3.6.10 \(\int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx\) [510]

Optimal. Leaf size=18 \[ -\frac {(2+b x)^{3/2}}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, 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} \begin {gather*} -\frac {(b x+2)^{3/2}}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + b*x]/x^(5/2),x]

[Out]

-1/3*(2 + b*x)^(3/2)/x^(3/2)

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

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {(2+b x)^{3/2}}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + b*x]/x^(5/2),x]

[Out]

-1/3*(2 + b*x)^(3/2)/x^(3/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(26\) vs. \(2(12)=24\).
time = 0.12, size = 27, normalized size = 1.50

method	result	size
gosper	\(-\frac {\left (b x +2\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}}\)	\(13\)
meijerg	\(-\frac {2 \sqrt {2}\, \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}}\)	\(17\)
risch	\(-\frac {x^{2} b^{2}+4 b x +4}{3 x^{\frac {3}{2}} \sqrt {b x +2}}\)	\(26\)
default	\(-\frac {2 \sqrt {b x +2}}{3 x^{\frac {3}{2}}}-\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.67 \begin {gather*} -\frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]
time = 0.51, size = 12, normalized size = 0.67 \begin {gather*} -\frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x + 2)^(3/2)/x^(3/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.72, size = 37, normalized size = 2.06 \begin {gather*} - \frac {b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{3} - \frac {2 \sqrt {b} \sqrt {1 + \frac {2}{b x}}}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
time = 1.07, size = 29, normalized size = 1.61 \begin {gather*} -\frac {{\left (b x + 2\right )}^{\frac {3}{2}} b^{4}}{3 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.21, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b\,x+2}\,\left (\frac {b\,x}{3}+\frac {2}{3}\right )}{x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + 2)^(1/2)/x^(5/2),x)

[Out]

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

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