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

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

[Out]

(-2*(a + b*x)^(3/2))/(3*a*x^(3/2))

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Rubi [A]  time = 0.0019194, 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} \[ -\frac{2 (a+b x)^{3/2}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x]/x^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0061355, size = 21, normalized size = 1. \[ -\frac{2 (a+b x)^{3/2}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x]/x^(5/2),x]

[Out]

(-2*(a + b*x)^(3/2))/(3*a*x^(3/2))

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,a} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04145, size = 20, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, a x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b*x + a)^(3/2)/(a*x^(3/2))

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Fricas [A]  time = 1.49494, size = 46, normalized size = 2.19 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, a x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b*x + a)^(3/2)/(a*x^(3/2))

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Sympy [B]  time = 2.20224, size = 41, normalized size = 1.95 \begin{align*} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.44095, size = 45, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{4}}{3 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}} a{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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