Optimal. Leaf size=31 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \]
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Rubi [A] time = 0.0701096, antiderivative size = 31, 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 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.5415, size = 29, normalized size = 0.94 \[ \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{2} e} \]
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)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0129638, size = 22, normalized size = 0.71 \[ \frac{x (d+e x)^3}{\left (c (d+e x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.003, size = 32, normalized size = 1. \[{ \left ( ex+d \right ) ^{3}x \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]
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)^(3/2),x)
[Out]
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Maxima [A] time = 0.685543, size = 86, normalized size = 2.77 \[ \frac{e x^{2}}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac{d^{2}}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]
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)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227647, size = 51, normalized size = 1.65 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c^{2} e x + c^{2} d} \]
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)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.42263, size = 42, normalized size = 1.35 \[ \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{2} e} & \text{for}\: e \neq 0 \\\frac{d^{3} x}{\left (c d^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
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)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265238, size = 72, normalized size = 2.32 \[ \frac{2 \, C_{0} d e^{\left (-1\right )} +{\left (2 \, C_{0} + \frac{x e}{c}\right )} x - \frac{d^{2} e^{\left (-1\right )}}{c}}{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \]
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)^(3/2),x, algorithm="giac")
[Out]