3.621 \(\int \frac{1}{x^{5/2} (2+b x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0073547, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(2 + b*x)^(3/2)),x]

[Out]

1/(x^(3/2)*Sqrt[2 + b*x]) - (2*Sqrt[2 + b*x])/(3*x^(3/2)) + (2*b*Sqrt[2 + b*x])/(3*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^{5/2} (2+b x)^{3/2}} \, dx &=\frac{1}{x^{3/2} \sqrt{2+b x}}+2 \int \frac{1}{x^{5/2} \sqrt{2+b x}} \, dx\\ &=\frac{1}{x^{3/2} \sqrt{2+b x}}-\frac{2 \sqrt{2+b x}}{3 x^{3/2}}-\frac{1}{3} (2 b) \int \frac{1}{x^{3/2} \sqrt{2+b x}} \, dx\\ &=\frac{1}{x^{3/2} \sqrt{2+b x}}-\frac{2 \sqrt{2+b x}}{3 x^{3/2}}+\frac{2 b \sqrt{2+b x}}{3 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.006948, size = 32, normalized size = 0.6 \[ \frac{2 b^2 x^2+2 b x-1}{3 x^{3/2} \sqrt{b x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(2 + b*x)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 27, normalized size = 0.5 \begin{align*}{\frac{2\,{b}^{2}{x}^{2}+2\,bx-1}{3}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x+2)^(3/2),x)

[Out]

1/3*(2*b^2*x^2+2*b*x-1)/x^(3/2)/(b*x+2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.07154, size = 55, normalized size = 1.04 \begin{align*} \frac{b^{2} \sqrt{x}}{4 \, \sqrt{b x + 2}} + \frac{\sqrt{b x + 2} b}{2 \, \sqrt{x}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{12 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x+2)^(3/2),x, algorithm="maxima")

[Out]

1/4*b^2*sqrt(x)/sqrt(b*x + 2) + 1/2*sqrt(b*x + 2)*b/sqrt(x) - 1/12*(b*x + 2)^(3/2)/x^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.48534, size = 90, normalized size = 1.7 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} + 2 \, b x - 1\right )} \sqrt{b x + 2} \sqrt{x}}{3 \,{\left (b x^{3} + 2 \, x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x+2)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 13.9313, size = 170, normalized size = 3.21 \begin{align*} \frac{2 b^{\frac{15}{2}} x^{3} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac{6 b^{\frac{13}{2}} x^{2} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac{3 b^{\frac{11}{2}} x \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} - \frac{2 b^{\frac{9}{2}} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(5/2)/(b*x+2)**(3/2),x)

[Out]

2*b**(15/2)*x**3*sqrt(1 + 2/(b*x))/(3*b**6*x**3 + 12*b**5*x**2 + 12*b**4*x) + 6*b**(13/2)*x**2*sqrt(1 + 2/(b*x
))/(3*b**6*x**3 + 12*b**5*x**2 + 12*b**4*x) + 3*b**(11/2)*x*sqrt(1 + 2/(b*x))/(3*b**6*x**3 + 12*b**5*x**2 + 12
*b**4*x) - 2*b**(9/2)*sqrt(1 + 2/(b*x))/(3*b**6*x**3 + 12*b**5*x**2 + 12*b**4*x)

________________________________________________________________________________________

Giac [B]  time = 1.08034, size = 116, normalized size = 2.19 \begin{align*} \frac{b^{\frac{7}{2}}}{{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}{\left | b \right |}} + \frac{{\left (5 \,{\left (b x + 2\right )} b^{2}{\left | b \right |} - 12 \, b^{2}{\left | b \right |}\right )} \sqrt{b x + 2}}{12 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x+2)^(3/2),x, algorithm="giac")

[Out]

b^(7/2)/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)*abs(b)) + 1/12*(5*(b*x + 2)*b^2*abs(b) -
12*b^2*abs(b))*sqrt(b*x + 2)/((b*x + 2)*b - 2*b)^(3/2)