\(\int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx\) [620]

Optimal result

Integrand size = 15, antiderivative size = 32 \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=\frac {1}{\sqrt {x} \sqrt {2+b x}}-\frac {\sqrt {2+b x}}{\sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=\frac {1}{\sqrt {x} \sqrt {b x+2}}-\frac {\sqrt {b x+2}}{\sqrt {x}} \]

[In]

Int[1/(x^(3/2)*(2 + b*x)^(3/2)),x]

[Out]

1/(Sqrt[x]*Sqrt[2 + b*x]) - Sqrt[2 + b*x]/Sqrt[x]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {1}{\sqrt {x} \sqrt {2+b x}}+\int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx \\ & = \frac {1}{\sqrt {x} \sqrt {2+b x}}-\frac {\sqrt {2+b x}}{\sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=\frac {-1-b x}{\sqrt {x} \sqrt {2+b x}} \]

[In]

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

[Out]

(-1 - b*x)/(Sqrt[x]*Sqrt[2 + b*x])

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56

method	result	size
gosper	\(-\frac {b x +1}{\sqrt {x}\, \sqrt {b x +2}}\)	\(18\)
meijerg	\(-\frac {\sqrt {2}\, \left (b x +1\right )}{2 \sqrt {x}\, \sqrt {\frac {b x}{2}+1}}\)	\(22\)
default	\(-\frac {1}{\sqrt {x}\, \sqrt {b x +2}}-\frac {b \sqrt {x}}{\sqrt {b x +2}}\)	\(27\)
risch	\(-\frac {\sqrt {b x +2}}{2 \sqrt {x}}-\frac {b \sqrt {x}}{2 \sqrt {b x +2}}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=-\frac {\sqrt {b x + 2} {\left (b x + 1\right )} \sqrt {x}}{b x^{2} + 2 \, x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=- \frac {\sqrt {b}}{\sqrt {1 + \frac {2}{b x}}} - \frac {1}{\sqrt {b} x \sqrt {1 + \frac {2}{b x}}} \]

[In]

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

[Out]

-sqrt(b)/sqrt(1 + 2/(b*x)) - 1/(sqrt(b)*x*sqrt(1 + 2/(b*x)))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=-\frac {b \sqrt {x}}{2 \, \sqrt {b x + 2}} - \frac {\sqrt {b x + 2}}{2 \, \sqrt {x}} \]

[In]

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

[Out]

-1/2*b*sqrt(x)/sqrt(b*x + 2) - 1/2*sqrt(b*x + 2)/sqrt(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (24) = 48\).

Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31 \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=-\frac {\sqrt {b x + 2} b^{2}}{2 \, \sqrt {{\left (b x + 2\right )} b - 2 \, b} {\left | b \right |}} - \frac {2 \, b^{\frac {5}{2}}}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} {\left | b \right |}} \]

[In]

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

[Out]

-1/2*sqrt(b*x + 2)*b^2/(sqrt((b*x + 2)*b - 2*b)*abs(b)) - 2*b^(5/2)/(((sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*
b - 2*b))^2 + 2*b)*abs(b))

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^{3/2} (2+b x)^{3/2}} \, dx=-\frac {b\,x+1}{\sqrt {x}\,\sqrt {b\,x+2}} \]

[In]

int(1/(x^(3/2)*(b*x + 2)^(3/2)),x)

[Out]

-(b*x + 1)/(x^(1/2)*(b*x + 2)^(1/2))