\(\int \frac {1}{(a+b \sqrt {x})^8 x} \, dx\) [2229]

Optimal result

Integrand size = 15, antiderivative size = 143 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=\frac {2}{7 a \left (a+b \sqrt {x}\right )^7}+\frac {1}{3 a^2 \left (a+b \sqrt {x}\right )^6}+\frac {2}{5 a^3 \left (a+b \sqrt {x}\right )^5}+\frac {1}{2 a^4 \left (a+b \sqrt {x}\right )^4}+\frac {2}{3 a^5 \left (a+b \sqrt {x}\right )^3}+\frac {1}{a^6 \left (a+b \sqrt {x}\right )^2}+\frac {2}{a^7 \left (a+b \sqrt {x}\right )}-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {\log (x)}{a^8} \]

[Out]

ln(x)/a^8-2*ln(a+b*x^(1/2))/a^8+2/7/a/(a+b*x^(1/2))^7+1/3/a^2/(a+b*x^(1/2))^6+2/5/a^3/(a+b*x^(1/2))^5+1/2/a^4/
(a+b*x^(1/2))^4+2/3/a^5/(a+b*x^(1/2))^3+1/a^6/(a+b*x^(1/2))^2+2/a^7/(a+b*x^(1/2))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46} \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {\log (x)}{a^8}+\frac {2}{a^7 \left (a+b \sqrt {x}\right )}+\frac {1}{a^6 \left (a+b \sqrt {x}\right )^2}+\frac {2}{3 a^5 \left (a+b \sqrt {x}\right )^3}+\frac {1}{2 a^4 \left (a+b \sqrt {x}\right )^4}+\frac {2}{5 a^3 \left (a+b \sqrt {x}\right )^5}+\frac {1}{3 a^2 \left (a+b \sqrt {x}\right )^6}+\frac {2}{7 a \left (a+b \sqrt {x}\right )^7} \]

[In]

Int[1/((a + b*Sqrt[x])^8*x),x]

[Out]

2/(7*a*(a + b*Sqrt[x])^7) + 1/(3*a^2*(a + b*Sqrt[x])^6) + 2/(5*a^3*(a + b*Sqrt[x])^5) + 1/(2*a^4*(a + b*Sqrt[x
])^4) + 2/(3*a^5*(a + b*Sqrt[x])^3) + 1/(a^6*(a + b*Sqrt[x])^2) + 2/(a^7*(a + b*Sqrt[x])) - (2*Log[a + b*Sqrt[
x]])/a^8 + Log[x]/a^8

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{x (a+b x)^8} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a^8 x}-\frac {b}{a (a+b x)^8}-\frac {b}{a^2 (a+b x)^7}-\frac {b}{a^3 (a+b x)^6}-\frac {b}{a^4 (a+b x)^5}-\frac {b}{a^5 (a+b x)^4}-\frac {b}{a^6 (a+b x)^3}-\frac {b}{a^7 (a+b x)^2}-\frac {b}{a^8 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{7 a \left (a+b \sqrt {x}\right )^7}+\frac {1}{3 a^2 \left (a+b \sqrt {x}\right )^6}+\frac {2}{5 a^3 \left (a+b \sqrt {x}\right )^5}+\frac {1}{2 a^4 \left (a+b \sqrt {x}\right )^4}+\frac {2}{3 a^5 \left (a+b \sqrt {x}\right )^3}+\frac {1}{a^6 \left (a+b \sqrt {x}\right )^2}+\frac {2}{a^7 \left (a+b \sqrt {x}\right )}-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {\log (x)}{a^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=\frac {\frac {a \left (1089 a^6+4683 a^5 b \sqrt {x}+9639 a^4 b^2 x+11165 a^3 b^3 x^{3/2}+7490 a^2 b^4 x^2+2730 a b^5 x^{5/2}+420 b^6 x^3\right )}{\left (a+b \sqrt {x}\right )^7}-420 \log \left (a+b \sqrt {x}\right )+210 \log (x)}{210 a^8} \]

[In]

Integrate[1/((a + b*Sqrt[x])^8*x),x]

[Out]

((a*(1089*a^6 + 4683*a^5*b*Sqrt[x] + 9639*a^4*b^2*x + 11165*a^3*b^3*x^(3/2) + 7490*a^2*b^4*x^2 + 2730*a*b^5*x^
(5/2) + 420*b^6*x^3))/(a + b*Sqrt[x])^7 - 420*Log[a + b*Sqrt[x]] + 210*Log[x])/(210*a^8)

