Optimal. Leaf size=210 \[ -\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{25}{8} d^5 e^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{13}{2} d^5 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.313565, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1807, 813, 815, 844, 217, 203, 266, 63, 208} \[ -\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{25}{8} d^5 e^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{13}{2} d^5 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 1807
Rule 813
Rule 815
Rule 844
Rule 217
Rule 203
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^4} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{5/2} \left (-9 d^4 e-5 d^3 e^2 x-3 d^2 e^3 x^2\right )}{x^3} \, dx}{3 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}+\frac{\int \frac{\left (10 d^5 e^2-39 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^2} \, dx}{6 d^4}\\ &=-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{\int \frac{\left (78 d^6 e^3+100 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{12 d^4}\\ &=-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}+\frac{\int \frac{\left (-312 d^8 e^5-300 d^7 e^6 x\right ) \sqrt{d^2-e^2 x^2}}{x} \, dx}{48 d^4 e^2}\\ &=-\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{\int \frac{624 d^{10} e^7+300 d^9 e^8 x}{x \sqrt{d^2-e^2 x^2}} \, dx}{96 d^4 e^4}\\ &=-\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{1}{2} \left (13 d^6 e^3\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx-\frac{1}{8} \left (25 d^5 e^4\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{1}{4} \left (13 d^6 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )-\frac{1}{8} \left (25 d^5 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=-\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{25}{8} d^5 e^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{1}{2} \left (13 d^6 e\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )\\ &=-\frac{1}{8} d^3 e^3 (52 d+25 e x) \sqrt{d^2-e^2 x^2}-\frac{1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac{e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac{25}{8} d^5 e^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{13}{2} d^5 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}
Mathematica [C] time = 0.293106, size = 251, normalized size = 1.2 \[ -\frac{d^7 \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{3 x^3 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 d^5 e^2 \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{e^2 x^2}{d^2}\right )}{x \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 e^3 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )}{7 d^2}+\frac{1}{15} e^3 \left (\sqrt{d^2-e^2 x^2} \left (-11 d^2 e^2 x^2+23 d^4+3 e^4 x^4\right )-15 d^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 277, normalized size = 1.3 \begin{align*} -{\frac{13\,{e}^{3}}{10} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{13\,{e}^{3}{d}^{2}}{6} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{13\,{e}^{3}{d}^{4}}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{13\,{e}^{3}{d}^{6}}{2}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{d}{3\,{x}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{5\,{e}^{2}}{3\,dx} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{5\,{e}^{4}x}{3\,d} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{25\,d{e}^{4}x}{12} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{25\,{d}^{3}{e}^{4}x}{8}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{25\,{d}^{5}{e}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{3\,e}{2\,{x}^{2}} \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 [A] time = 1.99381, size = 386, normalized size = 1.84 \begin{align*} \frac{750 \, d^{5} e^{3} x^{3} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 780 \, d^{5} e^{3} x^{3} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) - 656 \, d^{5} e^{3} x^{3} +{\left (24 \, e^{7} x^{7} + 90 \, d e^{6} x^{6} + 32 \, d^{2} e^{5} x^{5} - 345 \, d^{3} e^{4} x^{4} - 656 \, d^{4} e^{3} x^{3} - 80 \, d^{5} e^{2} x^{2} - 180 \, d^{6} e x - 40 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{120 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 19.0665, size = 926, normalized size = 4.41 \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 [A] time = 1.15548, size = 429, normalized size = 2.04 \begin{align*} -\frac{25}{8} \, d^{5} \arcsin \left (\frac{x e}{d}\right ) e^{3} \mathrm{sgn}\left (d\right ) + \frac{13}{2} \, d^{5} e^{3} \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) + \frac{{\left (d^{5} e^{8} + \frac{9 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{5} e^{6}}{x} + \frac{9 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} e^{4}}{x^{2}}\right )} x^{3} e}{24 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3}} - \frac{1}{24} \,{\left (\frac{9 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{5} e^{16}}{x} + \frac{9 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} e^{14}}{x^{2}} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{5} e^{12}}{x^{3}}\right )} e^{\left (-15\right )} - \frac{1}{120} \,{\left (656 \, d^{4} e^{3} +{\left (345 \, d^{3} e^{4} - 2 \,{\left (16 \, d^{2} e^{5} + 3 \,{\left (4 \, x e^{7} + 15 \, d e^{6}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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