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

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

[Out]

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

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Rubi [A]  time = 0.0379828, antiderivative size = 25, normalized size of antiderivative = 1., 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} \[ -\frac{4 \sqrt{a x+b \sqrt{x}}}{b \sqrt{x}} \]

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*}

Mathematica [A]  time = 0.0080369, size = 25, normalized size = 1. \[ -\frac{4 \sqrt{a x+b \sqrt{x}}}{b \sqrt{x}} \]

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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Maple [C]  time = 0.011, size = 160, normalized size = 6.4 \begin{align*} -{\frac{1}{{b}^{2}x}\sqrt{b\sqrt{x}+ax} \left ( 4\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{a}-2\,\sqrt{b\sqrt{x}+ax}{a}^{3/2}x-\ln \left ({\frac{1}{2} \left ( 2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b \right ){\frac{1}{\sqrt{a}}}} \right ) xab-2\,{a}^{3/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }x+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b \right ){\frac{1}{\sqrt{a}}}} \right ) xab \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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Fricas [A]  time = 2.43133, size = 51, normalized size = 2.04 \begin{align*} -\frac{4 \, \sqrt{a x + b \sqrt{x}}}{b \sqrt{x}} \end{align*}

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

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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Giac [A]  time = 1.22164, size = 34, normalized size = 1.36 \begin{align*} \frac{4}{\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}} \end{align*}

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)))