3.1751 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x^7} \, dx\)

Optimal. Leaf size=116 \[ -\frac{2 a^5}{3 b^6 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{10 a^4}{b^6 \sqrt{a+\frac{b}{x}}}+\frac{20 a^3 \sqrt{a+\frac{b}{x}}}{b^6}-\frac{20 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}+\frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6} \]

[Out]

(-2*a^5)/(3*b^6*(a + b/x)^(3/2)) + (10*a^4)/(b^6*Sqrt[a + b/x]) + (20*a^3*Sqrt[a + b/x])/b^6 - (20*a^2*(a + b/
x)^(3/2))/(3*b^6) + (2*a*(a + b/x)^(5/2))/b^6 - (2*(a + b/x)^(7/2))/(7*b^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0483134, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{2 a^5}{3 b^6 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{10 a^4}{b^6 \sqrt{a+\frac{b}{x}}}+\frac{20 a^3 \sqrt{a+\frac{b}{x}}}{b^6}-\frac{20 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}+\frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^7),x]

[Out]

(-2*a^5)/(3*b^6*(a + b/x)^(3/2)) + (10*a^4)/(b^6*Sqrt[a + b/x]) + (20*a^3*Sqrt[a + b/x])/b^6 - (20*a^2*(a + b/
x)^(3/2))/(3*b^6) + (2*a*(a + b/x)^(5/2))/b^6 - (2*(a + b/x)^(7/2))/(7*b^6)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^7} \, dx &=-\operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a^5}{b^5 (a+b x)^{5/2}}+\frac{5 a^4}{b^5 (a+b x)^{3/2}}-\frac{10 a^3}{b^5 \sqrt{a+b x}}+\frac{10 a^2 \sqrt{a+b x}}{b^5}-\frac{5 a (a+b x)^{3/2}}{b^5}+\frac{(a+b x)^{5/2}}{b^5}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^5}{3 b^6 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{10 a^4}{b^6 \sqrt{a+\frac{b}{x}}}+\frac{20 a^3 \sqrt{a+\frac{b}{x}}}{b^6}-\frac{20 a^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}+\frac{2 a \left (a+\frac{b}{x}\right )^{5/2}}{b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6}\\ \end{align*}

Mathematica [A]  time = 0.042633, size = 80, normalized size = 0.69 \[ \frac{2 \left (-16 a^2 b^3 x^2+96 a^3 b^2 x^3+384 a^4 b x^4+256 a^5 x^5+6 a b^4 x-3 b^5\right )}{21 b^6 x^4 \sqrt{a+\frac{b}{x}} (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^7),x]

[Out]

(2*(-3*b^5 + 6*a*b^4*x - 16*a^2*b^3*x^2 + 96*a^3*b^2*x^3 + 384*a^4*b*x^4 + 256*a^5*x^5))/(21*b^6*Sqrt[a + b/x]
*x^4*(b + a*x))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 77, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 256\,{a}^{5}{x}^{5}+384\,{a}^{4}b{x}^{4}+96\,{a}^{3}{b}^{2}{x}^{3}-16\,{a}^{2}{b}^{3}{x}^{2}+6\,a{b}^{4}x-3\,{b}^{5} \right ) }{21\,{x}^{6}{b}^{6}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x^7,x)

[Out]

2/21*(a*x+b)*(256*a^5*x^5+384*a^4*b*x^4+96*a^3*b^2*x^3-16*a^2*b^3*x^2+6*a*b^4*x-3*b^5)/x^6/b^6/((a*x+b)/x)^(5/
2)

________________________________________________________________________________________

Maxima [A]  time = 1.00255, size = 132, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}}}{7 \, b^{6}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a}{b^{6}} - \frac{20 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{2}}{3 \, b^{6}} + \frac{20 \, \sqrt{a + \frac{b}{x}} a^{3}}{b^{6}} + \frac{10 \, a^{4}}{\sqrt{a + \frac{b}{x}} b^{6}} - \frac{2 \, a^{5}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^7,x, algorithm="maxima")

[Out]

-2/7*(a + b/x)^(7/2)/b^6 + 2*(a + b/x)^(5/2)*a/b^6 - 20/3*(a + b/x)^(3/2)*a^2/b^6 + 20*sqrt(a + b/x)*a^3/b^6 +
10*a^4/(sqrt(a + b/x)*b^6) - 2/3*a^5/((a + b/x)^(3/2)*b^6)

