3.1770 \(\int (a+\frac{b}{x})^{5/2} x^{7/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{9 a}-\frac{4 b x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{63 a^2} \]

[Out]

(-4*b*(a + b/x)^(7/2)*x^(7/2))/(63*a^2) + (2*(a + b/x)^(7/2)*x^(9/2))/(9*a)

________________________________________________________________________________________

Rubi [A]  time = 0.01314, 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} \[ \frac{2 x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{9 a}-\frac{4 b x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{63 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*x^(7/2),x]

[Out]

(-4*b*(a + b/x)^(7/2)*x^(7/2))/(63*a^2) + (2*(a + b/x)^(7/2)*x^(9/2))/(9*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 )^{5/2} x^{7/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{9/2}}{9 a}-\frac{(2 b) \int \left (a+\frac{b}{x}\right )^{5/2} x^{5/2} \, dx}{9 a}\\ &=-\frac{4 b \left (a+\frac{b}{x}\right )^{7/2} x^{7/2}}{63 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{9/2}}{9 a}\\ \end{align*}

Mathematica [A]  time = 0.0156513, size = 38, normalized size = 0.79 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^3 (7 a x-2 b)}{63 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*x^(7/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^3*(-2*b + 7*a*x))/(63*a^2)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 33, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 7\,ax-2\,b \right ) }{63\,{a}^{2}} \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^(7/2),x)

[Out]

2/63*(a*x+b)*(7*a*x-2*b)*x^(5/2)*((a*x+b)/x)^(5/2)/a^2

________________________________________________________________________________________

Maxima [A]  time = 0.955044, size = 47, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (7 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} x^{\frac{9}{2}} - 9 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b x^{\frac{7}{2}}\right )}}{63 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(7/2),x, algorithm="maxima")

[Out]

2/63*(7*(a + b/x)^(9/2)*x^(9/2) - 9*(a + b/x)^(7/2)*b*x^(7/2))/a^2

________________________________________________________________________________________

Fricas [A]  time = 1.44855, size = 132, normalized size = 2.75 \begin{align*} \frac{2 \,{\left (7 \, a^{4} x^{4} + 19 \, a^{3} b x^{3} + 15 \, a^{2} b^{2} x^{2} + a b^{3} x - 2 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{63 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(7/2),x, algorithm="fricas")

[Out]

2/63*(7*a^4*x^4 + 19*a^3*b*x^3 + 15*a^2*b^2*x^2 + a*b^3*x - 2*b^4)*sqrt(x)*sqrt((a*x + b)/x)/a^2

________________________________________________________________________________________

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**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.22465, size = 212, normalized size = 4.42 \begin{align*} \frac{2}{15} \, b^{2}{\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{4}{105} \, a b{\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 ) + \frac{2}{315} \, a^{2}{\left (\frac{16 \, b^{\frac{9}{2}}}{a^{4}} + \frac{35 \,{\left (a x + b\right )}^{\frac{9}{2}} - 135 \,{\left (a x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{3}}{a^{4}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(7/2),x, algorithm="giac")

[Out]

2/15*b^2*(2*b^(5/2)/a^2 + (3*(a*x + b)^(5/2) - 5*(a*x + b)^(3/2)*b)/a^2)*sgn(x) - 4/105*a*b*(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) + 2/315*a^2*(16*b^(9/2)/a^4 +
(35*(a*x + b)^(9/2) - 135*(a*x + b)^(7/2)*b + 189*(a*x + b)^(5/2)*b^2 - 105*(a*x + b)^(3/2)*b^3)/a^4)*sgn(x)