3.6.73 \(\int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx\) [573]

Optimal. Leaf size=19 \[ -\frac {2 \sqrt {a+b x}}{a \sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 19, 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 {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 = 19, 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 [A]
time = 2.00, size = 20, normalized size = 1.05 \begin {gather*} \frac {-2 \sqrt {b} \sqrt {1+\frac {a}{b x}}}{a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-2 Sqrt[b] Sqrt[1 + a / (b x)] / a

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Maple [A]
time = 0.12, size = 16, normalized size = 0.84

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {x}}\)	\(16\)
default	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {x}}\)	\(16\)
risch	\(-\frac {2 \sqrt {b x +a}}{a \sqrt {x}}\)	\(16\)

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 = 15, normalized size = 0.79 \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 = 0.30, size = 15, normalized size = 0.79 \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 = 0.48, size = 19, normalized size = 1.00 \begin {gather*} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a} \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]

-2*sqrt(b)*sqrt(a/(b*x) + 1)/a

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Giac [A]
time = 0.00, size = 37, normalized size = 1.95 \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 = 0.35, size = 15, normalized size = 0.79 \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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