Optimal. Leaf size=201 \[ \frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5} \]
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Rubi [A] time = 0.160258, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5} \]
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^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx &=\int x^4 (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{\int x^3 \left (4 d^2 e-9 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{9 e^2}\\ &=-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{\int x^2 \left (27 d^3 e^2-32 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{72 e^4}\\ &=\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{\int x \left (64 d^4 e^3-189 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{504 e^6}\\ &=\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{d^5 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^4}\\ &=\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{\left (3 d^7\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{64 e^4}\\ &=\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{\left (3 d^9\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{128 e^4}\\ &=\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{\left (3 d^9\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^4}\\ &=\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5}\\ \end{align*}
Mathematica [A] time = 0.191343, size = 135, normalized size = 0.67 \[ \frac{\sqrt{d^2-e^2 x^2} \left (512 d^6 e^2 x^2-630 d^5 e^3 x^3+384 d^4 e^4 x^4+7560 d^3 e^5 x^5-6400 d^2 e^6 x^6-945 d^7 e x+1024 d^8-5040 d e^7 x^7+4480 e^8 x^8\right )+945 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{40320 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 330, normalized size = 1.6 \begin{align*} -{\frac{{x}^{2}}{9\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{d}^{2}}{63\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{dx}{8\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{d}^{3}x}{16\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{d}^{5}x}{64\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{45\,{d}^{7}x}{128\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{45\,{d}^{9}}{128\,{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}^{4}}{5\,{e}^{5}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{5}x}{4\,{e}^{4}} \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}^{7}x}{8\,{e}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+{\frac{3\,{d}^{9}}{8\,{e}^{4}}\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 = 1.79688, size = 323, normalized size = 1.61 \begin{align*} -\frac{1890 \, d^{9} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (4480 \, e^{8} x^{8} - 5040 \, d e^{7} x^{7} - 6400 \, d^{2} e^{6} x^{6} + 7560 \, d^{3} e^{5} x^{5} + 384 \, d^{4} e^{4} x^{4} - 630 \, d^{5} e^{3} x^{3} + 512 \, d^{6} e^{2} x^{2} - 945 \, d^{7} e x + 1024 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40320 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 22.2807, size = 833, normalized size = 4.14 \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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