Optimal. Leaf size=172 \[ \frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0639118, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 659
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{8 \int \frac{1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{11 d}\\ &=-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 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}{99 d^2}\\ &=-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{33 d^3}\\ &=\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{165 d^5}\\ &=\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{495 d^7 \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}{495 d^7}\\ &=\frac{16 x}{165 d^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8}{99 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{64 x}{495 d^7 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{128 x}{495 d^9 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0825094, size = 126, normalized size = 0.73 \[ \frac{\sqrt{d^2-e^2 x^2} \left (680 d^6 e^2 x^2+400 d^5 e^3 x^3-720 d^4 e^4 x^4-832 d^3 e^5 x^5+64 d^2 e^6 x^6+120 d^7 e x-125 d^8+384 d e^7 x^7+128 e^8 x^8\right )}{495 d^9 e (d-e x)^3 (d+e x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 121, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( -128\,{e}^{8}{x}^{8}-384\,{e}^{7}{x}^{7}d-64\,{e}^{6}{x}^{6}{d}^{2}+832\,{e}^{5}{x}^{5}{d}^{3}+720\,{e}^{4}{x}^{4}{d}^{4}-400\,{e}^{3}{x}^{3}{d}^{5}-680\,{e}^{2}{x}^{2}{d}^{6}-120\,x{d}^{7}e+125\,{d}^{8} \right ) }{495\,e{d}^{9} \left ( ex+d \right ) ^{2}} \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 = 5.7211, size = 575, normalized size = 3.34 \begin{align*} -\frac{125 \, e^{9} x^{9} + 375 \, d e^{8} x^{8} - 1000 \, d^{3} e^{6} x^{6} - 750 \, d^{4} e^{5} x^{5} + 750 \, d^{5} e^{4} x^{4} + 1000 \, d^{6} e^{3} x^{3} - 375 \, d^{8} e x - 125 \, d^{9} +{\left (128 \, e^{8} x^{8} + 384 \, d e^{7} x^{7} + 64 \, d^{2} e^{6} x^{6} - 832 \, d^{3} e^{5} x^{5} - 720 \, d^{4} e^{4} x^{4} + 400 \, d^{5} e^{3} x^{3} + 680 \, d^{6} e^{2} x^{2} + 120 \, d^{7} e x - 125 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{495 \,{\left (d^{9} e^{10} x^{9} + 3 \, d^{10} e^{9} x^{8} - 8 \, d^{12} e^{7} x^{6} - 6 \, d^{13} e^{6} x^{5} + 6 \, d^{14} e^{5} x^{4} + 8 \, d^{15} e^{4} x^{3} - 3 \, d^{17} e^{2} x - d^{18} 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 )^{3}}\, 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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