin{align*} -{\frac{1}{{e}^{6}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 ) ^{-5}}-4\,{\frac{1}{{e}^{5}{d}^{2}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{20}{3\,{e}^{4}{d}^{3}} \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}}-8\,{\frac{1}{{e}^{3}{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{9/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}-8\,{\frac{1}{e{d}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{7/2}}-{\frac{28\,x}{3\,{d}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{35\,x}{3\,d} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{35\,dx}{2}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{35\,{d}^{3}}{2}\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23631, size = 244, normalized size = 1.85 \begin{align*} -\frac{166 \, d^{3} e x + 166 \, d^{4} - 210 \,{\left (d^{3} e x + d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (2 \, e^{3} x^{3} - 13 \, d e^{2} x^{2} + 55 \, d^{2} e x + 166 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{2} x + d 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 (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}{\left (d + e x\right )^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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