3.2826 \(\int \sqrt{\frac{c}{(a+b x)^3}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]

[Out]

(-2*Sqrt[c/(a + b*x)^3]*(a + b*x))/b

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Rubi [A]  time = 0.0226791, antiderivative size = 23, 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 \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c/(a + b*x)^3],x]

[Out]

(-2*Sqrt[c/(a + b*x)^3]*(a + b*x))/b

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Rubi in Sympy [A]  time = 1.98856, size = 20, normalized size = 0.87 \[ - \frac{2 \sqrt{\frac{c}{\left (a + b x\right )^{3}}} \left (a + b x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(c/(a + b*x)**3)*(a + b*x)/b

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Mathematica [A]  time = 0.0121398, size = 23, normalized size = 1. \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c/(a + b*x)^3]*(a + b*x))/b

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Maple [A]  time = 0.004, size = 22, normalized size = 1. \[ -2\,{\frac{bx+a}{b}\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a)^3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.41481, size = 32, normalized size = 1.39 \[ -\frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2164, size = 58, normalized size = 2.52 \[ -\frac{2 \,{\left (b x + a\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.10123, size = 97, normalized size = 4.22 \[ \begin{cases} - \frac{2 a \sqrt{c} \sqrt{\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}}}{b} - 2 \sqrt{c} x \sqrt{\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}} & \text{for}\: b \neq 0 \\x \sqrt{\frac{c}{a^{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a*sqrt(c)*sqrt(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))/
b - 2*sqrt(c)*x*sqrt(1/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3)), Ne(b, 0
)), (x*sqrt(c/a**3), True))

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GIAC/XCAS [A]  time = 0.218709, size = 68, normalized size = 2.96 \[ -\frac{2 \, c{\rm sign}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ){\rm sign}\left (b x + a\right )}{\sqrt{b c x + a c} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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