________________________________________________________________________________________

Fricas [A]  time = 1.43244, size = 197, normalized size = 1.7 \begin{align*} \frac{2 \,{\left (256 \, a^{5} x^{5} + 384 \, a^{4} b x^{4} + 96 \, a^{3} b^{2} x^{3} - 16 \, a^{2} b^{3} x^{2} + 6 \, a b^{4} x - 3 \, b^{5}\right )} \sqrt{\frac{a x + b}{x}}}{21 \,{\left (a^{2} b^{6} x^{5} + 2 \, a b^{7} x^{4} + b^{8} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^7,x, algorithm="fricas")

[Out]

2/21*(256*a^5*x^5 + 384*a^4*b*x^4 + 96*a^3*b^2*x^3 - 16*a^2*b^3*x^2 + 6*a*b^4*x - 3*b^5)*sqrt((a*x + b)/x)/(a^
2*b^6*x^5 + 2*a*b^7*x^4 + b^8*x^3)

________________________________________________________________________________________

Sympy [B]  time = 10.2626, size = 9263, normalized size = 79.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x**7,x)

[Out]

512*a**(43/2)*b**(91/2)*x**18*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) +
2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(
27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)
*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*
x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21
*a**(7/2)*b**66*x**(7/2)) + 7424*a**(41/2)*b**(93/2)*x**17*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315
*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)
*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58
*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(1
7/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315
*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 50112*a**(39/2)*b**(95/2)*x**16*sqrt(a*x/b + 1)/(21*a
**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**
54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(
25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2)
+ 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a
**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 208800*a**(37/2)*b**(97
/2)*x**15*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53
*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2
) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 1
05105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**
(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(
7/2)) + 600300*a**(35/2)*b**(99/2)*x**14*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x
**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) +
63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 13513
5*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(1
5/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x*
*(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 1260630*a**(33/2)*b**(101/2)*x**13*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*
x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) +
28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*
a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17
/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*
x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 1995994*a**(31/2)*b**(103/2)*x**12*sqr
t(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) +
9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a*
*(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/
2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*
x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 24236
68*a**(29/2)*b**(105/2)*x**11*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) +
2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(
27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)
*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*
x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21
*a**(7/2)*b**66*x**(7/2)) + 2271984*a**(27/2)*b**(107/2)*x**10*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) +
315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(2
9/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b
**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x
**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) +
315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 1640210*a**(25/2)*b**(109/2)*x**9*sqrt(a*x/b + 1)
/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/
2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**5
7*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**
(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) +
2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 900614*a**(23/2)*
b**(111/2)*x**8*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)
*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x*
*(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/
2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 95
55*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**6
6*x**(7/2)) + 365976*a**(21/2)*b**(113/2)*x**7*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b
**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(2
9/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) +
135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665
*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b*
*65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 103740*a**(19/2)*b**(115/2)*x**6*sqrt(a*x/b + 1)/(21*a**(37/2)*b*
*51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2
) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135
135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a*
*(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b*
*64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) + 17178*a**(17/2)*b**(117/2)*x**5*sq
rt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) +
9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a
**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19
/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63
*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 90*a
**(15/2)*b**(119/2)*x**4*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*
a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)
*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**5
9*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(1
5/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(
7/2)*b**66*x**(7/2)) - 1004*a**(13/2)*b**(121/2)*x**3*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(
35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**5
5*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(
23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2)
+ 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(
9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 344*a**(11/2)*b**(123/2)*x**2*sqrt(a*x/b + 1)/(21*a**(37/2
)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(
31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) +
135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 6306
3*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2
)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 66*a**(9/2)*b**(125/2)*x*sqrt(
a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 95
55*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(
25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)
*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x*
*(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 6*a**(7
/2)*b**(127/2)*sqrt(a*x/b + 1)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*
b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**
(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2
) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 955
5*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66
*x**(7/2)) - 512*a**22*b**45*x**(37/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a*
*(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b
**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*
x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/
2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/
2)*b**66*x**(7/2)) - 7680*a**21*b**46*x**(35/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2)
+ 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a*
*(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/
2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**6
2*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) +
21*a**(7/2)*b**66*x**(7/2)) - 53760*a**20*b**47*x**(33/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*
x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2)
+ 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 1351
35*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(
15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x
**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 232960*a**19*b**48*x**(31/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(3
5/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55
*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(2
3/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) +
28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9
/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 698880*a**18*b**49*x**(29/2)/(21*a**(37/2)*b**51*x**(37/2)
+ 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(
29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*
b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*
x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2)
+ 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 1537536*a**17*b**50*x**(27/2)/(21*a**(37/2)*b**5
1*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2)
+ 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 13513
5*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(
17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**6
4*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 2562560*a**16*b**51*x**(25/2)/(21*a*
*(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**5
4*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(2
5/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2)
+ 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a*
*(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 3294720*a**15*b**52*x**(
23/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a*
*(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)
*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**6
0*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/
2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 3294720*a**1
4*b**53*x**(21/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/
2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 1051
05*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a*
*(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b
**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) -
2562560*a**13*b**54*x**(19/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b
**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(
27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2)
+ 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555
*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*
x**(7/2)) - 1537536*a**12*b**55*x**(17/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205
*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2
)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**
59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(
15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**
(7/2)*b**66*x**(7/2)) - 698880*a**11*b**56*x**(15/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*b**52*x**(3
5/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(29/2) + 630
63*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2) + 135135*a*
*(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 28665*a**(15/2)
*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b**65*x**(9/
2) + 21*a**(7/2)*b**66*x**(7/2)) - 232960*a**10*b**57*x**(13/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a**(35/2)*
b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b**55*x**(
29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x**(23/2)
+ 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/2) + 2866
5*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a**(9/2)*b
**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 53760*a**9*b**58*x**(11/2)/(21*a**(37/2)*b**51*x**(37/2) + 315*a
**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(29/2)*b
**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*b**58*x
**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*x**(17/
2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2) + 315*a
**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 7680*a**8*b**59*x**(9/2)/(21*a**(37/2)*b**51*x**(37/2)
+ 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 28665*a**(
29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(23/2)*
b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*b**61*
x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(11/2)
+ 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2)) - 512*a**7*b**60*x**(7/2)/(21*a**(37/2)*b**51*x**(
37/2) + 315*a**(35/2)*b**52*x**(35/2) + 2205*a**(33/2)*b**53*x**(33/2) + 9555*a**(31/2)*b**54*x**(31/2) + 2866
5*a**(29/2)*b**55*x**(29/2) + 63063*a**(27/2)*b**56*x**(27/2) + 105105*a**(25/2)*b**57*x**(25/2) + 135135*a**(
23/2)*b**58*x**(23/2) + 135135*a**(21/2)*b**59*x**(21/2) + 105105*a**(19/2)*b**60*x**(19/2) + 63063*a**(17/2)*
b**61*x**(17/2) + 28665*a**(15/2)*b**62*x**(15/2) + 9555*a**(13/2)*b**63*x**(13/2) + 2205*a**(11/2)*b**64*x**(
11/2) + 315*a**(9/2)*b**65*x**(9/2) + 21*a**(7/2)*b**66*x**(7/2))

