3.2229 \(\int \frac{1}{(a+b \sqrt{x})^8 x} \, dx\)
Optimal. Leaf size=143 \[ \frac{2}{a^7 \left (a+b \sqrt{x}\right )}+\frac{1}{a^6 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^5 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a^4 \left (a+b \sqrt{x}\right )^4}+\frac{2}{5 a^3 \left (a+b \sqrt{x}\right )^5}+\frac{1}{3 a^2 \left (a+b \sqrt{x}\right )^6}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{\log (x)}{a^8}+\frac{2}{7 a \left (a+b \sqrt{x}\right )^7} \]
[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
________________________________________________________________________________________
Rubi [A] time = 0.088623, antiderivative size = 143, 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, 44} \[ \frac{2}{a^7 \left (a+b \sqrt{x}\right )}+\frac{1}{a^6 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^5 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a^4 \left (a+b \sqrt{x}\right )^4}+\frac{2}{5 a^3 \left (a+b \sqrt{x}\right )^5}+\frac{1}{3 a^2 \left (a+b \sqrt{x}\right )^6}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{\log (x)}{a^8}+\frac{2}{7 a \left (a+b \sqrt{x}\right )^7} \]
Antiderivative was successfully verified.
[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 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 44
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] && L
tQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^8 x} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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] time = 0.130149, size = 106, normalized size = 0.74 \[ \frac{\frac{a \left (11165 a^3 b^3 x^{3/2}+7490 a^2 b^4 x^2+9639 a^4 b^2 x+4683 a^5 b \sqrt{x}+1089 a^6+2730 a b^5 x^{5/2}+420 b^6 x^3\right )}{\left (a+b \sqrt{x}\right )^7}-420 \log \left (a+b \sqrt{x}\right )+210 \log (x)}{210 a^8} \]
Antiderivative was successfully verified.
[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)
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Maple [A] time = 0.01, size = 118, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{{a}^{8}}}-2\,{\frac{\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}}+2\,{\frac{1}{{a}^{7} \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x/(a+b*x^(1/2))^8,x)
[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))
________________________________________________________________________________________
Maxima [A] time = 1.00387, size = 220, normalized size = 1.54 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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Fricas [B] time = 1.39418, size = 919, normalized size = 6.43 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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 [A] time = 14.0691, size = 2576, normalized size = 18.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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) + 630*a**7*sqrt(x)/(2
10*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*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) - 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) + 1470*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 + 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) - 8820*a**5*b**2*x**(3/2)*log(a/b + sqrt(x))/(210*a**15*sqrt(x) + 1470*a**1
4*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 + 14
70*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 + 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 + 147
0*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) - 4900*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) + 210*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) - 8575*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) + 210*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) + 2
10*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) - 6909*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) - 2940*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) - 2793*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) + 2
10*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 + 73
50*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 + 735
0*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) - 459*b**7*x**4/(
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))
________________________________________________________________________________________
Giac [A] time = 1.12069, size = 138, normalized size = 0.97 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)