Optimal. Leaf size=182 \[ -\frac{21 d^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{9 d (d+e x)^3 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{(d+e x)^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{63 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}-\frac{63 d^4 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{21 d^3 (d+e x) \sqrt{d^2-e^2 x^2}}{8 e} \]
[Out]
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Rubi [A] time = 0.252128, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{21 d^2 (d+e x)^2 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{9 d (d+e x)^3 \sqrt{d^2-e^2 x^2}}{20 e}-\frac{(d+e x)^4 \sqrt{d^2-e^2 x^2}}{5 e}+\frac{63 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}-\frac{63 d^4 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{21 d^3 (d+e x) \sqrt{d^2-e^2 x^2}}{8 e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5/Sqrt[d^2 - e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 33.5986, size = 155, normalized size = 0.85 \[ \frac{63 d^{5} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{8 e} - \frac{63 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{8 e} - \frac{21 d^{3} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{8 e} - \frac{21 d^{2} \left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}}{20 e} - \frac{9 d \left (d + e x\right )^{3} \sqrt{d^{2} - e^{2} x^{2}}}{20 e} - \frac{\left (d + e x\right )^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.110441, size = 92, normalized size = 0.51 \[ \frac{315 d^5 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (488 d^4+275 d^3 e x+144 d^2 e^2 x^2+50 d e^3 x^3+8 e^4 x^4\right )}{40 e} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5/Sqrt[d^2 - e^2*x^2],x]
[Out]
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Maple [A] time = 0.031, size = 144, normalized size = 0.8 \[{\frac{63\,{d}^{5}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{{e}^{3}{x}^{4}}{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{18\,e{d}^{2}{x}^{2}}{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{61\,{d}^{4}}{5\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{5\,d{e}^{2}{x}^{3}}{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{55\,{d}^{3}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.777208, size = 184, normalized size = 1.01 \[ -\frac{1}{5} \, \sqrt{-e^{2} x^{2} + d^{2}} e^{3} x^{4} - \frac{5}{4} \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{2} x^{3} - \frac{18}{5} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e x^{2} + \frac{63 \, d^{5} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}}} - \frac{55}{8} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x - \frac{61 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4}}{5 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/sqrt(-e^2*x^2 + d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226991, size = 489, normalized size = 2.69 \[ -\frac{8 \, e^{10} x^{10} + 50 \, d e^{9} x^{9} + 40 \, d^{2} e^{8} x^{8} - 375 \, d^{3} e^{7} x^{7} - 1160 \, d^{4} e^{6} x^{6} - 2175 \, d^{5} e^{5} x^{5} + 6900 \, d^{7} e^{3} x^{3} + 1600 \, d^{8} e^{2} x^{2} - 4400 \, d^{9} e x + 630 \,{\left (5 \, d^{6} e^{4} x^{4} - 20 \, d^{8} e^{2} x^{2} + 16 \, d^{10} -{\left (d^{5} e^{4} x^{4} - 12 \, d^{7} e^{2} x^{2} + 16 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 5 \,{\left (8 \, d e^{8} x^{8} + 50 \, d^{2} e^{7} x^{7} + 112 \, d^{3} e^{6} x^{6} + 75 \, d^{4} e^{5} x^{5} - 160 \, d^{5} e^{4} x^{4} - 940 \, d^{6} e^{3} x^{3} - 320 \, d^{7} e^{2} x^{2} + 880 \, d^{8} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40 \,{\left (5 \, d e^{5} x^{4} - 20 \, d^{3} e^{3} x^{2} + 16 \, d^{5} e -{\left (e^{5} x^{4} - 12 \, d^{2} e^{3} x^{2} + 16 \, d^{4} e\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/sqrt(-e^2*x^2 + d^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 26.8653, size = 646, normalized size = 3.55 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**5/(-e**2*x**2+d**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.22879, size = 99, normalized size = 0.54 \[ \frac{63}{8} \, d^{5} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{40} \,{\left (488 \, d^{4} e^{\left (-1\right )} +{\left (275 \, d^{3} + 2 \,{\left (72 \, d^{2} e +{\left (4 \, x e^{3} + 25 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/sqrt(-e^2*x^2 + d^2),x, algorithm="giac")
[Out]