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

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

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Rubi [A]  time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {4 \left (2 a \sqrt {x}+b\right )}{b^2 \sqrt {a x+b \sqrt {x}}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.04, size = 45, normalized size = 1.50 \begin {gather*} -\frac {4 \left (2 a \sqrt {x}+b\right ) \sqrt {a x+b \sqrt {x}}}{a b^2 x+b^3 \sqrt {x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.28, size = 46, normalized size = 1.53 \begin {gather*} -\frac {4 \left (2 a \sqrt {x}+b\right ) \sqrt {a x+b \sqrt {x}}}{b^2 \sqrt {x} \left (a \sqrt {x}+b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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^2*(b + a*Sqrt[x])*Sqrt[x])

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fricas [B]  time = 0.63, size = 54, normalized size = 1.80 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.18, size = 26, normalized size = 0.87 \begin {gather*} -\frac {4 \, {\left (\frac {2 \, a \sqrt {x}}{b^{2}} + \frac {1}{b}\right )}}{\sqrt {a x + b \sqrt {x}}} \end {gather*}

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))

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maple [B]  time = 0.06, size = 111, normalized size = 3.70 \begin {gather*} -\frac {4 \sqrt {a x +b \sqrt {x}}\, \left (\left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{2} x -\left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{2} x +2 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a b \sqrt {x}+\left (a x +b \sqrt {x}\right )^{\frac {3}{2}} b^{2}\right )}{\sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} \sqrt {x}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {x}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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