\(\int \frac {1}{\sqrt {x} (b \sqrt {x}+a x)^{3/2}} \, dx\) [127]

Optimal result

Integrand size = 21, antiderivative size = 30 \[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \left (b+2 a \sqrt {x}\right )}{b^2 \sqrt {b \sqrt {x}+a x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2038, 627} \[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \left (2 a \sqrt {x}+b\right )}{b^2 \sqrt {a x+b \sqrt {x}}} \]

[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 627

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 2038

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*} \text {integral}& = 2 \text {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] (verified)

Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \left (b+2 a \sqrt {x}\right ) \sqrt {b \sqrt {x}+a x}}{b^2 \left (b+a \sqrt {x}\right ) \sqrt {x}} \]

[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^2*(b + a*Sqrt[x])*Sqrt[x])

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {4 \left (b +2 a \sqrt {x}\right )}{b^{2} \sqrt {b \sqrt {x}+a x}}\)	\(25\)
default	\(-\frac {4 \sqrt {b \sqrt {x}+a x}\, \left (x \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a^{2}+2 \sqrt {x}\, \left (b \sqrt {x}+a x \right )^{\frac {3}{2}} a b -\left (\sqrt {x}\, \left (a \sqrt {x}+b \right )\right )^{\frac {3}{2}} a^{2} x +\left (b \sqrt {x}+a x \right )^{\frac {3}{2}} b^{2}\right )}{\sqrt {\sqrt {x}\, \left (a \sqrt {x}+b \right )}\, b^{3} x \left (a \sqrt {x}+b \right )^{2}}\)	\(111\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).

Time = 0.46 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\frac {4 \, {\left (a b x - {\left (2 \, a^{2} x - b^{2}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{a^{2} b^{2} x^{2} - b^{4} x} \]

[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)

Sympy [F]

\[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {x} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} \sqrt {x}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=-\frac {4 \, {\left (\frac {2 \, a \sqrt {x}}{b^{2}} + \frac {1}{b}\right )}}{\sqrt {a x + b \sqrt {x}}} \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {x}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]

[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)