Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c^2 e} \]
[Out]
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Rubi [A] time = 0.0715444, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c^2 e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 18.958, size = 31, normalized size = 0.91 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0114931, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4}{3 e \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
[Out]
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Maple [A] time = 0.004, size = 49, normalized size = 1.4 \[{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) \left ( ex+d \right ) }{3}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.692959, size = 197, normalized size = 5.79 \[ \frac{4 \, c^{2} d^{3} e^{4} \log \left (x + \frac{d}{e}\right )}{3 \, \left (c e^{2}\right )^{\frac{5}{2}}} - \frac{4 \, c d^{2} e^{3} x}{3 \, \left (c e^{2}\right )^{\frac{3}{2}}} + \frac{2 \, d e^{2} x^{2}}{3 \, \sqrt{c e^{2}}} - \frac{4}{3} \, d^{3} \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) + \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} e x^{2}}{3 \, c} + \frac{7 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d^{2}}{3 \, c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210899, size = 74, normalized size = 2.18 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )}}{3 \,{\left (c e x + c d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.33505, size = 114, normalized size = 3.35 \[ \begin{cases} \frac{d^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c e} + \frac{2 d x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c} + \frac{e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 c} & \text{for}\: e \neq 0 \\\frac{d^{3} x}{\sqrt{c d^{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.291255, size = 68, normalized size = 2. \[ \frac{1}{3} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (x{\left (\frac{x e}{c} + \frac{2 \, d}{c}\right )} + \frac{d^{2} e^{\left (-1\right )}}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")
[Out]