3.581 \(\int \frac{1}{x^{3/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{a \sqrt{x} \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{a^2 \sqrt{x}} \]

[Out]

2/(a*Sqrt[x]*Sqrt[a + b*x]) - (4*Sqrt[a + b*x])/(a^2*Sqrt[x])

________________________________________________________________________________________

Rubi [A]  time = 0.0047896, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{2}{a \sqrt{x} \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{a^2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + b*x)^(3/2)),x]

[Out]

2/(a*Sqrt[x]*Sqrt[a + b*x]) - (4*Sqrt[a + b*x])/(a^2*Sqrt[x])

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}{x^{3/2} (a+b x)^{3/2}} \, dx &=\frac{2}{a \sqrt{x} \sqrt{a+b x}}+\frac{2 \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{a}\\ &=\frac{2}{a \sqrt{x} \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{a^2 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0095566, size = 25, normalized size = 0.64 \[ -\frac{2 (a+2 b x)}{a^2 \sqrt{x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + b*x)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.6 \begin{align*} -2\,{\frac{2\,bx+a}{{a}^{2}\sqrt{x}\sqrt{bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(b*x+a)^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.12064, size = 43, normalized size = 1.1 \begin{align*} -\frac{2 \, b \sqrt{x}}{\sqrt{b x + a} a^{2}} - \frac{2 \, \sqrt{b x + a}}{a^{2} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.95174, size = 78, normalized size = 2. \begin{align*} -\frac{2 \,{\left (2 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}}{a^{2} b x^{2} + a^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 2.79672, size = 41, normalized size = 1.05 \begin{align*} - \frac{2}{a \sqrt{b} x \sqrt{\frac{a}{b x} + 1}} - \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*sqrt(b)*x*sqrt(a/(b*x) + 1)) - 4*sqrt(b)/(a**2*sqrt(a/(b*x) + 1))

________________________________________________________________________________________

Giac [B]  time = 1.07059, size = 111, normalized size = 2.85 \begin{align*} -\frac{4 \, b^{\frac{5}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a{\left | b \right |}} - \frac{2 \, \sqrt{b x + a} b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a^{2}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*b^(5/2)/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a*abs(b)) - 2*sqrt(b*x + a)*b^2/(sqrt(
(b*x + a)*b - a*b)*a^2*abs(b))