3.18.41 \(\int \frac {1}{(a+\frac {b}{x})^{3/2} x^7} \, dx\) [1741]

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]

20/3*a^3*(a+b/x)^(3/2)/b^6-4*a^2*(a+b/x)^(5/2)/b^6+10/7*a*(a+b/x)^(7/2)/b^6-2/9*(a+b/x)^(9/2)/b^6-2*a^5/b^6/(a
+b/x)^(1/2)-10*a^4*(a+b/x)^(1/2)/b^6

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Rubi [A]
time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} -\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 {gather*}

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 45

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])

Rule 272

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]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^7} \, dx &=-\text {Subst}\left (\int \frac {x^5}{(a+b x)^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\text {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*}

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Mathematica [A]
time = 0.05, size = 82, normalized size = 0.71 \begin {gather*} -\frac {2 \sqrt {\frac {b+a x}{x}} \left (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\right )}{63 b^6 x^4 (b+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(3/2)*x^7),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.07, size = 541, normalized size = 4.66

method	result	size
gosper	\(-\frac {2 \left (a x +b \right ) \left (256 a^{5} x^{5}+128 a^{4} b \,x^{4}-32 a^{3} b^{2} x^{3}+16 a^{2} x^{2} b^{3}-10 a \,b^{4} x +7 b^{5}\right )}{63 x^{6} b^{6} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\)	\(77\)
trager	\(-\frac {2 \left (256 a^{5} x^{5}+128 a^{4} b \,x^{4}-32 a^{3} b^{2} x^{3}+16 a^{2} x^{2} b^{3}-10 a \,b^{4} x +7 b^{5}\right ) \sqrt {-\frac {-a x -b}{x}}}{63 x^{4} b^{6} \left (a x +b \right )}\)	\(83\)
risch	\(-\frac {2 \left (a x +b \right ) \left (193 a^{4} x^{4}-65 a^{3} b \,x^{3}+33 a^{2} b^{2} x^{2}-17 a \,b^{3} x +7 b^{4}\right )}{63 b^{6} x^{5} \sqrt {\frac {a x +b}{x}}}-\frac {2 a^{5}}{b^{6} \sqrt {\frac {a x +b}{x}}}\)	\(86\)
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (126 \sqrt {x \left (a x +b \right )}\, a^{\frac {15}{2}} x^{8}+126 \sqrt {a \,x^{2}+b x}\, a^{\frac {15}{2}} x^{8}-63 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{7} b \,x^{8}+63 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{7} b \,x^{8}+63 \left (x \left (a x +b \right )\right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{6}+252 \sqrt {x \left (a x +b \right )}\, a^{\frac {13}{2}} b \,x^{7}-315 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{6}+252 \sqrt {a \,x^{2}+b x}\, a^{\frac {13}{2}} b \,x^{7}-126 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b^{2} x^{7}+126 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{6} b^{2} x^{7}+126 \sqrt {x \left (a x +b \right )}\, a^{\frac {11}{2}} b^{2} x^{6}-508 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} b \,x^{5}+126 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b^{2} x^{6}-63 \ln \left (\frac {2 \sqrt {x \left (a x +b \right )}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{3} x^{6}+63 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{5} b^{3} x^{6}-128 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2} x^{4}+32 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} x^{3}-16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4} x^{2}+10 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x -7 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{6}\right )}{63 x^{5} \sqrt {x \left (a x +b \right )}\, b^{7} \left (a x +b \right )^{2} \sqrt {a}}\)	\(541\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x*b)^(3/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

2/63*((a*x+b)/x)^(1/2)/x^5*(126*(x*(a*x+b))^(1/2)*a^(15/2)*x^8+126*(a*x^2+b*x)^(1/2)*a^(15/2)*x^8-63*ln(1/2*(2
*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^7*b*x^8+63*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2)
)*a^7*b*x^8+63*(x*(a*x+b))^(3/2)*a^(13/2)*x^6+252*(x*(a*x+b))^(1/2)*a^(13/2)*b*x^7-315*(a*x^2+b*x)^(3/2)*a^(13
/2)*x^6+252*(a*x^2+b*x)^(1/2)*a^(13/2)*b*x^7-126*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^6*b^2
*x^7+126*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^6*b^2*x^7+126*(x*(a*x+b))^(1/2)*a^(11/2)*b^2*
x^6-508*(a*x^2+b*x)^(3/2)*a^(11/2)*b*x^5+126*(a*x^2+b*x)^(1/2)*a^(11/2)*b^2*x^6-63*ln(1/2*(2*(x*(a*x+b))^(1/2)
*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*b^3*x^6+63*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*b^3*x^6-12
8*(a*x^2+b*x)^(3/2)*a^(9/2)*b^2*x^4+32*(a*x^2+b*x)^(3/2)*a^(7/2)*b^3*x^3-16*(a*x^2+b*x)^(3/2)*a^(5/2)*b^4*x^2+
10*(a*x^2+b*x)^(3/2)*a^(3/2)*b^5*x-7*(a*x^2+b*x)^(3/2)*a^(1/2)*b^6)/(x*(a*x+b))^(1/2)/b^7/(a*x+b)^2/a^(1/2)

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Maxima [A]
time = 0.30, size = 98, normalized size = 0.84 \begin {gather*} -\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 {gather*}

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)

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Fricas [A]
time = 0.37, size = 83, normalized size = 0.72 \begin {gather*} -\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 {gather*}

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)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 9534 vs. \(2 (100) = 200\).
time = 5.99, size = 9534, normalized size = 82.19 \begin {gather*} \text {Too large to display} \end {gather*}

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/...

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(3/2)/x^7,x, algorithm="giac")

[Out]

integrate(1/((a + b/x)^(3/2)*x^7), x)

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Mupad [B]
time = 1.44, size = 112, normalized size = 0.97 \begin {gather*} \frac {34\,a\,\sqrt {a+\frac {b}{x}}}{63\,b^3\,x^3}-\frac {2\,\sqrt {a+\frac {b}{x}}}{9\,b^2\,x^4}-\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {386\,a^4}{63\,b^5}+\frac {512\,a^5\,x}{63\,b^6}\right )}{b+a\,x}-\frac {22\,a^2\,\sqrt {a+\frac {b}{x}}}{21\,b^4\,x^2}+\frac {130\,a^3\,\sqrt {a+\frac {b}{x}}}{63\,b^5\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b/x)^(3/2)),x)

[Out]

(34*a*(a + b/x)^(1/2))/(63*b^3*x^3) - (2*(a + b/x)^(1/2))/(9*b^2*x^4) - ((a + b/x)^(1/2)*((386*a^4)/(63*b^5) +
(512*a^5*x)/(63*b^6)))/(b + a*x) - (22*a^2*(a + b/x)^(1/2))/(21*b^4*x^2) + (130*a^3*(a + b/x)^(1/2))/(63*b^5*
x)

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