3.2811 \(\int \frac{1}{\left (c (a+b x)^3\right )^{5/2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2}{13 b c^2 (a+b x)^5 \sqrt{c (a+b x)^3}} \]

[Out]

-2/(13*b*c^2*(a + b*x)^5*Sqrt[c*(a + b*x)^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.033819, 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 \[ -\frac{2}{13 b c^2 (a+b x)^5 \sqrt{c (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]  Int[(c*(a + b*x)^3)^(-5/2),x]

[Out]

-2/(13*b*c^2*(a + b*x)^5*Sqrt[c*(a + b*x)^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.28985, size = 51, normalized size = 1.7 \[ - \frac{2 \left (3 a + 3 b x\right )}{39 b \left (a^{3} c + 3 a^{2} b c x + 3 a b^{2} c x^{2} + b^{3} c x^{3}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*(b*x+a)**3)**(5/2),x)

[Out]

-2*(3*a + 3*b*x)/(39*b*(a**3*c + 3*a**2*b*c*x + 3*a*b**2*c*x**2 + b**3*c*x**3)**
(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0282359, size = 25, normalized size = 0.83 \[ -\frac{2 (a+b x)}{13 b \left (c (a+b x)^3\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*(a + b*x)^3)^(-5/2),x]

[Out]

(-2*(a + b*x))/(13*b*(c*(a + b*x)^3)^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.7 \[ -{\frac{2\,bx+2\,a}{13\,b} \left ( c \left ( bx+a \right ) ^{3} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*(b*x+a)^3)^(5/2),x)

[Out]

-2/13*(b*x+a)/b/(c*(b*x+a)^3)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.45222, size = 115, normalized size = 3.83 \[ -\frac{2 \, \sqrt{c}}{13 \,{\left (b^{6} c^{3} x^{5} + 5 \, a b^{5} c^{3} x^{4} + 10 \, a^{2} b^{4} c^{3} x^{3} + 10 \, a^{3} b^{3} c^{3} x^{2} + 5 \, a^{4} b^{2} c^{3} x + a^{5} b c^{3}\right )}{\left (b x + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^3*c)^(-5/2),x, algorithm="maxima")

[Out]

-2/13*sqrt(c)/((b^6*c^3*x^5 + 5*a*b^5*c^3*x^4 + 10*a^2*b^4*c^3*x^3 + 10*a^3*b^3*
c^3*x^2 + 5*a^4*b^2*c^3*x + a^5*b*c^3)*(b*x + a)^(3/2))

_______________________________________________________________________________________

Fricas [A]  time = 0.215959, size = 204, normalized size = 6.8 \[ -\frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{13 \,{\left (b^{9} c^{3} x^{8} + 8 \, a b^{8} c^{3} x^{7} + 28 \, a^{2} b^{7} c^{3} x^{6} + 56 \, a^{3} b^{6} c^{3} x^{5} + 70 \, a^{4} b^{5} c^{3} x^{4} + 56 \, a^{5} b^{4} c^{3} x^{3} + 28 \, a^{6} b^{3} c^{3} x^{2} + 8 \, a^{7} b^{2} c^{3} x + a^{8} b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^3*c)^(-5/2),x, algorithm="fricas")

[Out]

-2/13*sqrt(b^3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)/(b^9*c^3*x^8 + 8*a*b
^8*c^3*x^7 + 28*a^2*b^7*c^3*x^6 + 56*a^3*b^6*c^3*x^5 + 70*a^4*b^5*c^3*x^4 + 56*a
^5*b^4*c^3*x^3 + 28*a^6*b^3*c^3*x^2 + 8*a^7*b^2*c^3*x + a^8*b*c^3)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c \left (a + b x\right )^{3}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*(b*x+a)**3)**(5/2),x)

[Out]

Integral((c*(a + b*x)**3)**(-5/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.528523, 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)^3*c)^(-5/2),x, algorithm="giac")

[Out]

sage0*x