\(\int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx\) [613]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=-\frac {\sqrt {2+b x}}{3 x^{3/2}}+\frac {b \sqrt {2+b x}}{3 \sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 38, 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} \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=\frac {b \sqrt {b x+2}}{3 \sqrt {x}}-\frac {\sqrt {b x+2}}{3 x^{3/2}} \]

[In]

Int[1/(x^(5/2)*Sqrt[2 + b*x]),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=\frac {(-1+b x) \sqrt {2+b x}}{3 x^{3/2}} \]

[In]

Integrate[1/(x^(5/2)*Sqrt[2 + b*x]),x]

[Out]

((-1 + b*x)*Sqrt[2 + b*x])/(3*x^(3/2))

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.47

method	result	size
gosper	\(\frac {\sqrt {b x +2}\, \left (b x -1\right )}{3 x^{\frac {3}{2}}}\)	\(18\)
meijerg	\(-\frac {\sqrt {2}\, \left (-b x +1\right ) \sqrt {\frac {b x}{2}+1}}{3 x^{\frac {3}{2}}}\)	\(23\)
risch	\(\frac {b^{2} x^{2}+b x -2}{3 x^{\frac {3}{2}} \sqrt {b x +2}}\)	\(25\)
default	\(-\frac {\sqrt {b x +2}}{3 x^{\frac {3}{2}}}+\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\)	\(27\)

[In]

int(1/x^(5/2)/(b*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=\frac {\sqrt {b x + 2} {\left (b x - 1\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate(1/x^(5/2)/(b*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(b*x + 2)*(b*x - 1)/x^(3/2)

Sympy [A] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=\frac {b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{3} - \frac {\sqrt {b} \sqrt {1 + \frac {2}{b x}}}{3 x} \]

[In]

integrate(1/x**(5/2)/(b*x+2)**(1/2),x)

[Out]

b**(3/2)*sqrt(1 + 2/(b*x))/3 - sqrt(b)*sqrt(1 + 2/(b*x))/(3*x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=\frac {\sqrt {b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{6 \, x^{\frac {3}{2}}} \]

[In]

integrate(1/x^(5/2)/(b*x+2)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(b*x + 2)*b/sqrt(x) - 1/6*(b*x + 2)^(3/2)/x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=\frac {{\left ({\left (b x + 2\right )} b^{3} - 3 \, b^{3}\right )} \sqrt {b x + 2} b}{3 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}} {\left | b \right |}} \]

[In]

integrate(1/x^(5/2)/(b*x+2)^(1/2),x, algorithm="giac")

[Out]

1/3*((b*x + 2)*b^3 - 3*b^3)*sqrt(b*x + 2)*b/(((b*x + 2)*b - 2*b)^(3/2)*abs(b))

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx=\frac {\sqrt {b\,x+2}\,\left (\frac {b\,x}{3}-\frac {1}{3}\right )}{x^{3/2}} \]

[In]

int(1/(x^(5/2)*(b*x + 2)^(1/2)),x)

[Out]

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