3.2821 \(\int \frac{1}{\sqrt{\frac{c}{(a+b x)^2}}} \, dx\)

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

[Out]

(a + b*x)/(2*b*Sqrt[c/(a + b*x)^2])

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Rubi [A]  time = 0.0223095, 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 \[ \frac{a+b x}{2 b \sqrt{\frac{c}{(a+b x)^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c/(a + b*x)^2],x]

[Out]

(a + b*x)/(2*b*Sqrt[c/(a + b*x)^2])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c/(a + b*x)**2)*(a + b*x)*Integral(x, (x, a + b*x))/(b*c)

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Mathematica [A]  time = 0.0219579, size = 32, normalized size = 1.28 \[ \frac{x (2 a+b x)}{2 (a+b x) \sqrt{\frac{c}{(a+b x)^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[c/(a + b*x)^2],x]

[Out]

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

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Maple [A]  time = 0.005, size = 29, normalized size = 1.2 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}{\frac{1}{\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.3959, size = 20, normalized size = 0.8 \[ \frac{b x^{2} + 2 \, a x}{2 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)^2),x, algorithm="maxima")

[Out]

1/2*(b*x^2 + 2*a*x)/sqrt(c)

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Fricas [A]  time = 0.21445, size = 65, normalized size = 2.6 \[ \frac{{\left (b^{2} x^{3} + 3 \, a b x^{2} + 2 \, a^{2} x\right )} \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(c/(a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.21713, size = 45, normalized size = 1.8 \[ \frac{b c^{\frac{3}{2}} x^{2}{\rm sign}\left (b x + a\right ) + 2 \, a c^{\frac{3}{2}} x{\rm sign}\left (b x + a\right )}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)^2),x, algorithm="giac")

[Out]

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