3.2819 \(\int \left (\frac{c}{(a+b x)^2}\right )^{3/2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{c \sqrt{\frac{c}{(a+b x)^2}}}{2 b (a+b x)} \]

[Out]

-(c*Sqrt[c/(a + b*x)^2])/(2*b*(a + b*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.0249929, antiderivative size = 28, 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 \[ -\frac{c \sqrt{\frac{c}{(a+b x)^2}}}{2 b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^2)^(3/2),x]

[Out]

-(c*Sqrt[c/(a + b*x)^2])/(2*b*(a + b*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.14249, size = 22, normalized size = 0.79 \[ - \frac{c \sqrt{\frac{c}{\left (a + b x\right )^{2}}}}{2 b \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c/(b*x+a)**2)**(3/2),x)

[Out]

-c*sqrt(c/(a + b*x)**2)/(2*b*(a + b*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0135289, size = 25, normalized size = 0.89 \[ -\frac{(a+b x) \left (\frac{c}{(a+b x)^2}\right )^{3/2}}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c/(a + b*x)^2)^(3/2),x]

[Out]

-((c/(a + b*x)^2)^(3/2)*(a + b*x))/(2*b)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 22, normalized size = 0.8 \[ -{\frac{bx+a}{2\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a)^2)^(3/2),x)

[Out]

-1/2*(b*x+a)/b*(c/(b*x+a)^2)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.43811, size = 36, normalized size = 1.29 \[ -\frac{c^{\frac{3}{2}}}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*c^(3/2)/(b^3*x^2 + 2*a*b^2*x + a^2*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.214771, size = 49, normalized size = 1.75 \[ -\frac{c \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \,{\left (b^{2} x + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^2)^(3/2),x, algorithm="fricas")

[Out]

-1/2*c*sqrt(c/(b^2*x^2 + 2*a*b*x + a^2))/(b^2*x + a*b)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (\frac{c}{\left (a + b x\right )^{2}}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x+a)**2)**(3/2),x)

[Out]

Integral((c/(a + b*x)**2)**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215048, size = 28, normalized size = 1. \[ -\frac{c^{\frac{3}{2}}{\rm sign}\left (b x + a\right )}{2 \,{\left (b x + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^2)^(3/2),x, algorithm="giac")

[Out]

-1/2*c^(3/2)*sign(b*x + a)/((b*x + a)^2*b)