3.6.2 \(\int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx\) [502]

Optimal. Leaf size=22 \[ -\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 22, 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 (a-b x)^{3/2}}{3 a x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - b*x]/x^(5/2),x]

[Out]

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

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Mathematica [A]
time = 0.06, size = 22, normalized size = 1.00 \begin {gather*} -\frac {2 (a-b x)^{3/2}}{3 a x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - b*x]/x^(5/2),x]

[Out]

(-2*(a - b*x)^(3/2))/(3*a*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 = 2.65, size = 90, normalized size = 4.09 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-a+b x\right ) \sqrt {\frac {a-b x}{b x}}}{3 a x},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {I 2 b^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}}{3 a}-\frac {2 I \sqrt {b} \sqrt {1-\frac {a}{b x}}}{3 x}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(16)=32\).
time = 0.11, size = 52, normalized size = 2.36

method	result	size
gosper	\(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\)	\(17\)
risch	\(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}}}{3 a \,x^{\frac {3}{2}}}\)	\(17\)
default	\(-\frac {\sqrt {-b x +a}}{x^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{2}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 16, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (-b x + a\right )}^{\frac {3}{2}}}{3 \, a x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-b*x + a)^(3/2)/(a*x^(3/2))

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Fricas [A]
time = 0.30, size = 23, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (b x - a\right )} \sqrt {-b x + a}}{3 \, a x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.82, size = 88, normalized size = 4.00 \begin {gather*} \begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 x} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{3 x} + \frac {2 i b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{3 a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (16) = 32\).
time = 0.01, size = 83, normalized size = 3.77 \begin {gather*} -\frac {\frac {1}{9}\cdot 3\cdot 2 b^{2} b^{3} \sqrt {a-b x} \sqrt {a-b x} \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}}{\left |b\right | b a \left (a b-b \left (a-b x\right )\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.24, size = 21, normalized size = 0.95 \begin {gather*} \frac {\left (\frac {2\,b\,x}{3\,a}-\frac {2}{3}\right )\,\sqrt {a-b\,x}}{x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x)^(1/2)/x^(5/2),x)

[Out]

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

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