3.1771 \(\int (a+\frac{b}{x})^{5/2} x^{5/2} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]

[Out]

(2*(a + b/x)^(7/2)*x^(7/2))/(7*a)

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Rubi [A]  time = 0.0057093, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*x^(5/2),x]

[Out]

(2*(a + b/x)^(7/2)*x^(7/2))/(7*a)

Rule 264

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

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^{5/2} x^{5/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{7/2}}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.0165905, size = 23, normalized size = 1. \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{7 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*x^(5/2),x]

[Out]

(2*(a + b/x)^(7/2)*x^(7/2))/(7*a)

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Maple [A]  time = 0.003, size = 25, normalized size = 1.1 \begin{align*}{\frac{2\,ax+2\,b}{7\,a} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(a*x+b)*((a*x+b)/x)^(5/2)*x^(5/2)/a

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Maxima [A]  time = 0.953487, size = 23, normalized size = 1. \begin{align*} \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}}}{7 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(a + b/x)^(7/2)*x^(7/2)/a

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Fricas [B]  time = 1.49879, size = 101, normalized size = 4.39 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} + 3 \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{7 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(a^3*x^3 + 3*a^2*b*x^2 + 3*a*b^2*x + b^3)*sqrt(x)*sqrt((a*x + b)/x)/a

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.20944, size = 81, normalized size = 3.52 \begin{align*} \frac{2 \,{\left (15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} + 14 \,{\left (3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b\right )} b\right )}}{105 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 70*(a*x + b)^(3/2)*b^2 + 14*(3*(a*x + b)^(5/2) - 5*(a*x + b
)^(3/2)*b)*b)/a