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

Optimal. Leaf size=15 \[ \frac {\sqrt {x}}{\sqrt {b x+2}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \frac {\sqrt {x}}{\sqrt {b x+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(2 + b*x)^(3/2)),x]

[Out]

Sqrt[x]/Sqrt[2 + b*x]

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)^{3/2}} \, dx &=\frac {\sqrt {x}}{\sqrt {2+b x}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\sqrt {x}}{\sqrt {b x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[x]/Sqrt[2 + b*x]

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fricas [A]  time = 0.43, size = 11, normalized size = 0.73 \[ \frac {\sqrt {x}}{\sqrt {b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)/sqrt(b*x + 2)

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giac [B]  time = 1.22, size = 44, normalized size = 2.93 \[ \frac {4 \, b^{\frac {3}{2}}}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 12, normalized size = 0.80 \[ \frac {\sqrt {x}}{\sqrt {b x +2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 11, normalized size = 0.73 \[ \frac {\sqrt {x}}{\sqrt {b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)/sqrt(b*x + 2)

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mupad [B]  time = 0.31, size = 11, normalized size = 0.73 \[ \frac {\sqrt {x}}{\sqrt {b\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(b*x + 2)^(3/2)),x)

[Out]

x^(1/2)/(b*x + 2)^(1/2)

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sympy [A]  time = 0.86, size = 15, normalized size = 1.00 \[ \frac {1}{\sqrt {b} \sqrt {1 + \frac {2}{b x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(sqrt(b)*sqrt(1 + 2/(b*x)))

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