Optimal. Leaf size=67 \[ \frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0227085, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ \frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{(d+e x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{3 d}\\ &=\frac{(d+e x)^4}{3 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac{(d+e x)^5}{15 d^2 e \left (d^2-e^2 x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0687201, size = 49, normalized size = 0.73 \[ \frac{(4 d-e x) (d+e x)^2}{15 d^2 e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( ex+d \right ) ^{5} \left ( -ex+d \right ) \left ( -ex+4\,d \right ) }{15\,{d}^{2}e} \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 = 1.17327, size = 166, normalized size = 2.48 \begin{align*} \frac{e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{11 \, d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d^{3}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} - \frac{x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} - \frac{x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08619, size = 212, normalized size = 3.16 \begin{align*} \frac{4 \, e^{3} x^{3} - 12 \, d e^{2} x^{2} + 12 \, d^{2} e x - 4 \, d^{3} +{\left (e^{2} x^{2} - 3 \, d e x - 4 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{4} x^{3} - 3 \, d^{3} e^{3} x^{2} + 3 \, d^{4} e^{2} x - d^{5} 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 )^{4}}{\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 [A] time = 1.34625, size = 95, normalized size = 1.42 \begin{align*} -\frac{{\left (4 \, d^{3} e^{\left (-1\right )} +{\left (15 \, d^{2} -{\left (x{\left (\frac{x^{2} e^{4}}{d^{2}} - 10 \, e^{2}\right )} - 20 \, d e\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\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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