3.6.18 \(\int \frac {\sqrt {2-b x}}{x^{5/2}} \, dx\) [518]

Optimal. Leaf size=19 \[ -\frac {(2-b x)^{3/2}}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37} \begin {gather*} -\frac {(2-b x)^{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 = 0.05, size = 19, 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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.59, size = 77, normalized size = 4.05 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\sqrt {b} \left (-2+b x\right ) \sqrt {\frac {2-b x}{b x}}}{3 x},\frac {1}{\text {Abs}\left [b x\right ]}>\frac {1}{2}\right \}\right \},\frac {I b^{\frac {3}{2}} \sqrt {1-\frac {2}{b x}}}{3}-\frac {2 I \sqrt {b} \sqrt {1-\frac {2}{b x}}}{3 x}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[2 - b*x]/x^(5/2),x]')

[Out]

Piecewise[{{Sqrt[b] (-2 + b x) Sqrt[(2 - b x) / (b x)] / (3 x), 1 / Abs[b x] > 1 / 2}}, I b ^ (3 / 2) Sqrt[1 -
2 / (b x)] / 3 - 2 I Sqrt[b] Sqrt[1 - 2 / (b x)] / (3 x)]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(13)=26\).
time = 0.14, size = 29, normalized size = 1.53

method	result	size
gosper	\(-\frac {\left (-b x +2\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}}\)	\(14\)
meijerg	\(-\frac {2 \sqrt {2}\, \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}}\)	\(17\)
default	\(-\frac {2 \sqrt {-b x +2}}{3 x^{\frac {3}{2}}}+\frac {b \sqrt {-b x +2}}{3 \sqrt {x}}\)	\(29\)
risch	\(-\frac {\sqrt {\left (-b x +2\right ) x}\, \left (x^{2} b^{2}-4 b x +4\right )}{3 x^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {-x \left (b x -2\right )}}\)	\(47\)

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 = 0.27, size = 13, normalized size = 0.68 \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 = 0.32, size = 18, normalized size = 0.95 \begin {gather*} \frac {{\left (b x - 2\right )} \sqrt {-b x + 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)*sqrt(-b*x + 2)/x^(3/2)

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Sympy [C] Result contains complex when optimal does not.
time = 0.80, size = 83, normalized size = 4.37 \begin {gather*} \begin {cases} \frac {b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}}{3} - \frac {2 \sqrt {b} \sqrt {-1 + \frac {2}{b x}}}{3 x} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\\frac {i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}}{3} - \frac {2 i \sqrt {b} \sqrt {1 - \frac {2}{b x}}}{3 x} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b**(3/2)*sqrt(-1 + 2/(b*x))/3 - 2*sqrt(b)*sqrt(-1 + 2/(b*x))/(3*x), 1/Abs(b*x) > 1/2), (I*b**(3/2)*
sqrt(1 - 2/(b*x))/3 - 2*I*sqrt(b)*sqrt(1 - 2/(b*x))/(3*x), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).
time = 0.01, size = 80, normalized size = 4.21 \begin {gather*} -\frac {3\cdot 2 b^{2} b^{3} \sqrt {-b x+2} \sqrt {-b x+2} \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}}{\left |b\right | b\cdot 18 \left (-b \left (-b x+2\right )+2 b\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.22, size = 18, normalized size = 0.95 \begin {gather*} \frac {\sqrt {2-b\,x}\,\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((2 - b*x)^(1/2)/x^(5/2),x)

[Out]

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

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