Optimal. Leaf size=196 \[ -\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{27 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4} \]
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Rubi [A] time = 0.516622, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {852, 1805, 1807, 807, 266, 63, 208} \[ -\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{27 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx &=\int \frac{(d-e x)^4}{x^6 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-d^4+4 d^3 e x-7 d^2 e^2 x^2+8 d e^3 x^3-8 e^4 x^4+\frac{8 e^5 x^5}{d}}{x^6 \sqrt{d^2-e^2 x^2}} \, dx}{d^2}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{\int \frac{-20 d^5 e+39 d^4 e^2 x-40 d^3 e^3 x^2+40 d^2 e^4 x^3-40 d e^5 x^4}{x^5 \sqrt{d^2-e^2 x^2}} \, dx}{5 d^4}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{\int \frac{-156 d^6 e^2+220 d^5 e^3 x-160 d^4 e^4 x^2+160 d^3 e^5 x^3}{x^4 \sqrt{d^2-e^2 x^2}} \, dx}{20 d^6}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{\int \frac{-660 d^7 e^3+792 d^6 e^4 x-480 d^5 e^5 x^2}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{60 d^8}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{\int \frac{-1584 d^8 e^4+1620 d^7 e^5 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{120 d^{10}}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{\left (27 e^5\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^3}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}-\frac{\left (27 e^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^3}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{\left (27 e^3\right ) \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^3}\\ &=-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{27 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}\\ \end{align*}
Mathematica [A] time = 0.371896, size = 118, normalized size = 0.6 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (16 d^3 e^2 x^2-29 d^2 e^3 x^3-8 d^4 e x+2 d^5+77 d e^4 x^4+212 e^5 x^5\right )}{x^5 (d+e x)}-135 e^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+135 e^5 \log (x)}{10 d^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 628, normalized size = 3.2 \begin{align*} \text{result too large to display} \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{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{4} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62542, size = 305, normalized size = 1.56 \begin{align*} -\frac{80 \, e^{6} x^{6} + 80 \, d e^{5} x^{5} + 135 \,{\left (e^{6} x^{6} + d e^{5} x^{5}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (212 \, e^{5} x^{5} + 77 \, d e^{4} x^{4} - 29 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} - 8 \, d^{4} e x + 2 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\left (d^{4} e x^{6} + d^{5} x^{5}\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{5}{2}}}{x^{6} \left (d + e x\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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