Optimal. Leaf size=148 \[ \frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7} \]
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Rubi [A] time = 0.137417, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {850, 819, 641, 217, 203} \[ \frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7} \]
Antiderivative was successfully verified.
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Rule 850
Rule 819
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^6}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^6 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x^4 \left (5 d^3-6 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{x^2 \left (15 d^5-24 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{15 d^7-48 d^6 e x}{\sqrt{d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^6}\\ &=\frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}\\ &=\frac{x^5 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^3 (5 d-6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x (5 d-8 e x)}{5 e^6 \sqrt{d^2-e^2 x^2}}-\frac{16 \sqrt{d^2-e^2 x^2}}{5 e^7}-\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^7}\\ \end{align*}
Mathematica [A] time = 0.206691, size = 115, normalized size = 0.78 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (-87 d^3 e^2 x^2-52 d^2 e^3 x^3+33 d^4 e x+48 d^5+38 d e^4 x^4+15 e^5 x^5\right )}{(d-e x)^2 (d+e x)^3}+15 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 288, normalized size = 2. \begin{align*} -{\frac{{x}^{4}}{{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+5\,{\frac{{d}^{2}{x}^{2}}{{e}^{5} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{3/2}}}-3\,{\frac{{d}^{4}}{{e}^{7} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{3/2}}}-{\frac{d{x}^{3}}{3\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,dx}{3\,{e}^{6}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{d}{{e}^{6}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{2\,{d}^{3}x}{3\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{{d}^{5}}{5\,{e}^{8}} \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}^{3}x}{15\,{e}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,dx}{15\,{e}^{6}}{\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 [A] time = 1.80288, size = 543, normalized size = 3.67 \begin{align*} -\frac{48 \, d e^{5} x^{5} + 48 \, d^{2} e^{4} x^{4} - 96 \, d^{3} e^{3} x^{3} - 96 \, d^{4} e^{2} x^{2} + 48 \, d^{5} e x + 48 \, d^{6} - 30 \,{\left (d e^{5} x^{5} + d^{2} e^{4} x^{4} - 2 \, d^{3} e^{3} x^{3} - 2 \, d^{4} e^{2} x^{2} + d^{5} e x + d^{6}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{5} x^{5} + 38 \, d e^{4} x^{4} - 52 \, d^{2} e^{3} x^{3} - 87 \, d^{3} e^{2} x^{2} + 33 \, d^{4} e x + 48 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{12} x^{5} + d e^{11} x^{4} - 2 \, d^{2} e^{10} x^{3} - 2 \, d^{3} e^{9} x^{2} + d^{4} e^{8} x + d^{5} e^{7}\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^{6}}{\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}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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