3.2818 \(\int \left (\frac{c}{(a+b x)^2}\right )^{5/2} \, dx\)

Optimal. Leaf size=30 \[ -\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3} \]

[Out]

-(c^2*Sqrt[c/(a + b*x)^2])/(4*b*(a + b*x)^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0266223, 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 \[ -\frac{c^2 \sqrt{\frac{c}{(a+b x)^2}}}{4 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x)^2)^(5/2),x]

[Out]

-(c^2*Sqrt[c/(a + b*x)^2])/(4*b*(a + b*x)^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.56969, size = 26, normalized size = 0.87 \[ - \frac{c^{2} \sqrt{\frac{c}{\left (a + b x\right )^{2}}}}{4 b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c/(b*x+a)**2)**(5/2),x)

[Out]

-c**2*sqrt(c/(a + b*x)**2)/(4*b*(a + b*x)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0172298, size = 25, normalized size = 0.83 \[ -\frac{(a+b x) \left (\frac{c}{(a+b x)^2}\right )^{5/2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c/(a + b*x)^2)^(5/2),x]

[Out]

-((c/(a + b*x)^2)^(5/2)*(a + b*x))/(4*b)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 22, normalized size = 0.7 \[ -{\frac{bx+a}{4\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a)^2)^(5/2),x)

[Out]

-1/4*(b*x+a)/b*(c/(b*x+a)^2)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.39646, size = 66, normalized size = 2.2 \[ -\frac{c^{\frac{5}{2}}}{4 \,{\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/4*c^(5/2)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.218031, size = 81, normalized size = 2.7 \[ -\frac{c^{2} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{4 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^2)^(5/2),x, algorithm="fricas")

[Out]

-1/4*c^2*sqrt(c/(b^2*x^2 + 2*a*b*x + a^2))/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x
+ a^3*b)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x+a)**2)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215595, size = 1, normalized size = 0.03 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a)^2)^(5/2),x, algorithm="giac")

[Out]

+Infinity