Optimal. Leaf size=118 \[ -\frac{d (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{6 e^5}-\frac{4 x^2 \sqrt{d^2-e^2 x^2}}{3 e^3}+\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{3 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5} \]
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Rubi [A] time = 0.0961943, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {850, 819, 833, 780, 217, 203} \[ -\frac{d (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{6 e^5}-\frac{4 x^2 \sqrt{d^2-e^2 x^2}}{3 e^3}+\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{3 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5} \]
Antiderivative was successfully verified.
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Rule 850
Rule 819
Rule 833
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^4}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=\int \frac{x^4 (d-e x)}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{x^2 \left (3 d^3-4 d^2 e x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x^2 \sqrt{d^2-e^2 x^2}}{3 e^3}+\frac{\int \frac{x \left (8 d^4 e-9 d^3 e^2 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{3 d^2 e^4}\\ &=\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x^2 \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{d (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{6 e^5}-\frac{\left (3 d^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^4}\\ &=\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x^2 \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{d (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{6 e^5}-\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^4}\\ &=\frac{x^3 (d-e x)}{e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x^2 \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{d (16 d-9 e x) \sqrt{d^2-e^2 x^2}}{6 e^5}-\frac{3 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^5}\\ \end{align*}
Mathematica [A] time = 0.095506, size = 91, normalized size = 0.77 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-7 d^2 e x-16 d^3+d e^2 x^2-2 e^3 x^3\right )-9 d^3 (d+e x) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{6 e^5 (d+e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 147, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2}}{3\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{5\,{d}^{2}}{3\,{e}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{dx}{2\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,{d}^{3}}{2\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{d}^{3}}{{e}^{6}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}} \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 = 1.69335, size = 236, normalized size = 2. \begin{align*} -\frac{16 \, d^{3} e x + 16 \, d^{4} - 18 \,{\left (d^{3} e x + d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (2 \, e^{3} x^{3} - d e^{2} x^{2} + 7 \, d^{2} e x + 16 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{6} x + d e^{5}\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^{4}}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, 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: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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