Maple [A] (verified)

Time = 3.64 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\ln \left (x \right )}{a^{8}}-\frac {2 \ln \left (a +b \sqrt {x}\right )}{a^{8}}+\frac {2}{7 a \left (a +b \sqrt {x}\right )^{7}}+\frac {1}{3 a^{2} \left (a +b \sqrt {x}\right )^{6}}+\frac {2}{5 a^{3} \left (a +b \sqrt {x}\right )^{5}}+\frac {1}{2 a^{4} \left (a +b \sqrt {x}\right )^{4}}+\frac {2}{3 a^{5} \left (a +b \sqrt {x}\right )^{3}}+\frac {1}{a^{6} \left (a +b \sqrt {x}\right )^{2}}+\frac {2}{a^{7} \left (a +b \sqrt {x}\right )}\)	\(118\)
default	\(\frac {\ln \left (x \right )}{a^{8}}-\frac {2 \ln \left (a +b \sqrt {x}\right )}{a^{8}}+\frac {2}{7 a \left (a +b \sqrt {x}\right )^{7}}+\frac {1}{3 a^{2} \left (a +b \sqrt {x}\right )^{6}}+\frac {2}{5 a^{3} \left (a +b \sqrt {x}\right )^{5}}+\frac {1}{2 a^{4} \left (a +b \sqrt {x}\right )^{4}}+\frac {2}{3 a^{5} \left (a +b \sqrt {x}\right )^{3}}+\frac {1}{a^{6} \left (a +b \sqrt {x}\right )^{2}}+\frac {2}{a^{7} \left (a +b \sqrt {x}\right )}\)	\(118\)

[In]

int(1/x/(a+b*x^(1/2))^8,x,method=_RETURNVERBOSE)

[Out]

ln(x)/a^8-2*ln(a+b*x^(1/2))/a^8+2/7/a/(a+b*x^(1/2))^7+1/3/a^2/(a+b*x^(1/2))^6+2/5/a^3/(a+b*x^(1/2))^5+1/2/a^4/
(a+b*x^(1/2))^4+2/3/a^5/(a+b*x^(1/2))^3+1/a^6/(a+b*x^(1/2))^2+2/a^7/(a+b*x^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (117) = 234\).

Time = 0.31 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.78 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=-\frac {210 \, a^{2} b^{12} x^{6} - 1365 \, a^{4} b^{10} x^{5} + 3745 \, a^{6} b^{8} x^{4} - 5530 \, a^{8} b^{6} x^{3} + 5964 \, a^{10} b^{4} x^{2} - 273 \, a^{12} b^{2} x + 1089 \, a^{14} + 420 \, {\left (b^{14} x^{7} - 7 \, a^{2} b^{12} x^{6} + 21 \, a^{4} b^{10} x^{5} - 35 \, a^{6} b^{8} x^{4} + 35 \, a^{8} b^{6} x^{3} - 21 \, a^{10} b^{4} x^{2} + 7 \, a^{12} b^{2} x - a^{14}\right )} \log \left (b \sqrt {x} + a\right ) - 420 \, {\left (b^{14} x^{7} - 7 \, a^{2} b^{12} x^{6} + 21 \, a^{4} b^{10} x^{5} - 35 \, a^{6} b^{8} x^{4} + 35 \, a^{8} b^{6} x^{3} - 21 \, a^{10} b^{4} x^{2} + 7 \, a^{12} b^{2} x - a^{14}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (105 \, a b^{13} x^{6} - 700 \, a^{3} b^{11} x^{5} + 1981 \, a^{5} b^{9} x^{4} - 3072 \, a^{7} b^{7} x^{3} + 2891 \, a^{9} b^{5} x^{2} - 980 \, a^{11} b^{3} x + 735 \, a^{13} b\right )} \sqrt {x}}{210 \, {\left (a^{8} b^{14} x^{7} - 7 \, a^{10} b^{12} x^{6} + 21 \, a^{12} b^{10} x^{5} - 35 \, a^{14} b^{8} x^{4} + 35 \, a^{16} b^{6} x^{3} - 21 \, a^{18} b^{4} x^{2} + 7 \, a^{20} b^{2} x - a^{22}\right )}} \]

[In]

integrate(1/x/(a+b*x^(1/2))^8,x, algorithm="fricas")

[Out]

