Optimal. Leaf size=30 \[ -\frac{1}{2 b c (a+b x) \sqrt{c (a+b x)^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0304208, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{2 b c (a+b x) \sqrt{c (a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Int[(c*(a + b*x)^2)^(-3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 2.7823, size = 37, normalized size = 1.23 \[ - \frac{2 a + 2 b x}{4 b \left (a^{2} c + 2 a b c x + b^{2} c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*(b*x+a)**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0149022, size = 25, normalized size = 0.83 \[ -\frac{a+b x}{2 b \left (c (a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*(a + b*x)^2)^(-3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.005, size = 22, normalized size = 0.7 \[ -{\frac{bx+a}{2\,b} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*(b*x+a)^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34432, size = 24, normalized size = 0.8 \[ -\frac{1}{2 \, \left (b^{2} c\right )^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)^2*c)^(-3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.215344, size = 93, normalized size = 3.1 \[ -\frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \,{\left (b^{4} c^{2} x^{3} + 3 \, a b^{3} c^{2} x^{2} + 3 \, a^{2} b^{2} c^{2} x + a^{3} b c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)^2*c)^(-3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c \left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*(b*x+a)**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.554574, size = 4, normalized size = 0.13 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)^2*c)^(-3/2),x, algorithm="giac")
[Out]