________________________________________________________________________________________

Giac [A]  time = 1.22558, size = 203, normalized size = 1.75 \begin{align*} -\frac{2}{21} \, b{\left (\frac{7 \,{\left (a^{5} - \frac{15 \,{\left (a x + b\right )} a^{4}}{x}\right )} x}{{\left (a x + b\right )} b^{7} \sqrt{\frac{a x + b}{x}}} - \frac{210 \, a^{3} b^{42} \sqrt{\frac{a x + b}{x}} - \frac{70 \,{\left (a x + b\right )} a^{2} b^{42} \sqrt{\frac{a x + b}{x}}}{x} + \frac{21 \,{\left (a x + b\right )}^{2} a b^{42} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{3 \,{\left (a x + b\right )}^{3} b^{42} \sqrt{\frac{a x + b}{x}}}{x^{3}}}{b^{49}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^7,x, algorithm="giac")

[Out]

-2/21*b*(7*(a^5 - 15*(a*x + b)*a^4/x)*x/((a*x + b)*b^7*sqrt((a*x + b)/x)) - (210*a^3*b^42*sqrt((a*x + b)/x) -
70*(a*x + b)*a^2*b^42*sqrt((a*x + b)/x)/x + 21*(a*x + b)^2*a*b^42*sqrt((a*x + b)/x)/x^2 - 3*(a*x + b)^3*b^42*s
qrt((a*x + b)/x)/x^3)/b^49)