3.1741 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} x^7} \, dx\)
Optimal. Leaf size=116 \[ -\frac{2 a^5}{b^6 \sqrt{a+\frac{b}{x}}}-\frac{10 a^4 \sqrt{a+\frac{b}{x}}}{b^6}+\frac{20 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{b^6}+\frac{10 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^6} \]
[Out]
(-2*a^5)/(b^6*Sqrt[a + b/x]) - (10*a^4*Sqrt[a + b/x])/b^6 + (20*a^3*(a + b/x)^(3/2))/(3*b^6) - (4*a^2*(a + b/x
)^(5/2))/b^6 + (10*a*(a + b/x)^(7/2))/(7*b^6) - (2*(a + b/x)^(9/2))/(9*b^6)
________________________________________________________________________________________
Rubi [A] time = 0.0487726, 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} \[ -\frac{2 a^5}{b^6 \sqrt{a+\frac{b}{x}}}-\frac{10 a^4 \sqrt{a+\frac{b}{x}}}{b^6}+\frac{20 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{b^6}+\frac{10 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^6} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b/x)^(3/2)*x^7),x]
[Out]
(-2*a^5)/(b^6*Sqrt[a + b/x]) - (10*a^4*Sqrt[a + b/x])/b^6 + (20*a^3*(a + b/x)^(3/2))/(3*b^6) - (4*a^2*(a + b/x
)^(5/2))/b^6 + (10*a*(a + b/x)^(7/2))/(7*b^6) - (2*(a + b/x)^(9/2))/(9*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 )^{3/2} x^7} \, dx &=-\operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a^5}{b^5 (a+b x)^{3/2}}+\frac{5 a^4}{b^5 \sqrt{a+b x}}-\frac{10 a^3 \sqrt{a+b x}}{b^5}+\frac{10 a^2 (a+b x)^{3/2}}{b^5}-\frac{5 a (a+b x)^{5/2}}{b^5}+\frac{(a+b x)^{7/2}}{b^5}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^5}{b^6 \sqrt{a+\frac{b}{x}}}-\frac{10 a^4 \sqrt{a+\frac{b}{x}}}{b^6}+\frac{20 a^3 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^6}-\frac{4 a^2 \left (a+\frac{b}{x}\right )^{5/2}}{b^6}+\frac{10 a \left (a+\frac{b}{x}\right )^{7/2}}{7 b^6}-\frac{2 \left (a+\frac{b}{x}\right )^{9/2}}{9 b^6}\\ \end{align*}
Mathematica [A] time = 0.0393952, size = 73, normalized size = 0.63 \[ -\frac{2 \left (16 a^2 b^3 x^2-32 a^3 b^2 x^3+128 a^4 b x^4+256 a^5 x^5-10 a b^4 x+7 b^5\right )}{63 b^6 x^5 \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b/x)^(3/2)*x^7),x]
[Out]
(-2*(7*b^5 - 10*a*b^4*x + 16*a^2*b^3*x^2 - 32*a^3*b^2*x^3 + 128*a^4*b*x^4 + 256*a^5*x^5))/(63*b^6*Sqrt[a + b/x
]*x^5)
________________________________________________________________________________________
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}+128\,{a}^{4}b{x}^{4}-32\,{a}^{3}{b}^{2}{x}^{3}+16\,{a}^{2}{b}^{3}{x}^{2}-10\,a{b}^{4}x+7\,{b}^{5} \right ) }{63\,{x}^{6}{b}^{6}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b/x)^(3/2)/x^7,x)
[Out]
-2/63*(a*x+b)*(256*a^5*x^5+128*a^4*b*x^4-32*a^3*b^2*x^3+16*a^2*b^3*x^2-10*a*b^4*x+7*b^5)/x^6/b^6/((a*x+b)/x)^(
3/2)
________________________________________________________________________________________
Maxima [A] time = 0.971163, size = 132, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}}}{9 \, b^{6}} + \frac{10 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a}{7 \, b^{6}} - \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} a^{2}}{b^{6}} + \frac{20 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{3}}{3 \, b^{6}} - \frac{10 \, \sqrt{a + \frac{b}{x}} a^{4}}{b^{6}} - \frac{2 \, a^{5}}{\sqrt{a + \frac{b}{x}} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="maxima")
[Out]
-2/9*(a + b/x)^(9/2)/b^6 + 10/7*(a + b/x)^(7/2)*a/b^6 - 4*(a + b/x)^(5/2)*a^2/b^6 + 20/3*(a + b/x)^(3/2)*a^3/b
^6 - 10*sqrt(a + b/x)*a^4/b^6 - 2*a^5/(sqrt(a + b/x)*b^6)
________________________________________________________________________________________
Fricas [A] time = 1.66459, size = 178, normalized size = 1.53 \begin{align*} -\frac{2 \,{\left (256 \, a^{5} x^{5} + 128 \, a^{4} b x^{4} - 32 \, a^{3} b^{2} x^{3} + 16 \, a^{2} b^{3} x^{2} - 10 \, a b^{4} x + 7 \, b^{5}\right )} \sqrt{\frac{a x + b}{x}}}{63 \,{\left (a b^{6} x^{5} + b^{7} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="fricas")
[Out]
-2/63*(256*a^5*x^5 + 128*a^4*b*x^4 - 32*a^3*b^2*x^3 + 16*a^2*b^3*x^2 - 10*a*b^4*x + 7*b^5)*sqrt((a*x + b)/x)/(
a*b^6*x^5 + b^7*x^4)
________________________________________________________________________________________
Sympy [B] time = 12.7699, size = 9534, normalized size = 82.19 \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)**(3/2)/x**7,x)
[Out]
-512*a**(47/2)*b**(91/2)*x**19*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) +
6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a
**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23
/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b*
*62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/
2) + 63*a**(9/2)*b**66*x**(9/2)) - 7424*a**(45/2)*b**(93/2)*x**18*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2
) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a
**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25
/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b
**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(
13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 50112*a**(43/2)*b**(95/2)*x**17*sqrt(a*x
/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665
*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(2
7/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*
b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x
**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 2088
00*a**(41/2)*b**(97/2)*x**16*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6
615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**
(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2
)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**6
2*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2)
+ 63*a**(9/2)*b**66*x**(9/2)) - 600300*a**(39/2)*b**(99/2)*x**15*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2
) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a
**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25
/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b
**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(
13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 1260630*a**(37/2)*b**(101/2)*x**14*sqrt(
