3.2827 \(\int \frac{1}{\sqrt{\frac{c}{(a+b x)^3}}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0220779, antiderivative size = 25, 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 \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c/(a + b*x)^3],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.52962, size = 24, normalized size = 0.96 \[ \frac{2 \sqrt{\frac{c}{\left (a + b x\right )^{3}}} \left (a + b x\right )^{4}}{5 b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c/(a + b*x)**3)*(a + b*x)**4/(5*b*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.025573, size = 25, normalized size = 1. \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[c/(a + b*x)^3],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.9 \[{\frac{2\,bx+2\,a}{5\,b}{\frac{1}{\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a)^3)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.54344, size = 36, normalized size = 1.44 \[ \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{5 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)^3),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.217922, size = 107, normalized size = 4.28 \[ \frac{2 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{5 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)^3),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(c/(a + b*x)**3), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219584, size = 217, normalized size = 8.68 \[ \frac{2 \,{\left (15 \, \sqrt{b c x + a c} a^{2} - \frac{10 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a}{c} + \frac{15 \, \sqrt{b c x + a c} a^{2} b^{8} c^{10} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a b^{8} c^{9} + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}} b^{8} c^{8}}{b^{8} c^{10}}\right )}}{15 \, b c{\rm sign}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ){\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)^3),x, algorithm="giac")

[Out]

2/15*(15*sqrt(b*c*x + a*c)*a^2 - 10*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/
2))*a/c + (15*sqrt(b*c*x + a*c)*a^2*b^8*c^10 - 10*(b*c*x + a*c)^(3/2)*a*b^8*c^9
+ 3*(b*c*x + a*c)^(5/2)*b^8*c^8)/(b^8*c^10))/(b*c*sign(b^3*x^3 + 3*a*b^2*x^2 + 3
*a^2*b*x + a^3)*sign(b*x + a))