3.1760 \(\int (a+\frac{b}{x})^{3/2} x^{9/2} \, dx\)

Optimal. Leaf size=100 \[ \frac{16 b^2 x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{231 a^3}-\frac{32 b^3 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{1155 a^4}-\frac{4 b x^{9/2} \left (a+\frac{b}{x}\right )^{5/2}}{33 a^2}+\frac{2 x^{11/2} \left (a+\frac{b}{x}\right )^{5/2}}{11 a} \]

[Out]

(-32*b^3*(a + b/x)^(5/2)*x^(5/2))/(1155*a^4) + (16*b^2*(a + b/x)^(5/2)*x^(7/2))/(231*a^3) - (4*b*(a + b/x)^(5/
2)*x^(9/2))/(33*a^2) + (2*(a + b/x)^(5/2)*x^(11/2))/(11*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0337823, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{16 b^2 x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{231 a^3}-\frac{32 b^3 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{1155 a^4}-\frac{4 b x^{9/2} \left (a+\frac{b}{x}\right )^{5/2}}{33 a^2}+\frac{2 x^{11/2} \left (a+\frac{b}{x}\right )^{5/2}}{11 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*x^(9/2),x]

[Out]

(-32*b^3*(a + b/x)^(5/2)*x^(5/2))/(1155*a^4) + (16*b^2*(a + b/x)^(5/2)*x^(7/2))/(231*a^3) - (4*b*(a + b/x)^(5/
2)*x^(9/2))/(33*a^2) + (2*(a + b/x)^(5/2)*x^(11/2))/(11*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^{9/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{11/2}}{11 a}-\frac{(6 b) \int \left (a+\frac{b}{x}\right )^{3/2} x^{7/2} \, dx}{11 a}\\ &=-\frac{4 b \left (a+\frac{b}{x}\right )^{5/2} x^{9/2}}{33 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{11/2}}{11 a}+\frac{\left (8 b^2\right ) \int \left (a+\frac{b}{x}\right )^{3/2} x^{5/2} \, dx}{33 a^2}\\ &=\frac{16 b^2 \left (a+\frac{b}{x}\right )^{5/2} x^{7/2}}{231 a^3}-\frac{4 b \left (a+\frac{b}{x}\right )^{5/2} x^{9/2}}{33 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{11/2}}{11 a}-\frac{\left (16 b^3\right ) \int \left (a+\frac{b}{x}\right )^{3/2} x^{3/2} \, dx}{231 a^3}\\ &=-\frac{32 b^3 \left (a+\frac{b}{x}\right )^{5/2} x^{5/2}}{1155 a^4}+\frac{16 b^2 \left (a+\frac{b}{x}\right )^{5/2} x^{7/2}}{231 a^3}-\frac{4 b \left (a+\frac{b}{x}\right )^{5/2} x^{9/2}}{33 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{5/2} x^{11/2}}{11 a}\\ \end{align*}

Mathematica [A]  time = 0.0177868, size = 60, normalized size = 0.6 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (-70 a^2 b x^2+105 a^3 x^3+40 a b^2 x-16 b^3\right )}{1155 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)*x^(9/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)^2*(-16*b^3 + 40*a*b^2*x - 70*a^2*b*x^2 + 105*a^3*x^3))/(1155*a^4)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 55, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 105\,{a}^{3}{x}^{3}-70\,{a}^{2}b{x}^{2}+40\,xa{b}^{2}-16\,{b}^{3} \right ) }{1155\,{a}^{4}}{x}^{{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(3/2)*x^(9/2),x)

[Out]

2/1155*(a*x+b)*(105*a^3*x^3-70*a^2*b*x^2+40*a*b^2*x-16*b^3)*x^(3/2)*((a*x+b)/x)^(3/2)/a^4

________________________________________________________________________________________

Maxima [A]  time = 0.977166, size = 93, normalized size = 0.93 \begin{align*} \frac{2 \,{\left (105 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} x^{\frac{11}{2}} - 385 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} b x^{\frac{9}{2}} + 495 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b^{2} x^{\frac{7}{2}} - 231 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b^{3} x^{\frac{5}{2}}\right )}}{1155 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(9/2),x, algorithm="maxima")

[Out]

2/1155*(105*(a + b/x)^(11/2)*x^(11/2) - 385*(a + b/x)^(9/2)*b*x^(9/2) + 495*(a + b/x)^(7/2)*b^2*x^(7/2) - 231*
(a + b/x)^(5/2)*b^3*x^(5/2))/a^4

________________________________________________________________________________________

Fricas [A]  time = 1.42238, size = 163, normalized size = 1.63 \begin{align*} \frac{2 \,{\left (105 \, a^{5} x^{5} + 140 \, a^{4} b x^{4} + 5 \, a^{3} b^{2} x^{3} - 6 \, a^{2} b^{3} x^{2} + 8 \, a b^{4} x - 16 \, b^{5}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{1155 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)*x^(9/2),x, algorithm="fricas")

[Out]

2/1155*(105*a^5*x^5 + 140*a^4*b*x^4 + 5*a^3*b^2*x^3 - 6*a^2*b^3*x^2 + 8*a*b^4*x - 16*b^5)*sqrt(x)*sqrt((a*x +
b)/x)/a^4

________________________________________________________________________________________

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**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.22699, size = 186, normalized size = 1.86 \begin{align*} \frac{2}{315} \, b{\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 ) - \frac{2}{3465} \, a{\left (\frac{128 \, b^{\frac{11}{2}}}{a^{5}} - \frac{315 \,{\left (a x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (a x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (a x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{4}}{a^{5}}\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^(9/2),x, algorithm="giac")

[Out]

2/315*b*(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) - 2/3465*a*(128*b^(11/2)/a^5 - (315*(a*x + b)^(11/2) - 1540*(a*x + b)^(9/2)*b + 2970*
(a*x + b)^(7/2)*b^2 - 2772*(a*x + b)^(5/2)*b^3 + 1155*(a*x + b)^(3/2)*b^4)/a^5)*sgn(x)