a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28
665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a*
*(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/
2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**6
3*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 1
996008*a**(35/2)*b**(103/2)*x**13*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2
) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 18918
9*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**
(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)
*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(
11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 2423850*a**(33/2)*b**(105/2)*x**12*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x
**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) +
85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405
*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(
19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**
64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 2273076*a**(31/2)*b**(107/2)*x**1
1*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/
2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 31
5315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*
a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/
2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/
2)) - 1644214*a**(29/2)*b**(109/2)*x**10*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x
**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2)
+ 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405
405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a*
*(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**
65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 910624*a**(27/2)*b**(111/2)*x**9*sqrt(a*x/b + 1)/(63*a**(39/2)*b*
*51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/
2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 4
05405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189
*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2
)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 383994*a**(25/2)*b**(113/2)*
x**8*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(
35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) +
315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 3153
15*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(
15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**
(9/2)) - 127764*a**(23/2)*b**(115/2)*x**7*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*
x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2)
+ 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 40
5405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a
**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b*
*65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 41202*a**(21/2)*b**(117/2)*x**6*sqrt(a*x/b + 1)/(63*a**(39/2)*b*
*51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/
2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 4
05405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189
*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2
)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 17928*a**(19/2)*b**(119/2)*x
**5*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(3
5/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) +
315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 31531
5*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(1
5/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(
9/2)) - 9006*a**(17/2)*b**(121/2)*x**4*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**
(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) +
189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 40540
5*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(
17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65
*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 3660*a**(15/2)*b**(123/2)*x**3*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*
x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) +
85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 40540
5*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**
(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b*
*64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) - 1026*a**(13/2)*b**(125/2)*x**2*s
qrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2)
+ 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 31531
5*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**
(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*
b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2))
- 176*a**(11/2)*b**(127/2)*x*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) +
6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a*
*(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/
2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**
62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2
) + 63*a**(9/2)*b**66*x**(9/2)) - 14*a**(9/2)*b**(129/2)*sqrt(a*x/b + 1)/(63*a**(39/2)*b**51*x**(39/2) + 945*a
**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*
b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58
*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(
19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 9
45*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 512*a**24*b**45*x**(39/2)/(63*a**(39/2)*b**51*x**
(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85
995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a
**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19
/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64
*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 7680*a**23*b**46*x**(37/2)/(63*a**(
39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54
*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(2
7/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2)
+ 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*
a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 53760*a**22*b**47*x*
