Optimal. Leaf size=209 \[ -\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}} \]
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Rubi [A] time = 0.207456, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1639, 793, 659, 192, 191} \[ -\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 793
Rule 659
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{3 d^2 e^2-5 d e^3 x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{8 e^4}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{(7 d) \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{104 e^2}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 e^2}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{49 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1287 d e^2}\\ &=-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{14 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^2 e^2}\\ &=\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{2145 d^4 e^2}\\ &=\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{6435 d^6 e^2}\\ &=\frac{14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{112 x}{6435 d^8 e^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.111982, size = 137, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (700 d^7 e^2 x^2+945 d^6 e^3 x^3-280 d^5 e^4 x^4-1358 d^4 e^5 x^5-672 d^3 e^6 x^6+392 d^2 e^7 x^7+800 d^8 e x+200 d^9+448 d e^8 x^8+112 e^9 x^9\right )}{6435 d^8 e^3 (d-e x)^3 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 132, normalized size = 0.6 \begin{align*}{\frac{ \left ( -ex+d \right ) \left ( 112\,{e}^{9}{x}^{9}+448\,{e}^{8}{x}^{8}d+392\,{e}^{7}{x}^{7}{d}^{2}-672\,{e}^{6}{x}^{6}{d}^{3}-1358\,{e}^{5}{x}^{5}{d}^{4}-280\,{e}^{4}{x}^{4}{d}^{5}+945\,{x}^{3}{d}^{6}{e}^{3}+700\,{x}^{2}{d}^{7}{e}^{2}+800\,x{d}^{8}e+200\,{d}^{9} \right ) }{6435\,{e}^{3}{d}^{8} \left ( ex+d \right ) ^{3}} \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 [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 = 4.99928, size = 713, normalized size = 3.41 \begin{align*} \frac{200 \, e^{10} x^{10} + 800 \, d e^{9} x^{9} + 600 \, d^{2} e^{8} x^{8} - 1600 \, d^{3} e^{7} x^{7} - 2800 \, d^{4} e^{6} x^{6} + 2800 \, d^{6} e^{4} x^{4} + 1600 \, d^{7} e^{3} x^{3} - 600 \, d^{8} e^{2} x^{2} - 800 \, d^{9} e x - 200 \, d^{10} -{\left (112 \, e^{9} x^{9} + 448 \, d e^{8} x^{8} + 392 \, d^{2} e^{7} x^{7} - 672 \, d^{3} e^{6} x^{6} - 1358 \, d^{4} e^{5} x^{5} - 280 \, d^{5} e^{4} x^{4} + 945 \, d^{6} e^{3} x^{3} + 700 \, d^{7} e^{2} x^{2} + 800 \, d^{8} e x + 200 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6435 \,{\left (d^{8} e^{13} x^{10} + 4 \, d^{9} e^{12} x^{9} + 3 \, d^{10} e^{11} x^{8} - 8 \, d^{11} e^{10} x^{7} - 14 \, d^{12} e^{9} x^{6} + 14 \, d^{14} e^{7} x^{4} + 8 \, d^{15} e^{6} x^{3} - 3 \, d^{16} e^{5} x^{2} - 4 \, d^{17} e^{4} x - d^{18} e^{3}\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{x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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