Optimal. Leaf size=122 \[ \frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]
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Rubi [A] time = 0.101513, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {850, 819, 778, 217, 203} \[ \frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]
Antiderivative was successfully verified.
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Rule 850
Rule 819
Rule 778
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^5}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^5 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x^3 \left (4 d^3-5 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{x \left (8 d^5-15 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}\\ &=\frac{x^4 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d-5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d-15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}\\ \end{align*}
Mathematica [A] time = 0.150955, size = 103, normalized size = 0.84 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-27 d^2 e^2 x^2-7 d^3 e x+8 d^4+8 d e^3 x^3+23 e^4 x^4\right )}{(d-e x)^2 (d+e x)^3}+15 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 259, normalized size = 2.1 \begin{align*}{\frac{{x}^{3}}{3\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,x}{3\,{e}^{5}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+{\frac{1}{{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d{x}^{2}}{{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{{d}^{3}}{3\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,{d}^{2}x}{3\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{{d}^{4}}{5\,{e}^{7}} \left ({\frac{d}{e}}+x \right ) ^{-1} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{d}^{2}x}{15\,{e}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,x}{15\,{e}^{5}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \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 [B] time = 1.71697, size = 497, normalized size = 4.07 \begin{align*} \frac{8 \, e^{5} x^{5} + 8 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} - 16 \, d^{3} e^{2} x^{2} + 8 \, d^{4} e x + 8 \, d^{5} - 30 \,{\left (e^{5} x^{5} + d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} - 2 \, d^{3} e^{2} x^{2} + d^{4} e x + d^{5}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (23 \, e^{4} x^{4} + 8 \, d e^{3} x^{3} - 27 \, d^{2} e^{2} x^{2} - 7 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{11} x^{5} + d e^{10} x^{4} - 2 \, d^{2} e^{9} x^{3} - 2 \, d^{3} e^{8} x^{2} + d^{4} e^{7} x + d^{5} e^{6}\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{x^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, 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*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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