Optimal. Leaf size=139 \[ \frac{16 x}{45 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0466924, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac{16 x}{45 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 659
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{9 d}\\ &=-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{3 d^2}\\ &=\frac{2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{15 d^4}\\ &=\frac{2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{45 d^6}\\ &=\frac{2 x}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{9 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 x}{45 d^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{45 d^8 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0757517, size = 115, normalized size = 0.83 \[ \frac{\sqrt{d^2-e^2 x^2} \left (60 d^5 e^2 x^2-10 d^4 e^3 x^3-80 d^3 e^4 x^4-24 d^2 e^5 x^5+25 d^6 e x-10 d^7+32 d e^6 x^6+16 e^7 x^7\right )}{45 d^8 e (d-e x)^3 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 110, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -16\,{e}^{7}{x}^{7}-32\,{e}^{6}{x}^{6}d+24\,{e}^{5}{x}^{5}{d}^{2}+80\,{e}^{4}{x}^{4}{d}^{3}+10\,{e}^{3}{x}^{3}{d}^{4}-60\,{e}^{2}{x}^{2}{d}^{5}-25\,x{d}^{6}e+10\,{d}^{7} \right ) }{ \left ( 45\,ex+45\,d \right ){d}^{8}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 [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 [B] time = 3.84771, size = 524, normalized size = 3.77 \begin{align*} -\frac{10 \, e^{8} x^{8} + 20 \, d e^{7} x^{7} - 20 \, d^{2} e^{6} x^{6} - 60 \, d^{3} e^{5} x^{5} + 60 \, d^{5} e^{3} x^{3} + 20 \, d^{6} e^{2} x^{2} - 20 \, d^{7} e x - 10 \, d^{8} +{\left (16 \, e^{7} x^{7} + 32 \, d e^{6} x^{6} - 24 \, d^{2} e^{5} x^{5} - 80 \, d^{3} e^{4} x^{4} - 10 \, d^{4} e^{3} x^{3} + 60 \, d^{5} e^{2} x^{2} + 25 \, d^{6} e x - 10 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{45 \,{\left (d^{8} e^{9} x^{8} + 2 \, d^{9} e^{8} x^{7} - 2 \, d^{10} e^{7} x^{6} - 6 \, d^{11} e^{6} x^{5} + 6 \, d^{13} e^{4} x^{3} + 2 \, d^{14} e^{3} x^{2} - 2 \, d^{15} e^{2} x - d^{16} 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}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )^{2}}\, 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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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