Optimal. Leaf size=205 \[ \frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0889444, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 659
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{9 \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{13 d}\\ &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{72 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^2}\\ &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{56 \int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^3}\\ &=-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{48 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 d^4}\\ &=\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{192 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{715 d^6}\\ &=\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{128 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{715 d^8}\\ &=\frac{48 x}{715 d^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{13 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{9}{143 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{143 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{715 d^8 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{128 x}{715 d^{10} \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0886175, size = 137, normalized size = 0.67 \[ \frac{\sqrt{d^2-e^2 x^2} \left (800 d^7 e^2 x^2+1080 d^6 e^3 x^3-320 d^5 e^4 x^4-1552 d^4 e^5 x^5-768 d^3 e^6 x^6+448 d^2 e^7 x^7-5 d^8 e x-180 d^9+512 d e^8 x^8+128 e^9 x^9\right )}{715 d^{10} e (d-e x)^3 (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 132, normalized size = 0.6 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -128\,{e}^{9}{x}^{9}-512\,{e}^{8}{x}^{8}d-448\,{e}^{7}{x}^{7}{d}^{2}+768\,{e}^{6}{x}^{6}{d}^{3}+1552\,{e}^{5}{x}^{5}{d}^{4}+320\,{e}^{4}{x}^{4}{d}^{5}-1080\,{e}^{3}{x}^{3}{d}^{6}-800\,{e}^{2}{x}^{2}{d}^{7}+5\,x{d}^{8}e+180\,{d}^{9} \right ) }{715\,e{d}^{10} \left ( ex+d \right ) ^{3}} \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 [A] time = 7.73555, size = 709, normalized size = 3.46 \begin{align*} -\frac{180 \, e^{10} x^{10} + 720 \, d e^{9} x^{9} + 540 \, d^{2} e^{8} x^{8} - 1440 \, d^{3} e^{7} x^{7} - 2520 \, d^{4} e^{6} x^{6} + 2520 \, d^{6} e^{4} x^{4} + 1440 \, d^{7} e^{3} x^{3} - 540 \, d^{8} e^{2} x^{2} - 720 \, d^{9} e x - 180 \, d^{10} +{\left (128 \, e^{9} x^{9} + 512 \, d e^{8} x^{8} + 448 \, d^{2} e^{7} x^{7} - 768 \, d^{3} e^{6} x^{6} - 1552 \, d^{4} e^{5} x^{5} - 320 \, d^{5} e^{4} x^{4} + 1080 \, d^{6} e^{3} x^{3} + 800 \, d^{7} e^{2} x^{2} - 5 \, d^{8} e x - 180 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{715 \,{\left (d^{10} e^{11} x^{10} + 4 \, d^{11} e^{10} x^{9} + 3 \, d^{12} e^{9} x^{8} - 8 \, d^{13} e^{8} x^{7} - 14 \, d^{14} e^{7} x^{6} + 14 \, d^{16} e^{5} x^{4} + 8 \, d^{17} e^{4} x^{3} - 3 \, d^{18} e^{3} x^{2} - 4 \, d^{19} e^{2} x - d^{20} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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