*(35/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665
*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(2
7/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*
b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x
**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 2329
60*a**21*b**48*x**(33/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*
x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/
2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) +
315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*
a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66
*x**(9/2)) + 698880*a**20*b**49*x**(31/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615
*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29
/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b
**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x
**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) +
63*a**(9/2)*b**66*x**(9/2)) + 1537536*a**19*b**50*x**(29/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**5
2*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/
2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) +
405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995
*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*
b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 2562560*a**18*b**51*x**(27/2)/(63*a**(39/2)*b**51*x**(39/2) +
945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(3
1/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*
b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61
*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2
) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 3294720*a**17*b**52*x**(25/2)/(63*a**(39/2)*
b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(3
3/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) +
405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 1891
89*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13
/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 3294720*a**16*b**53*x**(23
/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**
(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)
*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**6
0*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(1
5/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 2562560*
a**15*b**54*x**(21/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**
(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2)
+ 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315
315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**
(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x*
*(9/2)) + 1537536*a**14*b**55*x**(19/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a
**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2
)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**
59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**
(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63
*a**(9/2)*b**66*x**(9/2)) + 698880*a**13*b**56*x**(17/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x
**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2)
+ 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405
405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a*
*(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**
65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 232960*a**12*b**57*x**(15/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*
a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)
*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**5
8*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**(19/2)*b**61*x**
(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b**64*x**(13/2) +
945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 53760*a**11*b**58*x**(13/2)/(63*a**(39/2)*b**51*
x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b**54*x**(33/2) +
85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x**(27/2) + 40540
5*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/2) + 189189*a**
(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 6615*a**(13/2)*b*
*64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 7680*a**10*b**59*x**(11/2)/(63*a
**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*a**(33/2)*b*
*54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27/2)*b**57*x*
*(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b**60*x**(21/
2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x**(15/2) + 66
15*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2)) + 512*a**9*b**60*x*
*(9/2)/(63*a**(39/2)*b**51*x**(39/2) + 945*a**(37/2)*b**52*x**(37/2) + 6615*a**(35/2)*b**53*x**(35/2) + 28665*
a**(33/2)*b**54*x**(33/2) + 85995*a**(31/2)*b**55*x**(31/2) + 189189*a**(29/2)*b**56*x**(29/2) + 315315*a**(27
/2)*b**57*x**(27/2) + 405405*a**(25/2)*b**58*x**(25/2) + 405405*a**(23/2)*b**59*x**(23/2) + 315315*a**(21/2)*b
**60*x**(21/2) + 189189*a**(19/2)*b**61*x**(19/2) + 85995*a**(17/2)*b**62*x**(17/2) + 28665*a**(15/2)*b**63*x*
*(15/2) + 6615*a**(13/2)*b**64*x**(13/2) + 945*a**(11/2)*b**65*x**(11/2) + 63*a**(9/2)*b**66*x**(9/2))
________________________________________________________________________________________
Giac [A] time = 1.27019, size = 211, normalized size = 1.82 \begin{align*} -\frac{2}{63} \, b{\left (\frac{63 \, a^{5}}{b^{7} \sqrt{\frac{a x + b}{x}}} + \frac{315 \, a^{4} b^{56} \sqrt{\frac{a x + b}{x}} - \frac{210 \,{\left (a x + b\right )} a^{3} b^{56} \sqrt{\frac{a x + b}{x}}}{x} + \frac{126 \,{\left (a x + b\right )}^{2} a^{2} b^{56} \sqrt{\frac{a x + b}{x}}}{x^{2}} - \frac{45 \,{\left (a x + b\right )}^{3} a b^{56} \sqrt{\frac{a x + b}{x}}}{x^{3}} + \frac{7 \,{\left (a x + b\right )}^{4} b^{56} \sqrt{\frac{a x + b}{x}}}{x^{4}}}{b^{63}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="giac")
[Out]
-2/63*b*(63*a^5/(b^7*sqrt((a*x + b)/x)) + (315*a^4*b^56*sqrt((a*x + b)/x) - 210*(a*x + b)*a^3*b^56*sqrt((a*x +
b)/x)/x + 126*(a*x + b)^2*a^2*b^56*sqrt((a*x + b)/x)/x^2 - 45*(a*x + b)^3*a*b^56*sqrt((a*x + b)/x)/x^3 + 7*(a
*x + b)^4*b^56*sqrt((a*x + b)/x)/x^4)/b^63)