Optimal. Leaf size=146 \[ -\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5} \]
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Rubi [A] time = 0.304573, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {852, 1805, 807, 266, 63, 208} \[ -\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5} \]
Antiderivative was successfully verified.
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Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 (d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx &=\int \frac{(d-e x)^3}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 d^3+15 d^2 e x-16 d e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{15 d^3-45 d^2 e x+42 d e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-15 d^3+45 d^2 e x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{(3 e) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{d^4}\\ &=-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}-\frac{(3 e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{3 \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 )}{d^4 e}\\ &=-\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end{align*}
Mathematica [A] time = 0.19203, size = 92, normalized size = 0.63 \[ -\frac{\frac{\sqrt{d^2-e^2 x^2} \left (39 d^2 e x+5 d^3+57 d e^2 x^2+24 e^3 x^3\right )}{x (d+e x)^3}-15 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+15 e \log (x)}{5 d^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 199, normalized size = 1.4 \begin{align*} 3\,{\frac{e}{{d}^{4}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}{x}} \right ) }-{\frac{19}{5\,{d}^{5}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{4}{5\,{d}^{4}e}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-2}}-{\frac{1}{5\,{e}^{2}{d}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-3}}-{\frac{1}{{d}^{5}x}\sqrt{-{x}^{2}{e}^{2}+{d}^{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}{\sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55484, size = 375, normalized size = 2.57 \begin{align*} -\frac{24 \, e^{4} x^{4} + 72 \, d e^{3} x^{3} + 72 \, d^{2} e^{2} x^{2} + 24 \, d^{3} e x + 15 \,{\left (e^{4} x^{4} + 3 \, d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} + d^{3} e x\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (24 \, e^{3} x^{3} + 57 \, d e^{2} x^{2} + 39 \, d^{2} e x + 5 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{5} e^{3} x^{4} + 3 \, d^{6} e^{2} x^{3} + 3 \, d^{7} e x^{2} + d^{8} x\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^{2} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16536, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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