-1/210*(210*a^2*b^12*x^6 - 1365*a^4*b^10*x^5 + 3745*a^6*b^8*x^4 - 5530*a^8*b^6*x^3 + 5964*a^10*b^4*x^2 - 273*a
^12*b^2*x + 1089*a^14 + 420*(b^14*x^7 - 7*a^2*b^12*x^6 + 21*a^4*b^10*x^5 - 35*a^6*b^8*x^4 + 35*a^8*b^6*x^3 - 2
1*a^10*b^4*x^2 + 7*a^12*b^2*x - a^14)*log(b*sqrt(x) + a) - 420*(b^14*x^7 - 7*a^2*b^12*x^6 + 21*a^4*b^10*x^5 -
35*a^6*b^8*x^4 + 35*a^8*b^6*x^3 - 21*a^10*b^4*x^2 + 7*a^12*b^2*x - a^14)*log(sqrt(x)) - 4*(105*a*b^13*x^6 - 70
0*a^3*b^11*x^5 + 1981*a^5*b^9*x^4 - 3072*a^7*b^7*x^3 + 2891*a^9*b^5*x^2 - 980*a^11*b^3*x + 735*a^13*b)*sqrt(x)
)/(a^8*b^14*x^7 - 7*a^10*b^12*x^6 + 21*a^12*b^10*x^5 - 35*a^14*b^8*x^4 + 35*a^16*b^6*x^3 - 21*a^18*b^4*x^2 + 7
*a^20*b^2*x - a^22)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2581 vs. \(2 (133) = 266\).

Time = 2.47 (sec) , antiderivative size = 2581, normalized size of antiderivative = 18.05 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=\text {Too large to display} \]

[In]

integrate(1/x/(a+b*x**(1/2))**8,x)

[Out]

Piecewise((zoo/x**4, Eq(a, 0) & Eq(b, 0)), (log(x)/a**8, Eq(b, 0)), (-1/(4*b**8*x**4), Eq(a, 0)), (210*a**7*sq
rt(x)*log(x)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**1
1*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) - 420*a**7*sqrt(x)*log(
a/b + sqrt(x))/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a*
*11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 1089*a**7*sqrt(x)/(
210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2
) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 1470*a**6*b*x*log(x)/(210*a**15*sqr
t(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**1
0*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) - 2940*a**6*b*x*log(a/b + sqrt(x))/(210*a**15*sqrt
(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10
*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 4683*a**6*b*x/(210*a**15*sqrt(x) + 1470*a**14*b*x
+ 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a*
*9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 4410*a**5*b**2*x**(3/2)*log(x)/(210*a**15*sqrt(x) + 1470*a**14*b*x +
4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*
b**6*x**(7/2) + 210*a**8*b**7*x**4) - 8820*a**5*b**2*x**(3/2)*log(a/b + sqrt(x))/(210*a**15*sqrt(x) + 1470*a**
14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1
470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 9639*a**5*b**2*x**(3/2)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4
410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b
**6*x**(7/2) + 210*a**8*b**7*x**4) + 7350*a**4*b**3*x**2*log(x)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**
13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**
(7/2) + 210*a**8*b**7*x**4) - 14700*a**4*b**3*x**2*log(a/b + sqrt(x))/(210*a**15*sqrt(x) + 1470*a**14*b*x + 44
10*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b*
*6*x**(7/2) + 210*a**8*b**7*x**4) + 11165*a**4*b**3*x**2/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2
*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) +
210*a**8*b**7*x**4) + 7350*a**3*b**4*x**(5/2)*log(x)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x*
*(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 21
0*a**8*b**7*x**4) - 14700*a**3*b**4*x**(5/2)*log(a/b + sqrt(x))/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**
13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**
(7/2) + 210*a**8*b**7*x**4) + 7490*a**3*b**4*x**(5/2)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x*
*(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 21
0*a**8*b**7*x**4) + 4410*a**2*b**5*x**3*log(x)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2)
+ 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*
b**7*x**4) - 8820*a**2*b**5*x**3*log(a/b + sqrt(x))/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(
3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*
a**8*b**7*x**4) + 2730*a**2*b**5*x**3/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a*
*12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4
) + 1470*a*b**6*x**(7/2)*log(x)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b*
*3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) - 29
40*a*b**6*x**(7/2)*log(a/b + sqrt(x))/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a*
*12*b**3*x**2 + 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4
) + 420*a*b**6*x**(7/2)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2
+ 7350*a**11*b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 210*b**7*x
**4*log(x)/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*
b**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) - 420*b**7*x**4*log(a/b +
sqrt(x))/(210*a**15*sqrt(x) + 1470*a**14*b*x + 4410*a**13*b**2*x**(3/2) + 7350*a**12*b**3*x**2 + 7350*a**11*b
**4*x**(5/2) + 4410*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=\frac {420 \, b^{6} x^{3} + 2730 \, a b^{5} x^{\frac {5}{2}} + 7490 \, a^{2} b^{4} x^{2} + 11165 \, a^{3} b^{3} x^{\frac {3}{2}} + 9639 \, a^{4} b^{2} x + 4683 \, a^{5} b \sqrt {x} + 1089 \, a^{6}}{210 \, {\left (a^{7} b^{7} x^{\frac {7}{2}} + 7 \, a^{8} b^{6} x^{3} + 21 \, a^{9} b^{5} x^{\frac {5}{2}} + 35 \, a^{10} b^{4} x^{2} + 35 \, a^{11} b^{3} x^{\frac {3}{2}} + 21 \, a^{12} b^{2} x + 7 \, a^{13} b \sqrt {x} + a^{14}\right )}} - \frac {2 \, \log \left (b \sqrt {x} + a\right )}{a^{8}} + \frac {\log \left (x\right )}{a^{8}} \]

