\(\int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx\) [586]

Optimal result

Integrand size = 15, antiderivative size = 21 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

[Out]

2/3*x^(3/2)/a/(b*x+a)^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

[In]

Int[Sqrt[x]/(a + b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

[In]

Integrate[Sqrt[x]/(a + b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76

method	result	size
gosper	\(\frac {2 x^{\frac {3}{2}}}{3 a \left (b x +a \right )^{\frac {3}{2}}}\)	\(16\)
default	\(-\frac {\sqrt {x}}{b \left (b x +a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {2 \sqrt {x}}{3 a \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {b x +a}}\right )}{2 b}\)	\(54\)

[In]

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

[Out]

2/3*x^(3/2)/a/(b*x+a)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} x^{\frac {3}{2}}}{3 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (17) = 34\).

Time = 0.81 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {2 x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x}{a}}} \]

[In]

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

[Out]

2*x**(3/2)/(3*a**(5/2)*sqrt(1 + b*x/a) + 3*a**(3/2)*b*x*sqrt(1 + b*x/a))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {2 \, x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a} \]

[In]

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

[Out]

2/3*x^(3/2)/((b*x + a)^(3/2)*a)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (15) = 30\).

Time = 0.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 4.10 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {4 \, {\left (3 \, {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} + a^{2} b^{\frac {5}{2}}\right )} {\left | b \right |}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{2}} \]

[In]

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

[Out]

4/3*(3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*sqrt(b) + a^2*b^(5/2))*abs(b)/(((sqrt(b*x + a)*sqrt
(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*b^2)

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {x}}{(a+b x)^{5/2}} \, dx=\frac {2\,x^{3/2}\,\sqrt {a+b\,x}}{3\,\left (a^3+2\,a^2\,b\,x+a\,b^2\,x^2\right )} \]

[In]

int(x^(1/2)/(a + b*x)^(5/2),x)

[Out]

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