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

Optimal. Leaf size=37 \[ \frac{\sqrt{x}}{3 \sqrt{b x+2}}+\frac{\sqrt{x}}{3 (b x+2)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0034164, antiderivative size = 37, 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} \[ \frac{\sqrt{x}}{3 \sqrt{b x+2}}+\frac{\sqrt{x}}{3 (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(2 + b*x)^(5/2)),x]

[Out]

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

Mathematica [A]  time = 0.0061133, size = 23, normalized size = 0.62 \[ \frac{\sqrt{x} (b x+3)}{3 (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 18, normalized size = 0.5 \begin{align*}{\frac{bx+3}{3}\sqrt{x} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00239, size = 32, normalized size = 0.86 \begin{align*} -\frac{{\left (b - \frac{3 \,{\left (b x + 2\right )}}{x}\right )} x^{\frac{3}{2}}}{6 \,{\left (b x + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4751, size = 80, normalized size = 2.16 \begin{align*} \frac{{\left (b x + 3\right )} \sqrt{b x + 2} \sqrt{x}}{3 \,{\left (b^{2} x^{2} + 4 \, b x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.98838, size = 75, normalized size = 2.03 \begin{align*} \frac{b x}{3 b^{\frac{3}{2}} x \sqrt{1 + \frac{2}{b x}} + 6 \sqrt{b} \sqrt{1 + \frac{2}{b x}}} + \frac{3}{3 b^{\frac{3}{2}} x \sqrt{1 + \frac{2}{b x}} + 6 \sqrt{b} \sqrt{1 + \frac{2}{b x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x/(3*b**(3/2)*x*sqrt(1 + 2/(b*x)) + 6*sqrt(b)*sqrt(1 + 2/(b*x))) + 3/(3*b**(3/2)*x*sqrt(1 + 2/(b*x)) + 6*sqr
t(b)*sqrt(1 + 2/(b*x)))

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Giac [B]  time = 1.09351, size = 107, normalized size = 2.89 \begin{align*} \frac{8 \,{\left (3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac{5}{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/(b*x+2)^(5/2)/x^(1/2),x, algorithm="giac")

[Out]

8/3*(3*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)*b^(5/2)/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x
+ 2)*b - 2*b))^2 + 2*b)^3*abs(b))