Optimal. Leaf size=103 \[ \frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3} \]
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Rubi [A] time = 0.0483763, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {655, 659, 651} \[ \frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3} \]
Antiderivative was successfully verified.
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Rule 655
Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\int \frac{1}{(d-e x)^3 \sqrt{d^2-e^2 x^2}} \, dx\\ &=\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \int \frac{1}{(d-e x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{5 d}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{2 \int \frac{1}{(d-e x) \sqrt{d^2-e^2 x^2}} \, dx}{15 d^2}\\ &=\frac{\sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^2 e (d-e x)^2}+\frac{2 \sqrt{d^2-e^2 x^2}}{15 d^3 e (d-e x)}\\ \end{align*}
Mathematica [A] time = 0.0673158, size = 58, normalized size = 0.56 \[ \frac{(d+e x) \left (7 d^2-6 d e x+2 e^2 x^2\right )}{15 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 55, normalized size = 0.5 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{4} \left ( 2\,{x}^{2}{e}^{2}-6\,dex+7\,{d}^{2} \right ) }{15\,{d}^{3}e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991146, size = 136, normalized size = 1.32 \begin{align*} \frac{e x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{4 \, d x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d} + \frac{2 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56251, size = 215, normalized size = 2.09 \begin{align*} \frac{7 \, e^{3} x^{3} - 21 \, d e^{2} x^{2} + 21 \, d^{2} e x - 7 \, d^{3} -{\left (2 \, e^{2} x^{2} - 6 \, d e x + 7 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{4} x^{3} - 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x - d^{6} 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 )^{3}}{\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 = 1.18938, size = 95, normalized size = 0.92 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left (7 \, d^{2} e^{\left (-1\right )} +{\left ({\left (x{\left (\frac{2 \, x^{2} e^{4}}{d^{3}} - \frac{5 \, e^{2}}{d}\right )} + 5 \, e\right )} x + 15 \, d\right )} x\right )}}{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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