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

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

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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]

-(2 + b*x)^(3/2)/(3*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 = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {(b x+2)^{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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IntegrateAlgebraic [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {(b x+2)^{3/2}}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(2 + b*x)^(3/2)/x^(3/2)

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fricas [A]  time = 0.78, 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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giac [B]  time = 1.15, 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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maple [A]  time = 0.00, size = 13, normalized size = 0.72 \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]

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

[Out]

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

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maxima [A]  time = 1.31, 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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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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sympy [B]  time = 1.45, 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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