Optimal. Leaf size=33 \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0094206, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {651} \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0944502, size = 41, normalized size = 1.24 \[ \frac{(d+e x)^3}{5 d e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 36, normalized size = 1.1 \begin{align*}{\frac{ \left ( ex+d \right ) ^{6} \left ( -ex+d \right ) }{5\,de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17891, size = 200, normalized size = 6.06 \begin{align*} \frac{e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, d e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, d^{3} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{d^{4}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.965, size = 197, normalized size = 5.97 \begin{align*} \frac{e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3} -{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d e^{4} x^{3} - 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x - d^{4} 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{\left (d + e x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30536, size = 103, normalized size = 3.12 \begin{align*} -\frac{{\left (d^{4} e^{\left (-1\right )} +{\left (5 \, d^{3} +{\left (10 \, d^{2} e +{\left (x{\left (\frac{x e^{4}}{d} + 5 \, e^{3}\right )} + 10 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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