Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{e}{4 \sqrt{4 x^2+12 x+9}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0473908, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2 d-3 e}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{e}{4 \sqrt{4 x^2+12 x+9}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.29605, size = 34, normalized size = 0.65 \[ - \frac{\left (d + e x\right )^{2} \left (8 x + 12\right )}{8 \left (2 d - 3 e\right ) \left (4 x^{2} + 12 x + 9\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(4*x**2+12*x+9)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0201292, size = 34, normalized size = 0.65 \[ \frac{-2 d-e (4 x+3)}{8 (2 x+3) \sqrt{(2 x+3)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 28, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 4\,ex+2\,d+3\,e \right ) }{8} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(4*x^2+12*x+9)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.827836, size = 49, normalized size = 0.94 \[ -\frac{e}{4 \, \sqrt{4 \, x^{2} + 12 \, x + 9}} - \frac{d}{4 \,{\left (2 \, x + 3\right )}^{2}} + \frac{3 \, e}{8 \,{\left (2 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(4*x^2 + 12*x + 9)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.201526, size = 34, normalized size = 0.65 \[ -\frac{4 \, e x + 2 \, d + 3 \, e}{8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(4*x^2 + 12*x + 9)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(4*x**2+12*x+9)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.591578, size = 4, normalized size = 0.08 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(4*x^2 + 12*x + 9)^(3/2),x, algorithm="giac")
[Out]