[In]

integrate(1/x/(a+b*x^(1/2))^8,x, algorithm="maxima")

[Out]

1/210*(420*b^6*x^3 + 2730*a*b^5*x^(5/2) + 7490*a^2*b^4*x^2 + 11165*a^3*b^3*x^(3/2) + 9639*a^4*b^2*x + 4683*a^5
*b*sqrt(x) + 1089*a^6)/(a^7*b^7*x^(7/2) + 7*a^8*b^6*x^3 + 21*a^9*b^5*x^(5/2) + 35*a^10*b^4*x^2 + 35*a^11*b^3*x
^(3/2) + 21*a^12*b^2*x + 7*a^13*b*sqrt(x) + a^14) - 2*log(b*sqrt(x) + a)/a^8 + log(x)/a^8

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=-\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{8}} + \frac {\log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{3} + 2730 \, a^{2} b^{5} x^{\frac {5}{2}} + 7490 \, a^{3} b^{4} x^{2} + 11165 \, a^{4} b^{3} x^{\frac {3}{2}} + 9639 \, a^{5} b^{2} x + 4683 \, a^{6} b \sqrt {x} + 1089 \, a^{7}}{210 \, {\left (b \sqrt {x} + a\right )}^{7} a^{8}} \]

[In]

integrate(1/x/(a+b*x^(1/2))^8,x, algorithm="giac")

[Out]

-2*log(abs(b*sqrt(x) + a))/a^8 + log(abs(x))/a^8 + 1/210*(420*a*b^6*x^3 + 2730*a^2*b^5*x^(5/2) + 7490*a^3*b^4*
x^2 + 11165*a^4*b^3*x^(3/2) + 9639*a^5*b^2*x + 4683*a^6*b*sqrt(x) + 1089*a^7)/((b*sqrt(x) + a)^7*a^8)

Mupad [B] (verification not implemented)

Time = 5.63 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x} \, dx=\frac {\frac {363}{70\,a}+\frac {223\,b\,\sqrt {x}}{10\,a^2}+\frac {459\,b^2\,x}{10\,a^3}+\frac {107\,b^4\,x^2}{3\,a^5}+\frac {319\,b^3\,x^{3/2}}{6\,a^4}+\frac {2\,b^6\,x^3}{a^7}+\frac {13\,b^5\,x^{5/2}}{a^6}}{a^7+b^7\,x^{7/2}+21\,a^5\,b^2\,x+7\,a\,b^6\,x^3+7\,a^6\,b\,\sqrt {x}+35\,a^3\,b^4\,x^2+35\,a^4\,b^3\,x^{3/2}+21\,a^2\,b^5\,x^{5/2}}-\frac {4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^8} \]

[In]

int(1/(x*(a + b*x^(1/2))^8),x)

[Out]

(363/(70*a) + (223*b*x^(1/2))/(10*a^2) + (459*b^2*x)/(10*a^3) + (107*b^4*x^2)/(3*a^5) + (319*b^3*x^(3/2))/(6*a
^4) + (2*b^6*x^3)/a^7 + (13*b^5*x^(5/2))/a^6)/(a^7 + b^7*x^(7/2) + 21*a^5*b^2*x + 7*a*b^6*x^3 + 7*a^6*b*x^(1/2
) + 35*a^3*b^4*x^2 + 35*a^4*b^3*x^(3/2) + 21*a^2*b^5*x^(5/2)) - (4*atanh((2*b*x^(1/2))/a + 1))/a^8