3.2799 \(\int \frac{1}{\left (c (a+b x)^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{1}{2 b c (a+b x) \sqrt{c (a+b x)^2}} \]

[Out]

-1/(2*b*c*(a + b*x)*Sqrt[c*(a + b*x)^2])

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Rubi [A]  time = 0.0304208, 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 \[ -\frac{1}{2 b c (a+b x) \sqrt{c (a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*(a + b*x)^2)^(-3/2),x]

[Out]

-1/(2*b*c*(a + b*x)*Sqrt[c*(a + b*x)^2])

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Rubi in Sympy [A]  time = 2.7823, size = 37, normalized size = 1.23 \[ - \frac{2 a + 2 b x}{4 b \left (a^{2} c + 2 a b c x + b^{2} c x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0149022, size = 25, normalized size = 0.83 \[ -\frac{a+b x}{2 b \left (c (a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 22, normalized size = 0.7 \[ -{\frac{bx+a}{2\,b} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*(b*x+a)^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.34432, size = 24, normalized size = 0.8 \[ -\frac{1}{2 \, \left (b^{2} c\right )^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((b^2*c)^(3/2)*(x + a/b)^2)

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Fricas [A]  time = 0.215344, size = 93, normalized size = 3.1 \[ -\frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{2 \,{\left (b^{4} c^{2} x^{3} + 3 \, a b^{3} c^{2} x^{2} + 3 \, a^{2} b^{2} c^{2} x + a^{3} b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c*(a + b*x)**2)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.554574, 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)^2*c)^(-3/2),x, algorithm="giac")

[Out]

sage0*x