3.1.63 \(\int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx\)

Optimal. Leaf size=25 \[ -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \]

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(b*Sqrt[x])

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx &=-\frac {4 \sqrt {b \sqrt {x}+a x}}{b \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(b*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.14, size = 25, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}}}{b \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(b*Sqrt[x])

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fricas [A]  time = 0.85, size = 19, normalized size = 0.76 \begin {gather*} -\frac {4 \, \sqrt {a x + b \sqrt {x}}}{b \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(a*x + b*sqrt(x))/(b*sqrt(x))

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giac [A]  time = 0.18, size = 25, normalized size = 1.00 \begin {gather*} \frac {4}{\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.06, size = 159, normalized size = 6.36 \begin {gather*} \frac {\sqrt {a x +b \sqrt {x}}\, \left (-a b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+a b x \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {3}{2}} x +2 \sqrt {a x +b \sqrt {x}}\, a^{\frac {3}{2}} x -4 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}\, b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a*x+b*x^(1/2))^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b \sqrt {x}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a*x + b*sqrt(x))*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x\,\sqrt {a\,x+b\,\sqrt {x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a*x + b*x^(1/2))^(1/2)),x)

[Out]

int(1/(x*(a*x + b*x^(1/2))^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {a x + b \sqrt {x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(a*x + b*sqrt(x))), x)

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