3.1762 \(\int (a+\frac{b}{x})^{3/2} x^{5/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{7 a}-\frac{4 b x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{35 a^2} \]

[Out]

(-4*b*(a + b/x)^(5/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(5/2)*x^(7/2))/(7*a)

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Rubi [A]  time = 0.0136551, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{2 x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{7 a}-\frac{4 b x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{35 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*x^(5/2),x]

[Out]

(-4*b*(a + b/x)^(5/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(5/2)*x^(7/2))/(7*a)

Rule 271

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

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 )^{3/2} x^{5/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{7/2}}{7 a}-\frac{(2 b) \int \left (a+\frac{b}{x}\right )^{3/2} x^{3/2} \, dx}{7 a}\\ &=-\frac{4 b \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}}{35 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{7/2}}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.0154217, size = 40, normalized size = 0.83 \[ \frac{2}{35} x^{5/2} \sqrt{a+\frac{b}{x}} \left (\frac{b}{a x}+1\right )^2 (5 a x-2 b) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)*x^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*(1 + b/(a*x))^2*x^(5/2)*(-2*b + 5*a*x))/35

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Maple [A]  time = 0.002, size = 33, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 5\,ax-2\,b \right ) }{35\,{a}^{2}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(3/2)*x^(5/2),x)

[Out]

2/35*(a*x+b)*(5*a*x-2*b)*x^(3/2)*((a*x+b)/x)^(3/2)/a^2

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Maxima [A]  time = 0.972391, size = 47, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (5 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 7 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}}\right )}}{35 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(5/2),x, algorithm="maxima")

[Out]

2/35*(5*(a + b/x)^(7/2)*x^(7/2) - 7*(a + b/x)^(5/2)*b*x^(5/2))/a^2

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Fricas [A]  time = 1.44455, size = 108, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (5 \, a^{3} x^{3} + 8 \, a^{2} b x^{2} + a b^{2} x - 2 \, b^{3}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{35 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(5/2),x, algorithm="fricas")

[Out]

2/35*(5*a^3*x^3 + 8*a^2*b*x^2 + a*b^2*x - 2*b^3)*sqrt(x)*sqrt((a*x + b)/x)/a^2

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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)**(3/2)*x**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.18023, size = 122, normalized size = 2.54 \begin{align*} \frac{2}{15} \, b{\left (\frac{2 \, b^{\frac{5}{2}}}{a^{2}} + \frac{3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 5 \,{\left (a x + b\right )}^{\frac{3}{2}} b}{a^{2}}\right )} \mathrm{sgn}\left (x\right ) - \frac{2}{105} \, a{\left (\frac{8 \, b^{\frac{7}{2}}}{a^{3}} - \frac{15 \,{\left (a x + b\right )}^{\frac{7}{2}} - 42 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2}}{a^{3}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(5/2),x, algorithm="giac")

[Out]

2/15*b*(2*b^(5/2)/a^2 + (3*(a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b)/a^2)*sgn(x) - 2/105*a*(8*b^(7/2)/a^3 - (15*(
a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 35*(a*x + b)^(3/2)*b^2)/a^3)*sgn(x)