3.27 \(\int \frac{1+a x}{x^2 \sqrt{a x} \sqrt{1-a x}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{10 a \sqrt{1-a x}}{3 \sqrt{a x}}-\frac{2 a \sqrt{1-a x}}{3 (a x)^{3/2}} \]

[Out]

(-2*a*Sqrt[1 - a*x])/(3*(a*x)^(3/2)) - (10*a*Sqrt[1 - a*x])/(3*Sqrt[a*x])

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Rubi [A]  time = 0.0620002, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{10 a \sqrt{1-a x}}{3 \sqrt{a x}}-\frac{2 a \sqrt{1-a x}}{3 (a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + a*x)/(x^2*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-2*a*Sqrt[1 - a*x])/(3*(a*x)^(3/2)) - (10*a*Sqrt[1 - a*x])/(3*Sqrt[a*x])

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Rubi in Sympy [A]  time = 7.65369, size = 41, normalized size = 0.91 \[ - \frac{10 a \sqrt{- a x + 1}}{3 \sqrt{a x}} - \frac{2 a \sqrt{- a x + 1}}{3 \left (a x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*a*sqrt(-a*x + 1)/(3*sqrt(a*x)) - 2*a*sqrt(-a*x + 1)/(3*(a*x)**(3/2))

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Mathematica [A]  time = 0.0342676, size = 29, normalized size = 0.64 \[ -\frac{2 \sqrt{-a x (a x-1)} (5 a x+1)}{3 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-(a*x*(-1 + a*x))]*(1 + 5*a*x))/(3*a*x^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.6 \[ -{\frac{10\,ax+2}{3\,x}\sqrt{-ax+1}{\frac{1}{\sqrt{ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(5*a*x+1)/x/(a*x)^(1/2)*(-a*x+1)^(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((a*x + 1)/(sqrt(a*x)*sqrt(-a*x + 1)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221137, size = 36, normalized size = 0.8 \[ -\frac{2 \,{\left (5 \, a x + 1\right )} \sqrt{a x} \sqrt{-a x + 1}}{3 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(5*a*x + 1)*sqrt(a*x)*sqrt(-a*x + 1)/(a*x^2)

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Sympy [A]  time = 26.0984, size = 107, normalized size = 2.38 \[ a \left (\begin{cases} - 2 \sqrt{-1 + \frac{1}{a x}} & \text{for}\: \left |{\frac{1}{a x}}\right | > 1 \\- 2 i \sqrt{1 - \frac{1}{a x}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{4 a \sqrt{-1 + \frac{1}{a x}}}{3} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{3 x} & \text{for}\: \left |{\frac{1}{a x}}\right | > 1 \\- \frac{4 i a \sqrt{1 - \frac{1}{a x}}}{3} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{3 x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Piecewise((-2*sqrt(-1 + 1/(a*x)), Abs(1/(a*x)) > 1), (-2*I*sqrt(1 - 1/(a*x)),
True)) + Piecewise((-4*a*sqrt(-1 + 1/(a*x))/3 - 2*sqrt(-1 + 1/(a*x))/(3*x), Abs(
1/(a*x)) > 1), (-4*I*a*sqrt(1 - 1/(a*x))/3 - 2*I*sqrt(1 - 1/(a*x))/(3*x), True))

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GIAC/XCAS [A]  time = 0.216529, size = 119, normalized size = 2.64 \[ -\frac{\frac{a^{2}{\left (\sqrt{-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac{3}{2}}} + \frac{21 \, a^{2}{\left (\sqrt{-a x + 1} - 1\right )}}{\sqrt{a x}} - \frac{{\left (a^{2} + \frac{21 \, a{\left (\sqrt{-a x + 1} - 1\right )}^{2}}{x}\right )} \left (a x\right )^{\frac{3}{2}}}{{\left (\sqrt{-a x + 1} - 1\right )}^{3}}}{12 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(a^2*(sqrt(-a*x + 1) - 1)^3/(a*x)^(3/2) + 21*a^2*(sqrt(-a*x + 1) - 1)/sqrt
(a*x) - (a^2 + 21*a*(sqrt(-a*x + 1) - 1)^2/x)*(a*x)^(3/2)/(sqrt(-a*x + 1) - 1)^3
)/a