\(\int (a+\frac {b}{x})^{5/2} x^{9/2} \, dx\) [1769]

Optimal result

Integrand size = 17, antiderivative size = 74 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=\frac {16 b^2 \left (a+\frac {b}{x}\right )^{7/2} x^{7/2}}{693 a^3}-\frac {8 b \left (a+\frac {b}{x}\right )^{7/2} x^{9/2}}{99 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{11 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=\frac {16 b^2 x^{7/2} \left (a+\frac {b}{x}\right )^{7/2}}{693 a^3}-\frac {8 b x^{9/2} \left (a+\frac {b}{x}\right )^{7/2}}{99 a^2}+\frac {2 x^{11/2} \left (a+\frac {b}{x}\right )^{7/2}}{11 a} \]

[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 270

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 277

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*} \text {integral}& = \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] (verified)

Time = 6.87 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.66 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (b+a x)^3 \left (8 b^2-28 a b x+63 a^2 x^2\right )}{693 a^3} \]

[In]

Integrate[(a + b/x)^(5/2)*x^(9/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.59

method	result	size
gosper	\(\frac {2 \left (a x +b \right ) \left (63 a^{2} x^{2}-28 a b x +8 b^{2}\right ) x^{\frac {5}{2}} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}{693 a^{3}}\)	\(44\)
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a x +b \right )^{3} \left (63 a^{2} x^{2}-28 a b x +8 b^{2}\right )}{693 a^{3}}\)	\(46\)
risch	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (63 a^{5} x^{5}+161 a^{4} b \,x^{4}+113 a^{3} b^{2} x^{3}+3 a^{2} b^{3} x^{2}-4 b^{4} x a +8 b^{5}\right )}{693 a^{3}}\)	\(72\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=\frac {2 \, {\left (63 \, a^{5} x^{5} + 161 \, a^{4} b x^{4} + 113 \, a^{3} b^{2} x^{3} + 3 \, a^{2} b^{3} x^{2} - 4 \, a b^{4} x + 8 \, b^{5}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{693 \, a^{3}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (63) = 126\).

Time = 131.38 (sec) , antiderivative size = 423, normalized size of antiderivative = 5.72 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=\frac {126 a^{7} b^{\frac {9}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} + \frac {574 a^{6} b^{\frac {11}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} + \frac {996 a^{5} b^{\frac {13}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} + \frac {780 a^{4} b^{\frac {15}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} + \frac {230 a^{3} b^{\frac {17}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} + \frac {6 a^{2} b^{\frac {19}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} + \frac {24 a b^{\frac {21}{2}} x \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} + \frac {16 b^{\frac {23}{2}} \sqrt {\frac {a x}{b} + 1}}{693 a^{5} b^{4} x^{2} + 1386 a^{4} b^{5} x + 693 a^{3} b^{6}} \]

[In]

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

[Out]

126*a**7*b**(9/2)*x**7*sqrt(a*x/b + 1)/(693*a**5*b**4*x**2 + 1386*a**4*b**5*x + 693*a**3*b**6) + 574*a**6*b**(
11/2)*x**6*sqrt(a*x/b + 1)/(693*a**5*b**4*x**2 + 1386*a**4*b**5*x + 693*a**3*b**6) + 996*a**5*b**(13/2)*x**5*s
qrt(a*x/b + 1)/(693*a**5*b**4*x**2 + 1386*a**4*b**5*x + 693*a**3*b**6) + 780*a**4*b**(15/2)*x**4*sqrt(a*x/b +
1)/(693*a**5*b**4*x**2 + 1386*a**4*b**5*x + 693*a**3*b**6) + 230*a**3*b**(17/2)*x**3*sqrt(a*x/b + 1)/(693*a**5
*b**4*x**2 + 1386*a**4*b**5*x + 693*a**3*b**6) + 6*a**2*b**(19/2)*x**2*sqrt(a*x/b + 1)/(693*a**5*b**4*x**2 + 1
386*a**4*b**5*x + 693*a**3*b**6) + 24*a*b**(21/2)*x*sqrt(a*x/b + 1)/(693*a**5*b**4*x**2 + 1386*a**4*b**5*x + 6
93*a**3*b**6) + 16*b**(23/2)*sqrt(a*x/b + 1)/(693*a**5*b**4*x**2 + 1386*a**4*b**5*x + 693*a**3*b**6)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.70 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=\frac {2 \, {\left (63 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} x^{\frac {11}{2}} - 154 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} b x^{\frac {9}{2}} + 99 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{2} x^{\frac {7}{2}}\right )}}{693 \, a^{3}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (56) = 112\).

Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.62 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=-\frac {2}{105} \, b^{2} {\left (\frac {8 \, b^{\frac {7}{2}}}{a^{3}} - \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} - 42 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}}{a^{3}}\right )} \mathrm {sgn}\left (x\right ) + \frac {4}{315} \, a b {\left (\frac {16 \, b^{\frac {9}{2}}}{a^{4}} + \frac {35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 135 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}}{a^{4}}\right )} \mathrm {sgn}\left (x\right ) - \frac {2}{3465} \, a^{2} {\left (\frac {128 \, b^{\frac {11}{2}}}{a^{5}} - \frac {315 \, {\left (a x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (a x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (a x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{4}}{a^{5}}\right )} \mathrm {sgn}\left (x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 6.60 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx=\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a^2\,x^{11/2}}{11}+\frac {226\,b^2\,x^{7/2}}{693}+\frac {2\,b^3\,x^{5/2}}{231\,a}-\frac {8\,b^4\,x^{3/2}}{693\,a^2}+\frac {16\,b^5\,\sqrt {x}}{693\,a^3}+\frac {46\,a\,b\,x^{9/2}}{99}\right ) \]

[In]

int(x^(9/2)*(a + b/x)^(5/2),x)

[Out]

(a + b/x)^(1/2)*((2*a^2*x^(11/2))/11 + (226*b^2*x^(7/2))/693 + (2*b^3*x^(5/2))/(231*a) - (8*b^4*x^(3/2))/(693*
a^2) + (16*b^5*x^(1/2))/(693*a^3) + (46*a*b*x^(9/2))/99)