Optimal. Leaf size=118 \[ \frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 x}{35 d^4 e^3 \sqrt{d^2-e^2 x^2}}-\frac{x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0773119, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {850, 819, 778, 192, 191} \[ \frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 x}{35 d^4 e^3 \sqrt{d^2-e^2 x^2}}-\frac{x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 850
Rule 819
Rule 778
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\int \frac{x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx\\ &=\frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{\int \frac{x \left (2 d^3-3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d^2 e^2}\\ &=\frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{3 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 e^3}\\ &=\frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{35 d^2 e^3}\\ &=\frac{x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{35 d^4 e^3 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.137547, size = 104, normalized size = 0.88 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (-5 d^4 e^2 x^2-5 d^3 e^3 x^3-5 d^2 e^4 x^4+2 d^5 e x+2 d^6+2 d e^5 x^5+2 e^6 x^6\right )}{35 d^4 e^4 (d-e x)^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 92, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( 2\,{e}^{6}{x}^{6}+2\,{e}^{5}{x}^{5}d-5\,{x}^{4}{d}^{2}{e}^{4}-5\,{x}^{3}{d}^{3}{e}^{3}-5\,{x}^{2}{d}^{4}{e}^{2}+2\,{d}^{5}xe+2\,{d}^{6} \right ) }{35\,{d}^{4}{e}^{4}} \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 [B] time = 2.13383, size = 475, normalized size = 4.03 \begin{align*} -\frac{2 \, e^{7} x^{7} + 2 \, d e^{6} x^{6} - 6 \, d^{2} e^{5} x^{5} - 6 \, d^{3} e^{4} x^{4} + 6 \, d^{4} e^{3} x^{3} + 6 \, d^{5} e^{2} x^{2} - 2 \, d^{6} e x - 2 \, d^{7} -{\left (2 \, e^{6} x^{6} + 2 \, d e^{5} x^{5} - 5 \, d^{2} e^{4} x^{4} - 5 \, d^{3} e^{3} x^{3} - 5 \, d^{4} e^{2} x^{2} + 2 \, d^{5} e x + 2 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{35 \,{\left (d^{4} e^{11} x^{7} + d^{5} e^{10} x^{6} - 3 \, d^{6} e^{9} x^{5} - 3 \, d^{7} e^{8} x^{4} + 3 \, d^{8} e^{7} x^{3} + 3 \, d^{9} e^{6} x^{2} - d^{10} e^{5} x - d^{11} e^{4}\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^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )}\, 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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