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

Optimal. Leaf size=21 \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{c x}} \]

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[c*x])

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Rubi [A]  time = 0.0210846, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2 \sqrt{a+b x}}{a \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[c*x]*Sqrt[a + b*x]),x]

[Out]

(-2*Sqrt[a + b*x])/(a*Sqrt[c*x])

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Rubi in Sympy [A]  time = 3.47961, size = 19, normalized size = 0.9 \[ - \frac{2 \sqrt{a + b x}}{a \sqrt{c x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a + b*x)/(a*sqrt(c*x))

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Mathematica [A]  time = 0.0227006, size = 23, normalized size = 1.1 \[ -\frac{2 c x \sqrt{a+b x}}{a (c x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 18, normalized size = 0.9 \[ -2\,{\frac{\sqrt{bx+a}}{a\sqrt{cx}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21755, size = 31, normalized size = 1.48 \[ -\frac{2 \, \sqrt{b x + a} \sqrt{c x}}{a c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + a)*sqrt(c*x)/(a*c*x)

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Sympy [A]  time = 4.9157, size = 24, normalized size = 1.14 \[ - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238733, size = 47, normalized size = 2.24 \[ -\frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b c - a b c} a{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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