Optimal. Leaf size=133 \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4} \]
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Rubi [A] time = 0.0552447, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^4 \sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}+\frac{3 \int \frac{1}{(d+e x)^3 \sqrt{d^2-e^2 x^2}} \, dx}{7 d}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}+\frac{6 \int \frac{1}{(d+e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{35 d^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}+\frac{2 \int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx}{35 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{7 d e (d+e x)^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{35 d^2 e (d+e x)^3}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^3 e (d+e x)^2}-\frac{2 \sqrt{d^2-e^2 x^2}}{35 d^4 e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0423093, size = 63, normalized size = 0.47 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (13 d^2 e x+12 d^3+8 d e^2 x^2+2 e^3 x^3\right )}{35 d^4 e (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 66, normalized size = 0.5 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{3}{x}^{3}+8\,{e}^{2}{x}^{2}d+13\,x{d}^{2}e+12\,{d}^{3} \right ) }{35\,e{d}^{4} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \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 = 2.28272, size = 288, normalized size = 2.17 \begin{align*} -\frac{12 \, e^{4} x^{4} + 48 \, d e^{3} x^{3} + 72 \, d^{2} e^{2} x^{2} + 48 \, d^{3} e x + 12 \, d^{4} +{\left (2 \, e^{3} x^{3} + 8 \, d e^{2} x^{2} + 13 \, d^{2} e x + 12 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{35 \,{\left (d^{4} e^{5} x^{4} + 4 \, d^{5} e^{4} x^{3} + 6 \, d^{6} e^{3} x^{2} + 4 \, d^{7} e^{2} x + d^{8} e\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}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \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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