3.1769 \(\int (a+\frac{b}{x})^{5/2} x^{9/2} \, dx\)

Optimal. Leaf size=74 \[ \frac{16 b^2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{693 a^3}-\frac{8 b x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{99 a^2}+\frac{2 x^{11/2} \left (a+\frac{b}{x}\right )^{7/2}}{11 a} \]

[Out]

(16*b^2*(a + b/x)^(7/2)*x^(7/2))/(693*a^3) - (8*b*(a + b/x)^(7/2)*x^(9/2))/(99*a^2) + (2*(a + b/x)^(7/2)*x^(11
/2))/(11*a)

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Rubi [A]  time = 0.0225016, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{16 b^2 x^{7/2} \left (a+\frac{b}{x}\right )^{7/2}}{693 a^3}-\frac{8 b x^{9/2} \left (a+\frac{b}{x}\right )^{7/2}}{99 a^2}+\frac{2 x^{11/2} \left (a+\frac{b}{x}\right )^{7/2}}{11 a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*x^(9/2),x]

[Out]

(16*b^2*(a + b/x)^(7/2)*x^(7/2))/(693*a^3) - (8*b*(a + b/x)^(7/2)*x^(9/2))/(99*a^2) + (2*(a + b/x)^(7/2)*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 )^{5/2} x^{9/2} \, dx &=\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{11/2}}{11 a}-\frac{(4 b) \int \left (a+\frac{b}{x}\right )^{5/2} x^{7/2} \, dx}{11 a}\\ &=-\frac{8 b \left (a+\frac{b}{x}\right )^{7/2} x^{9/2}}{99 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{11/2}}{11 a}+\frac{\left (8 b^2\right ) \int \left (a+\frac{b}{x}\right )^{5/2} x^{5/2} \, dx}{99 a^2}\\ &=\frac{16 b^2 \left (a+\frac{b}{x}\right )^{7/2} x^{7/2}}{693 a^3}-\frac{8 b \left (a+\frac{b}{x}\right )^{7/2} x^{9/2}}{99 a^2}+\frac{2 \left (a+\frac{b}{x}\right )^{7/2} x^{11/2}}{11 a}\\ \end{align*}

Mathematica [A]  time = 0.0176428, size = 51, normalized size = 0.69 \[ \frac{2}{693} x^{7/2} \sqrt{a+\frac{b}{x}} \left (\frac{b}{a x}+1\right )^3 \left (63 a^2 x^2-28 a b x+8 b^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*x^(9/2),x]

[Out]

(2*Sqrt[a + b/x]*(1 + b/(a*x))^3*x^(7/2)*(8*b^2 - 28*a*b*x + 63*a^2*x^2))/693

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Maple [A]  time = 0.004, size = 44, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 63\,{a}^{2}{x}^{2}-28\,xab+8\,{b}^{2} \right ) }{693\,{a}^{3}}{x}^{{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(5/2)*x^(9/2),x)

[Out]

2/693*(a*x+b)*(63*a^2*x^2-28*a*b*x+8*b^2)*x^(5/2)*((a*x+b)/x)^(5/2)/a^3

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Maxima [A]  time = 0.961441, size = 70, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (63 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} x^{\frac{11}{2}} - 154 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} b x^{\frac{9}{2}} + 99 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b^{2} x^{\frac{7}{2}}\right )}}{693 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(9/2),x, algorithm="maxima")

[Out]

2/693*(63*(a + b/x)^(11/2)*x^(11/2) - 154*(a + b/x)^(9/2)*b*x^(9/2) + 99*(a + b/x)^(7/2)*b^2*x^(7/2))/a^3

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Fricas [A]  time = 1.48765, size = 162, normalized size = 2.19 \begin{align*} \frac{2 \,{\left (63 \, a^{5} x^{5} + 161 \, a^{4} b x^{4} + 113 \, a^{3} b^{2} x^{3} + 3 \, a^{2} b^{3} x^{2} - 4 \, a b^{4} x + 8 \, b^{5}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{693 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)*x^(9/2),x, algorithm="fricas")

[Out]

2/693*(63*a^5*x^5 + 161*a^4*b*x^4 + 113*a^3*b^2*x^3 + 3*a^2*b^3*x^2 - 4*a*b^4*x + 8*b^5)*sqrt(x)*sqrt((a*x + b
)/x)/a^3

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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**(9/2),x)

[Out]

Timed out

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Giac [B]  time = 1.23779, size = 262, normalized size = 3.54 \begin{align*} -\frac{2}{105} \, b^{2}{\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{4}{315} \, a 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^{2}{\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)^(5/2)*x^(9/2),x, algorithm="giac")

[Out]

-2/105*b^2*(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) +
4/315*a*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^2*(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)