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

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]

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

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Rubi [A]  time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, size = 60, normalized size = 0.60 \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} (a x+b)^2 \left (105 a^3 x^3-70 a^2 b x^2+40 a b^2 x-16 b^3\right )}{1155 a^4} \]

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)

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fricas [A]  time = 0.68, size = 71, normalized size = 0.71 \[ \frac {2 \, {\left (105 \, a^{5} x^{5} + 140 \, a^{4} b x^{4} + 5 \, a^{3} b^{2} x^{3} - 6 \, a^{2} b^{3} x^{2} + 8 \, a b^{4} x - 16 \, b^{5}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{1155 \, a^{4}} \]

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

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giac [A]  time = 0.20, size = 138, normalized size = 1.38 \[ \frac {2}{315} \, b {\left (\frac {16 \, b^{\frac {9}{2}}}{a^{4}} + \frac {35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 135 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}}{a^{4}}\right )} \mathrm {sgn}\relax (x) - \frac {2}{3465} \, a {\left (\frac {128 \, b^{\frac {11}{2}}}{a^{5}} - \frac {315 \, {\left (a x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (a x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (a x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{4}}{a^{5}}\right )} \mathrm {sgn}\relax (x) \]

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)

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maple [A]  time = 0.00, size = 55, normalized size = 0.55 \[ \frac {2 \left (a x +b \right ) \left (105 a^{3} x^{3}-70 a^{2} b \,x^{2}+40 a \,b^{2} x -16 b^{3}\right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}}}{1155 a^{4}} \]

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

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maxima [A]  time = 0.99, size = 69, normalized size = 0.69 \[ \frac {2 \, {\left (105 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} x^{\frac {11}{2}} - 385 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} b x^{\frac {9}{2}} + 495 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{2} x^{\frac {7}{2}} - 231 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{3} x^{\frac {5}{2}}\right )}}{1155 \, a^{4}} \]

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

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mupad [B]  time = 1.46, size = 67, normalized size = 0.67 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a\,x^{11/2}}{11}+\frac {8\,b\,x^{9/2}}{33}+\frac {2\,b^2\,x^{7/2}}{231\,a}-\frac {4\,b^3\,x^{5/2}}{385\,a^2}+\frac {16\,b^4\,x^{3/2}}{1155\,a^3}-\frac {32\,b^5\,\sqrt {x}}{1155\,a^4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)*(a + b/x)^(3/2),x)

[Out]

(a + b/x)^(1/2)*((2*a*x^(11/2))/11 + (8*b*x^(9/2))/33 + (2*b^2*x^(7/2))/(231*a) - (4*b^3*x^(5/2))/(385*a^2) +
(16*b^4*x^(3/2))/(1155*a^3) - (32*b^5*x^(1/2))/(1155*a^4))

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sympy [B]  time = 171.29, size = 585, normalized size = 5.85 \[ \frac {210 a^{8} b^{\frac {19}{2}} x^{8} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {910 a^{7} b^{\frac {21}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {1480 a^{6} b^{\frac {23}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {1068 a^{5} b^{\frac {25}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} + \frac {290 a^{4} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {10 a^{3} b^{\frac {29}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {60 a^{2} b^{\frac {31}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {80 a b^{\frac {33}{2}} x \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} - \frac {32 b^{\frac {35}{2}} \sqrt {\frac {a x}{b} + 1}}{1155 a^{7} b^{9} x^{3} + 3465 a^{6} b^{10} x^{2} + 3465 a^{5} b^{11} x + 1155 a^{4} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

210*a**8*b**(19/2)*x**8*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155
*a**4*b**12) + 910*a**7*b**(21/2)*x**7*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5
*b**11*x + 1155*a**4*b**12) + 1480*a**6*b**(23/2)*x**6*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*
x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) + 1068*a**5*b**(25/2)*x**5*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 +
3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) + 290*a**4*b**(27/2)*x**4*sqrt(a*x/b + 1)/(1155*a*
*7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) - 10*a**3*b**(29/2)*x**3*sqrt(a*x/b
+ 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) - 60*a**2*b**(31/2)*x
**2*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12) - 80*a*
b**(33/2)*x*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b**12)
- 32*b**(35/2)*sqrt(a*x/b + 1)/(1155*a**7*b**9*x**3 + 3465*a**6*b**10*x**2 + 3465*a**5*b**11*x + 1155*a**4*b*
*12)

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