3.6.94 \(\int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx\) [594]

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

[Out]

-2*(-b*x+a)^(1/2)/a/x^(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 {a-b x}}{a \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[a - b*x]),x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a - b*x])/(a*Sqrt[x])

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.19, size = 55, normalized size = 2.75 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-2 \sqrt {b} \sqrt {-1+\frac {a}{b x}}}{a},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-2 I \sqrt {b} \sqrt {1-\frac {a}{b x}}}{a}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 17, normalized size = 0.85

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 16, normalized size = 0.80 \begin {gather*} -\frac {2 \, \sqrt {-b x + a}}{a \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

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Fricas [A]
time = 0.31, size = 16, normalized size = 0.80 \begin {gather*} -\frac {2 \, \sqrt {-b x + a}}{a \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

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Sympy [A]
time = 0.51, size = 46, normalized size = 2.30 \begin {gather*} \begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{a} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).
time = 0.00, size = 40, normalized size = 2.00 \begin {gather*} \frac {8 \sqrt {-b}}{2 \left (\left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2}-a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.40, size = 16, normalized size = 0.80 \begin {gather*} -\frac {2\,\sqrt {a-b\,x}}{a\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(a - b*x)^(1/2)),x)

[Out]

-(2*(a - b*x)^(1/2))/(a*x^(1/2))

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