Optimal. Leaf size=170 \[ \frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}-\frac{95 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3} \]
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Rubi [A] time = 0.394445, antiderivative size = 170, 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 = {852, 1805, 1807, 807, 266, 63, 208} \[ \frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}-\frac{95 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3} \]
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^5 (d+e x)^4} \, dx &=\int \frac{(d-e x)^4}{x^5 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{8 e^4 (d-e x)}{d^3 \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}{x^5 \sqrt{d^2-e^2 x^2}} \, dx}{d^2}\\ &=\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{\int \frac{-16 d^5 e+31 d^4 e^2 x-32 d^3 e^3 x^2+32 d^2 e^4 x^3}{x^4 \sqrt{d^2-e^2 x^2}} \, dx}{4 d^4}\\ &=\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{\int \frac{-93 d^6 e^2+128 d^5 e^3 x-96 d^4 e^4 x^2}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{12 d^6}\\ &=\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}+\frac{\int \frac{-256 d^7 e^3+285 d^6 e^4 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{24 d^8}\\ &=\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}+\frac{\left (95 e^4\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{8 d^2}\\ &=\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}+\frac{\left (95 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^2}\\ &=\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}-\frac{\left (95 e^2\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 )}{8 d^2}\\ &=\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}-\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x}-\frac{95 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3}\\ \end{align*}
Mathematica [A] time = 0.305262, size = 107, normalized size = 0.63 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-61 d^2 e^2 x^2+26 d^3 e x-6 d^4+163 d e^3 x^3+448 e^4 x^4\right )}{x^4 (d+e x)}-285 e^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+285 e^4 \log (x)}{24 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.093, size = 600, normalized size = 3.5 \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64302, size = 286, normalized size = 1.68 \begin{align*} \frac{192 \, e^{5} x^{5} + 192 \, d e^{4} x^{4} + 285 \,{\left (e^{5} x^{5} + d e^{4} x^{4}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (448 \, e^{4} x^{4} + 163 \, d e^{3} x^{3} - 61 \, d^{2} e^{2} x^{2} + 26 \, d^{3} e x - 6 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (d^{3} e x^{5} + d^{4} x^{4}\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^{5} \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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