3.121 \(\int \frac{1}{x^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=54 \[ \frac{8 a \sqrt{a x+b \sqrt{x}}}{3 b^2 \sqrt{x}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{3 b x} \]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(3*b*x) + (8*a*Sqrt[b*Sqrt[x] + a*x])/(3*b^2*Sqrt[x])

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Rubi [A]  time = 0.073614, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac{8 a \sqrt{a x+b \sqrt{x}}}{3 b^2 \sqrt{x}}-\frac{4 \sqrt{a x+b \sqrt{x}}}{3 b x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x])/(3*b*x) + (8*a*Sqrt[b*Sqrt[x] + a*x])/(3*b^2*Sqrt[x])

Rule 2016

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*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

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^{3/2} \sqrt{b \sqrt{x}+a x}} \, dx &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{3 b x}-\frac{(2 a) \int \frac{1}{x \sqrt{b \sqrt{x}+a x}} \, dx}{3 b}\\ &=-\frac{4 \sqrt{b \sqrt{x}+a x}}{3 b x}+\frac{8 a \sqrt{b \sqrt{x}+a x}}{3 b^2 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0466093, size = 37, normalized size = 0.69 \[ \frac{4 \left (2 a \sqrt{x}-b\right ) \sqrt{a x+b \sqrt{x}}}{3 b^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-b + 2*a*Sqrt[x])*Sqrt[b*Sqrt[x] + a*x])/(3*b^2*x)

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Maple [C]  time = 0.009, size = 194, normalized size = 3.6 \begin{align*}{\frac{1}{3\,{b}^{3}}\sqrt{b\sqrt{x}+ax} \left ( 12\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{3/2}{x}^{3/2}-6\,\sqrt{b\sqrt{x}+ax}{a}^{5/2}{x}^{5/2}-3\,\ln \left ( 1/2\,{\frac{2\,a\sqrt{x}+2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+b}{\sqrt{a}}} \right ){x}^{5/2}{a}^{2}b-6\,{a}^{5/2}{x}^{5/2}\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }\sqrt{a}+2\,a\sqrt{x}+b}{\sqrt{a}}} \right ){x}^{5/2}{a}^{2}b-4\, \left ( b\sqrt{x}+ax \right ) ^{3/2}b\sqrt{a}x \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }}}{\frac{1}{\sqrt{a}}}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b*x^(1/2)+a*x)^(1/2)*(12*(b*x^(1/2)+a*x)^(3/2)*a^(3/2)*x^(3/2)-6*(b*x^(1/2)+a*x)^(1/2)*a^(5/2)*x^(5/2)-3*
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^(5/2)*a^2*b-6*a^(5/2)*x^(5/2)*(x^(1/2)*(b+a*
x^(1/2)))^(1/2)+3*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^(5/2)*a^2*b-4*(b*x
^(1/2)+a*x)^(3/2)*b*a^(1/2)*x)/(x^(1/2)*(b+a*x^(1/2)))^(1/2)/b^3/a^(1/2)/x^(5/2)

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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^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.30499, size = 72, normalized size = 1.33 \begin{align*} \frac{4 \, \sqrt{a x + b \sqrt{x}}{\left (2 \, a \sqrt{x} - b\right )}}{3 \, b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{3}{2}} \sqrt{a x + b \sqrt{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**(3/2)*sqrt(a*x + b*sqrt(x))), x)

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Giac [A]  time = 1.27765, size = 72, normalized size = 1.33 \begin{align*} \frac{4 \,{\left (3 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} + b\right )}}{3 \,{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(3*sqrt(a)*(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) + b)/(sqrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x)))^3