3.1758 \(\int \left (a+\frac{b}{x}\right )^{3/2} x^{9/2} \, dx\)

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

[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 = 0.115986, 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 \[ -\frac{32 b^3 x^{5/2} \left (a+\frac{b}{x}\right )^{5/2}}{1155 a^4}+\frac{16 b^2 x^{7/2} \left (a+\frac{b}{x}\right )^{5/2}}{231 a^3}-\frac{4 b x^{9/2} \left (a+\frac{b}{x}\right )^{5/2}}{33 a^2}+\frac{2 x^{11/2} \left (a+\frac{b}{x}\right )^{5/2}}{11 a} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.60424, size = 87, normalized size = 0.87 \[ \frac{2 x^{\frac{11}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{11 a} - \frac{4 b x^{\frac{9}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{33 a^{2}} + \frac{16 b^{2} x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{231 a^{3}} - \frac{32 b^{3} x^{\frac{5}{2}} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{1155 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(3/2)*x**(9/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0600247, size = 60, normalized size = 0.6 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)^2 \left (105 a^3 x^3-70 a^2 b x^2+40 a b^2 x-16 b^3\right )}{1155 a^4} \]

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 = 0.007, size = 55, normalized size = 0.6 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 105\,{a}^{3}{x}^{3}-70\,{a}^{2}b{x}^{2}+40\,a{b}^{2}x-16\,{b}^{3} \right ) }{1155\,{a}^{4}}{x}^{{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \]

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 = 1.42921, size = 93, normalized size = 0.93 \[ \frac{2 \,{\left (105 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} x^{\frac{11}{2}} - 385 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} b x^{\frac{9}{2}} + 495 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} b^{2} x^{\frac{7}{2}} - 231 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b^{3} x^{\frac{5}{2}}\right )}}{1155 \, a^{4}} \]

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 = 0.238769, size = 96, normalized size = 0.96 \[ \frac{2 \,{\left (105 \, a^{5} x^{5} + 140 \, a^{4} b x^{4} + 5 \, a^{3} b^{2} x^{3} - 6 \, a^{2} b^{3} x^{2} + 8 \, a b^{4} x - 16 \, b^{5}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{1155 \, a^{4}} \]

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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.236617, size = 186, normalized size = 1.86 \[ \frac{2}{315} \, b{\left (\frac{16 \, b^{\frac{9}{2}}}{a^{4}} + \frac{35 \,{\left (a x + b\right )}^{\frac{9}{2}} - 135 \,{\left (a x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{3}}{a^{4}}\right )}{\rm sign}\left (x\right ) - \frac{2}{3465} \, a{\left (\frac{128 \, b^{\frac{11}{2}}}{a^{5}} - \frac{315 \,{\left (a x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (a x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (a x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (a x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{4}}{a^{5}}\right )}{\rm sign}\left (x\right ) \]

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)*sign(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)*sign(x)