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