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 = 0.0098901, 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, Rules used = {16, 78, 37} \[ -\frac{10 a \sqrt{1-a x}}{3 \sqrt{a x}}-\frac{2 a \sqrt{1-a x}}{3 (a x)^{3/2}} \]

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])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1+a x}{x^2 \sqrt{a x} \sqrt{1-a x}} \, dx &=a^2 \int \frac{1+a x}{(a x)^{5/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a \sqrt{1-a x}}{3 (a x)^{3/2}}+\frac{1}{3} \left (5 a^2\right ) \int \frac{1}{(a x)^{3/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a \sqrt{1-a x}}{3 (a x)^{3/2}}-\frac{10 a \sqrt{1-a x}}{3 \sqrt{a x}}\\ \end{align*}

Mathematica [A]  time = 0.012325, 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 = 0.003, size = 25, normalized size = 0.6 \begin{align*} -{\frac{10\,ax+2}{3\,x}\sqrt{-ax+1}{\frac{1}{\sqrt{ax}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45223, size = 69, normalized size = 1.53 \begin{align*} -\frac{2 \,{\left (5 \, a x + 1\right )} \sqrt{a x} \sqrt{-a x + 1}}{3 \, a x^{2}} \end{align*}

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, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 8.86366, size = 107, normalized size = 2.38 \begin{align*} a \left (\begin{cases} - 2 \sqrt{-1 + \frac{1}{a x}} & \text{for}\: \frac{1}{\left |{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}\: \frac{1}{\left |{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} \end{align*}

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)), 1/Abs(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), 1/Abs(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 [B]  time = 1.57237, size = 119, normalized size = 2.64 \begin{align*} -\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} \end{align*}

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, 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