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

Optimal. Leaf size=21 \[ \frac{2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

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Rubi [A]  time = 0.001626, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ \frac{2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[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*} \int \frac{\sqrt{x}}{(a+b x)^{5/2}} \, dx &=\frac{2 x^{3/2}}{3 a (a+b x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.005987, size = 21, normalized size = 1. \[ \frac{2 x^{3/2}}{3 a (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a + b*x)^(3/2))

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*}{\frac{2}{3\,a}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04428, size = 20, normalized size = 0.95 \begin{align*} \frac{2 \, x^{\frac{3}{2}}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.07273, size = 77, normalized size = 3.67 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 2.42667, size = 42, normalized size = 2. \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [B]  time = 1.15023, size = 116, normalized size = 5.52 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)