3.28 \(\int \frac{1+a x}{x^3 \sqrt{a x} \sqrt{1-a x}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{12 a^2 \sqrt{1-a x}}{5 \sqrt{a x}}-\frac{6 a^2 \sqrt{1-a x}}{5 (a x)^{3/2}}-\frac{2 a^2 \sqrt{1-a x}}{5 (a x)^{5/2}} \]

[Out]

(-2*a^2*Sqrt[1 - a*x])/(5*(a*x)^(5/2)) - (6*a^2*Sqrt[1 - a*x])/(5*(a*x)^(3/2)) - (12*a^2*Sqrt[1 - a*x])/(5*Sqr
t[a*x])

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Rubi [A]  time = 0.0190258, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {16, 78, 45, 37} \[ -\frac{12 a^2 \sqrt{1-a x}}{5 \sqrt{a x}}-\frac{6 a^2 \sqrt{1-a x}}{5 (a x)^{3/2}}-\frac{2 a^2 \sqrt{1-a x}}{5 (a x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + a*x)/(x^3*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-2*a^2*Sqrt[1 - a*x])/(5*(a*x)^(5/2)) - (6*a^2*Sqrt[1 - a*x])/(5*(a*x)^(3/2)) - (12*a^2*Sqrt[1 - a*x])/(5*Sqr
t[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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^3 \sqrt{a x} \sqrt{1-a x}} \, dx &=a^3 \int \frac{1+a x}{(a x)^{7/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^2 \sqrt{1-a x}}{5 (a x)^{5/2}}+\frac{1}{5} \left (9 a^3\right ) \int \frac{1}{(a x)^{5/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^2 \sqrt{1-a x}}{5 (a x)^{5/2}}-\frac{6 a^2 \sqrt{1-a x}}{5 (a x)^{3/2}}+\frac{1}{5} \left (6 a^3\right ) \int \frac{1}{(a x)^{3/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^2 \sqrt{1-a x}}{5 (a x)^{5/2}}-\frac{6 a^2 \sqrt{1-a x}}{5 (a x)^{3/2}}-\frac{12 a^2 \sqrt{1-a x}}{5 \sqrt{a x}}\\ \end{align*}

Mathematica [A]  time = 0.0146271, size = 37, normalized size = 0.51 \[ -\frac{2 \sqrt{-a x (a x-1)} \left (6 a^2 x^2+3 a x+1\right )}{5 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + a*x)/(x^3*Sqrt[a*x]*Sqrt[1 - a*x]),x]

[Out]

(-2*Sqrt[-(a*x*(-1 + a*x))]*(1 + 3*a*x + 6*a^2*x^2))/(5*a*x^3)

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Maple [A]  time = 0.004, size = 33, normalized size = 0.5 \begin{align*} -{\frac{12\,{a}^{2}{x}^{2}+6\,ax+2}{5\,{x}^{2}}\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^3/(a*x)^(1/2)/(-a*x+1)^(1/2),x)

[Out]

-2/5*(6*a^2*x^2+3*a*x+1)/x^2/(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^3/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.49853, size = 85, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (6 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt{a x} \sqrt{-a x + 1}}{5 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/x^3/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="fricas")

[Out]

-2/5*(6*a^2*x^2 + 3*a*x + 1)*sqrt(a*x)*sqrt(-a*x + 1)/(a*x^3)

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Sympy [C]  time = 12.7437, size = 189, normalized size = 2.59 \begin{align*} a \left (\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}\right ) + \begin{cases} - \frac{16 a^{2} \sqrt{-1 + \frac{1}{a x}}}{15} - \frac{8 a \sqrt{-1 + \frac{1}{a x}}}{15 x} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{5 x^{2}} & \text{for}\: \frac{1}{\left |{a x}\right |} > 1 \\- \frac{16 i a^{2} \sqrt{1 - \frac{1}{a x}}}{15} - \frac{8 i a \sqrt{1 - \frac{1}{a x}}}{15 x} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{5 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/x**3/(a*x)**(1/2)/(-a*x+1)**(1/2),x)

[Out]

a*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)) + Piecewise((-16*a**2*sqrt(-1 + 1/(a*x))/15 - 8*a*sqrt(-1 + 1/(a*x))
/(15*x) - 2*sqrt(-1 + 1/(a*x))/(5*x**2), 1/Abs(a*x) > 1), (-16*I*a**2*sqrt(1 - 1/(a*x))/15 - 8*I*a*sqrt(1 - 1/
(a*x))/(15*x) - 2*I*sqrt(1 - 1/(a*x))/(5*x**2), True))

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Giac [B]  time = 1.8686, size = 176, normalized size = 2.41 \begin{align*} -\frac{\frac{a^{3}{\left (\sqrt{-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac{5}{2}}} + \frac{15 \, a^{3}{\left (\sqrt{-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac{3}{2}}} + \frac{110 \, a^{3}{\left (\sqrt{-a x + 1} - 1\right )}}{\sqrt{a x}} - \frac{{\left (a^{3} + \frac{15 \, a^{2}{\left (\sqrt{-a x + 1} - 1\right )}^{2}}{x} + \frac{110 \, a{\left (\sqrt{-a x + 1} - 1\right )}^{4}}{x^{2}}\right )} \left (a x\right )^{\frac{5}{2}}}{{\left (\sqrt{-a x + 1} - 1\right )}^{5}}}{80 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/x^3/(a*x)^(1/2)/(-a*x+1)^(1/2),x, algorithm="giac")

[Out]

-1/80*(a^3*(sqrt(-a*x + 1) - 1)^5/(a*x)^(5/2) + 15*a^3*(sqrt(-a*x + 1) - 1)^3/(a*x)^(3/2) + 110*a^3*(sqrt(-a*x
+ 1) - 1)/sqrt(a*x) - (a^3 + 15*a^2*(sqrt(-a*x + 1) - 1)^2/x + 110*a*(sqrt(-a*x + 1) - 1)^4/x^2)*(a*x)^(5/2)/
(sqrt(-a*x + 1) - 1)^5)/a