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

Optimal. Leaf size=30 \[ -\frac{4 \left (2 a \sqrt{x}+b\right )}{b^2 \sqrt{a x+b \sqrt{x}}} \]

[Out]

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

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Rubi [A]  time = 0.0471669, antiderivative size = 30, 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 = {2013, 613} \[ -\frac{4 \left (2 a \sqrt{x}+b\right )}{b^2 \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2013

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[(a*x^Simplify[j/n]
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j
/n]] && EqQ[Simplify[m - n + 1], 0]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

Mathematica [A]  time = 0.0373334, size = 45, normalized size = 1.5 \[ -\frac{4 \left (2 a \sqrt{x}+b\right ) \sqrt{a x+b \sqrt{x}}}{a b^2 x+b^3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.01, size = 111, normalized size = 3.7 \begin{align*} -4\,{\frac{\sqrt{b\sqrt{x}+ax} \left ( \left ( b\sqrt{x}+ax \right ) ^{3/2}x{a}^{2}+2\, \left ( b\sqrt{x}+ax \right ) ^{3/2}\sqrt{x}ab- \left ( \sqrt{x} \left ( b+a\sqrt{x} \right ) \right ) ^{3/2}x{a}^{2}+ \left ( b\sqrt{x}+ax \right ) ^{3/2}{b}^{2} \right ) }{\sqrt{\sqrt{x} \left ( b+a\sqrt{x} \right ) }{b}^{3} \left ( b+a\sqrt{x} \right ) ^{2}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b \sqrt{x}\right )}^{\frac{3}{2}} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.2923, size = 109, normalized size = 3.63 \begin{align*} \frac{4 \,{\left (a b x -{\left (2 \, a^{2} x - b^{2}\right )} \sqrt{x}\right )} \sqrt{a x + b \sqrt{x}}}{a^{2} b^{2} x^{2} - b^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21724, size = 35, normalized size = 1.17 \begin{align*} -\frac{4 \,{\left (\frac{2 \, a \sqrt{x}}{b^{2}} + \frac{1}{b}\right )}}{\sqrt{a x + b \sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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