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

Optimal. Leaf size=38 \[ \frac{b \sqrt{b x+2}}{3 \sqrt{x}}-\frac{\sqrt{b x+2}}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0035665, antiderivative size = 38, 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} \[ \frac{b \sqrt{b x+2}}{3 \sqrt{x}}-\frac{\sqrt{b x+2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[2 + b*x]),x]

[Out]

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

Mathematica [A]  time = 0.0053017, size = 23, normalized size = 0.61 \[ \frac{(b x-1) \sqrt{b x+2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 18, normalized size = 0.5 \begin{align*}{\frac{bx-1}{3}\sqrt{bx+2}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.19424, size = 35, normalized size = 0.92 \begin{align*} \frac{\sqrt{b x + 2} b}{2 \, \sqrt{x}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{6 \, 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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.86946, size = 50, normalized size = 1.32 \begin{align*} \frac{\sqrt{b x + 2}{\left (b x - 1\right )}}{3 \, 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)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(b*x + 2)*(b*x - 1)/x^(3/2)

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Sympy [A]  time = 4.14656, size = 34, normalized size = 0.89 \begin{align*} \frac{b^{\frac{3}{2}} \sqrt{1 + \frac{2}{b x}}}{3} - \frac{\sqrt{b} \sqrt{1 + \frac{2}{b x}}}{3 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07499, size = 57, normalized size = 1.5 \begin{align*} \frac{{\left ({\left (b x + 2\right )} b^{3} - 3 \, b^{3}\right )} \sqrt{b x + 2} b}{3 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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