3.2809 \(\int \frac{1}{\sqrt{c (a+b x)^3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0288196, 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 \[ -\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c*(a + b*x)^3],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.32463, size = 51, normalized size = 2.22 \[ - \frac{2 \left (3 a + 3 b x\right )}{3 b \sqrt{a^{3} c + 3 a^{2} b c x + 3 a b^{2} c x^{2} + b^{3} c x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0147842, size = 23, normalized size = 1. \[ -\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*(b*x+a)^3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.36705, size = 36, normalized size = 1.57 \[ -\frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212834, size = 82, normalized size = 3.57 \[ -\frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{b^{3} c x^{2} + 2 \, a b^{2} c x + a^{2} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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GIAC/XCAS [A]  time = 0.217114, size = 31, normalized size = 1.35 \[ -\frac{2}{\sqrt{b c x + a c} b{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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