Optimal. Leaf size=29 \[ \frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]
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Rubi [A] time = 0.0664944, antiderivative size = 29, 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 \[ \frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]
Antiderivative was successfully verified.
[In] Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^3,x]
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Rubi in Sympy [A] time = 18.3049, size = 27, normalized size = 0.93 \[ \frac{c \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2)/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.0173351, size = 22, normalized size = 0.76 \[ \frac{x \left (c (d+e x)^2\right )^{3/2}}{(d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^3,x]
[Out]
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Maple [A] time = 0.003, size = 32, normalized size = 1.1 \[{\frac{x}{ \left ( ex+d \right ) ^{3}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218335, size = 43, normalized size = 1.48 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c x}{e x + d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.41378, size = 39, normalized size = 1.34 \[ c \left (\begin{cases} \frac{x \sqrt{c d^{2}}}{d} & \text{for}\: e = 0 \\\frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2)/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)/(e*x + d)^3,x, algorithm="giac")
[Out]