Optimal. Leaf size=172 \[ -\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{3 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^4} \]
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Rubi [A] time = 0.123488, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {850, 833, 780, 195, 217, 203} \[ -\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{3 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^4} \]
Antiderivative was successfully verified.
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Rule 850
Rule 833
Rule 780
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx &=\int x^3 (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{\int x^2 \left (3 d^2 e-8 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{8 e^2}\\ &=-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}+\frac{\int x \left (16 d^3 e^2-21 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{56 e^4}\\ &=-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{d^4 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^3}\\ &=-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{\left (3 d^6\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{64 e^3}\\ &=-\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{\left (3 d^8\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{128 e^3}\\ &=-\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{\left (3 d^8\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^3}\\ &=-\frac{3 d^6 x \sqrt{d^2-e^2 x^2}}{128 e^3}-\frac{d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac{d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac{x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac{d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac{3 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^4}\\ \end{align*}
Mathematica [A] time = 0.139402, size = 124, normalized size = 0.72 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-128 d^5 e^2 x^2+70 d^4 e^3 x^3+1024 d^3 e^4 x^4-840 d^2 e^5 x^5+105 d^6 e x-256 d^7-640 d e^6 x^6+560 e^7 x^7\right )-105 d^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{4480 e^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 305, normalized size = 1.8 \begin{align*} -{\frac{x}{8\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{d}^{2}x}{16\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{15\,{d}^{4}x}{64\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{45\,{d}^{6}x}{128\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{45\,{d}^{8}}{128\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d}{7\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{d}^{3}}{5\,{e}^{4}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{{d}^{4}x}{4\,{e}^{3}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{6}x}{8\,{e}^{3}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{3\,{d}^{8}}{8\,{e}^{3}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{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 = 1.65833, size = 288, normalized size = 1.67 \begin{align*} \frac{210 \, d^{8} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (560 \, e^{7} x^{7} - 640 \, d e^{6} x^{6} - 840 \, d^{2} e^{5} x^{5} + 1024 \, d^{3} e^{4} x^{4} + 70 \, d^{4} e^{3} x^{3} - 128 \, d^{5} e^{2} x^{2} + 105 \, d^{6} e x - 256 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{4480 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.617, size = 779, normalized size = 4.53 \begin{align*} \text{result too large to display} \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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