Optimal. Leaf size=215 \[ -\frac{128 e^2 \sqrt{d^2-e^2 x^2}}{15 d^9 x}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}+\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{7 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^9} \]
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Rubi [A] time = 0.209699, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {857, 823, 835, 807, 266, 63, 208} \[ -\frac{128 e^2 \sqrt{d^2-e^2 x^2}}{15 d^9 x}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}+\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{7 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^9} \]
Antiderivative was successfully verified.
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Rule 857
Rule 823
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-8 d e^2+7 e^3 x}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-48 d^3 e^4+35 d^2 e^5 x}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-192 d^5 e^6+105 d^4 e^7 x}{x^4 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{\int \frac{-315 d^6 e^7+384 d^5 e^8 x}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{45 d^{12} e^6}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{\int \frac{-768 d^7 e^8+315 d^6 e^9 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{90 d^{14} e^6}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{128 e^2 \sqrt{d^2-e^2 x^2}}{15 d^9 x}-\frac{\left (7 e^3\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^8}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{128 e^2 \sqrt{d^2-e^2 x^2}}{15 d^9 x}-\frac{\left (7 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^8}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{128 e^2 \sqrt{d^2-e^2 x^2}}{15 d^9 x}+\frac{(7 e) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{2 d^8}\\ &=\frac{8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 d-35 e x}{15 d^6 x^3 \sqrt{d^2-e^2 x^2}}-\frac{64 \sqrt{d^2-e^2 x^2}}{15 d^7 x^3}+\frac{7 e \sqrt{d^2-e^2 x^2}}{2 d^8 x^2}-\frac{128 e^2 \sqrt{d^2-e^2 x^2}}{15 d^9 x}+\frac{7 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^9}\\ \end{align*}
Mathematica [A] time = 0.182137, size = 148, normalized size = 0.69 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (75 d^5 e^2 x^2+236 d^4 e^3 x^3-244 d^3 e^4 x^4-489 d^2 e^5 x^5-5 d^6 e x+10 d^7+151 d e^6 x^6+256 e^7 x^7\right )}{x^3 (d-e x)^2 (d+e x)^3}-105 e^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+105 e^3 \log (x)}{30 d^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 326, normalized size = 1.5 \begin{align*} -{\frac{7\,{e}^{3}}{6\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,{e}^{3}}{2\,{d}^{8}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+{\frac{7\,{e}^{3}}{2\,{d}^{8}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{1}{3\,{d}^{3}{x}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-3\,{\frac{{e}^{2}}{{d}^{5}x \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{3/2}}}+4\,{\frac{{e}^{4}x}{{d}^{7} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{3/2}}}+8\,{\frac{{e}^{4}x}{{d}^{9}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}-{\frac{{e}^{2}}{5\,{d}^{5}} \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\,{e}^{4}x}{15\,{d}^{7}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{e}^{4}x}{15\,{d}^{9}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}+{\frac{e}{2\,{d}^{4}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4647, size = 626, normalized size = 2.91 \begin{align*} -\frac{116 \, e^{8} x^{8} + 116 \, d e^{7} x^{7} - 232 \, d^{2} e^{6} x^{6} - 232 \, d^{3} e^{5} x^{5} + 116 \, d^{4} e^{4} x^{4} + 116 \, d^{5} e^{3} x^{3} + 105 \,{\left (e^{8} x^{8} + d e^{7} x^{7} - 2 \, d^{2} e^{6} x^{6} - 2 \, d^{3} e^{5} x^{5} + d^{4} e^{4} x^{4} + d^{5} e^{3} x^{3}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (256 \, e^{7} x^{7} + 151 \, d e^{6} x^{6} - 489 \, d^{2} e^{5} x^{5} - 244 \, d^{3} e^{4} x^{4} + 236 \, d^{4} e^{3} x^{3} + 75 \, d^{5} e^{2} x^{2} - 5 \, d^{6} e x + 10 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{9} e^{5} x^{8} + d^{10} e^{4} x^{7} - 2 \, d^{11} e^{3} x^{6} - 2 \, d^{12} e^{2} x^{5} + d^{13} e x^{4} + d^{14} x^{3}\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{1}{x^{4} \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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