3.6.99 \(\int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx\) [599]

Optimal. Leaf size=20 \[ \frac {2 \sqrt {x}}{a \sqrt {a-b x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, 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 \sqrt {x}}{a \sqrt {a-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a - b*x)^(3/2)),x]

[Out]

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

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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x}}{a \sqrt {a-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x])/(a*Sqrt[a - b*x])

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.18, size = 55, normalized size = 2.75 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2}{a \sqrt {b} \sqrt {-1+\frac {a}{b x}}},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-2 I}{a \sqrt {b} \sqrt {1-\frac {a}{b x}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(Sqrt[x]*(a - b*x)^(3/2)),x]')

[Out]

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

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Maple [A]
time = 0.12, size = 17, normalized size = 0.85

method	result	size
gosper	\(\frac {2 \sqrt {x}}{a \sqrt {-b x +a}}\)	\(17\)
default	\(\frac {2 \sqrt {x}}{a \sqrt {-b x +a}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x+a)^(3/2)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, \sqrt {x}}{\sqrt {-b x + a} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)/(sqrt(-b*x + a)*a)

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Fricas [A]
time = 0.31, size = 25, normalized size = 1.25 \begin {gather*} -\frac {2 \, \sqrt {-b x + a} \sqrt {x}}{a b x - a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.50, size = 44, normalized size = 2.20 \begin {gather*} \begin {cases} \frac {2}{a \sqrt {b} \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i}{a \sqrt {b} \sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (16) = 32\).
time = 0.01, size = 59, normalized size = 2.95 \begin {gather*} -\frac {8 b \sqrt {-b}}{2 \left |b\right | \left (\left (\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right )^{2}-a b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.34, size = 24, normalized size = 1.20 \begin {gather*} \frac {2\,\sqrt {x}\,\sqrt {a-b\,x}}{a^2-a\,b\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(a - b*x)^(3/2)),x)

[Out]

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

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