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

Optimal. Leaf size=55 \[ -\frac{2 \sqrt{b x+2}}{3 \sqrt{x}}+\frac{2}{3 \sqrt{x} \sqrt{b x+2}}+\frac{1}{3 \sqrt{x} (b x+2)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0059313, antiderivative size = 55, 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} \[ -\frac{2 \sqrt{b x+2}}{3 \sqrt{x}}+\frac{2}{3 \sqrt{x} \sqrt{b x+2}}+\frac{1}{3 \sqrt{x} (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(2 + b*x)^(5/2)),x]

[Out]

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

Mathematica [A]  time = 0.0095435, size = 32, normalized size = 0.58 \[ \frac{-2 b^2 x^2-6 b x-3}{3 \sqrt{x} (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3 - 6*b*x - 2*b^2*x^2)/(3*Sqrt[x]*(2 + b*x)^(3/2))

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Maple [A]  time = 0.003, size = 27, normalized size = 0.5 \begin{align*} -{\frac{2\,{b}^{2}{x}^{2}+6\,bx+3}{3} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*b^2*x^2+6*b*x+3)/x^(1/2)/(b*x+2)^(3/2)

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Maxima [A]  time = 0.974572, size = 54, normalized size = 0.98 \begin{align*} \frac{{\left (b^{2} - \frac{6 \,{\left (b x + 2\right )} b}{x}\right )} x^{\frac{3}{2}}}{12 \,{\left (b x + 2\right )}^{\frac{3}{2}}} - \frac{\sqrt{b x + 2}}{4 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b^2 - 6*(b*x + 2)*b/x)*x^(3/2)/(b*x + 2)^(3/2) - 1/4*sqrt(b*x + 2)/sqrt(x)

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Fricas [A]  time = 1.62944, size = 105, normalized size = 1.91 \begin{align*} -\frac{{\left (2 \, b^{2} x^{2} + 6 \, b x + 3\right )} \sqrt{b x + 2} \sqrt{x}}{3 \,{\left (b^{2} x^{3} + 4 \, b x^{2} + 4 \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*b^2*x^2 + 6*b*x + 3)*sqrt(b*x + 2)*sqrt(x)/(b^2*x^3 + 4*b*x^2 + 4*x)

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Sympy [B]  time = 14.1047, size = 117, normalized size = 2.13 \begin{align*} - \frac{2 b^{\frac{13}{2}} x^{2} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac{6 b^{\frac{11}{2}} x \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} - \frac{3 b^{\frac{9}{2}} \sqrt{1 + \frac{2}{b x}}}{3 b^{6} x^{2} + 12 b^{5} x + 12 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.13735, size = 196, normalized size = 3.56 \begin{align*} -\frac{\sqrt{b x + 2} b^{2}}{4 \, \sqrt{{\left (b x + 2\right )} b - 2 \, b}{\left | b \right |}} - \frac{3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} b^{\frac{5}{2}} + 24 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac{7}{2}} + 20 \, b^{\frac{9}{2}}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(b*x + 2)*b^2/(sqrt((b*x + 2)*b - 2*b)*abs(b)) - 1/3*(3*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2
*b))^4*b^(5/2) + 24*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2*b^(7/2) + 20*b^(9/2))/(((sqrt(b*x + 2)
*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)^3*abs(b))