3.2227 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^8 x} \, dx\)
Optimal. Leaf size=143 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{\log (x)}{a^8}+\frac{2}{a^7 \left (a+b \sqrt{x}\right )}+\frac{1}{a^6 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^5 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a^4 \left (a+b \sqrt{x}\right )^4}+\frac{2}{5 a^3 \left (a+b \sqrt{x}\right )^5}+\frac{1}{3 a^2 \left (a+b \sqrt{x}\right )^6}+\frac{2}{7 a \left (a+b \sqrt{x}\right )^7} \]
[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
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Rubi [A] time = 0.185326, 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
\[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{\log (x)}{a^8}+\frac{2}{a^7 \left (a+b \sqrt{x}\right )}+\frac{1}{a^6 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^5 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a^4 \left (a+b \sqrt{x}\right )^4}+\frac{2}{5 a^3 \left (a+b \sqrt{x}\right )^5}+\frac{1}{3 a^2 \left (a+b \sqrt{x}\right )^6}+\frac{2}{7 a \left (a+b \sqrt{x}\right )^7} \]
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
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Rubi in Sympy [A] time = 29.8079, size = 138, normalized size = 0.97 \[ \frac{2}{7 a \left (a + b \sqrt{x}\right )^{7}} + \frac{1}{3 a^{2} \left (a + b \sqrt{x}\right )^{6}} + \frac{2}{5 a^{3} \left (a + b \sqrt{x}\right )^{5}} + \frac{1}{2 a^{4} \left (a + b \sqrt{x}\right )^{4}} + \frac{2}{3 a^{5} \left (a + b \sqrt{x}\right )^{3}} + \frac{1}{a^{6} \left (a + b \sqrt{x}\right )^{2}} + \frac{2}{a^{7} \left (a + b \sqrt{x}\right )} + \frac{2 \log{\left (\sqrt{x} \right )}}{a^{8}} - \frac{2 \log{\left (a + b \sqrt{x} \right )}}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(a+b*x**(1/2))**8,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*sq
rt(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(sqrt(x))/a**8 - 2*lo
g(a + b*sqrt(x))/a**8
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Mathematica [A] time = 0.147629, size = 106, normalized size = 0.74 \[ \frac{\frac{a \left (1089 a^6+4683 a^5 b \sqrt{x}+9639 a^4 b^2 x+11165 a^3 b^3 x^{3/2}+7490 a^2 b^4 x^2+2730 a b^5 x^{5/2}+420 b^6 x^3\right )}{\left (a+b \sqrt{x}\right )^7}-420 \log \left (a+b \sqrt{x}\right )+210 \log (x)}{210 a^8} \]
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) + 74
90*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 = 0.018, size = 118, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{{a}^{8}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{8}}}+{\frac{2}{7\,a} \left ( a+b\sqrt{x} \right ) ^{-7}}+{\frac{1}{3\,{a}^{2}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{2}{5\,{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-5}}+{\frac{1}{2\,{a}^{4}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{2}{3\,{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-3}}+{\frac{1}{{a}^{6}} \left ( a+b\sqrt{x} \right ) ^{-2}}+2\,{\frac{1}{{a}^{7} \left ( a+b\sqrt{x} \right ) }} \]
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))
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Maxima [A] time = 1.43753, size = 220, normalized size = 1.54 \[ \frac{420 \, b^{6} x^{3} + 2730 \, a b^{5} x^{\frac{5}{2}} + 7490 \, a^{2} b^{4} x^{2} + 11165 \, a^{3} b^{3} x^{\frac{3}{2}} + 9639 \, a^{4} b^{2} x + 4683 \, a^{5} b \sqrt{x} + 1089 \, a^{6}}{210 \,{\left (a^{7} b^{7} x^{\frac{7}{2}} + 7 \, a^{8} b^{6} x^{3} + 21 \, a^{9} b^{5} x^{\frac{5}{2}} + 35 \, a^{10} b^{4} x^{2} + 35 \, a^{11} b^{3} x^{\frac{3}{2}} + 21 \, a^{12} b^{2} x + 7 \, a^{13} b \sqrt{x} + a^{14}\right )}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{8}} + \frac{\log \left (x\right )}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*sqrt(x) + a)^8*x),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 [A] time = 0.258582, size = 412, normalized size = 2.88 \[ \frac{420 \, a b^{6} x^{3} + 7490 \, a^{3} b^{4} x^{2} + 9639 \, a^{5} b^{2} x + 1089 \, a^{7} - 420 \,{\left (7 \, a b^{6} x^{3} + 35 \, a^{3} b^{4} x^{2} + 21 \, a^{5} b^{2} x + a^{7} +{\left (b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 35 \, a^{4} b^{3} x + 7 \, a^{6} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 420 \,{\left (7 \, a b^{6} x^{3} + 35 \, a^{3} b^{4} x^{2} + 21 \, a^{5} b^{2} x + a^{7} +{\left (b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 35 \, a^{4} b^{3} x + 7 \, a^{6} b\right )} \sqrt{x}\right )} \log \left (\sqrt{x}\right ) + 7 \,{\left (390 \, a^{2} b^{5} x^{2} + 1595 \, a^{4} b^{3} x + 669 \, a^{6} b\right )} \sqrt{x}}{210 \,{\left (7 \, a^{9} b^{6} x^{3} + 35 \, a^{11} b^{4} x^{2} + 21 \, a^{13} b^{2} x + a^{15} +{\left (a^{8} b^{7} x^{3} + 21 \, a^{10} b^{5} x^{2} + 35 \, a^{12} b^{3} x + 7 \, a^{14} b\right )} \sqrt{x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*sqrt(x) + a)^8*x),x, algorithm="fricas")
[Out]
1/210*(420*a*b^6*x^3 + 7490*a^3*b^4*x^2 + 9639*a^5*b^2*x + 1089*a^7 - 420*(7*a*b
^6*x^3 + 35*a^3*b^4*x^2 + 21*a^5*b^2*x + a^7 + (b^7*x^3 + 21*a^2*b^5*x^2 + 35*a^
4*b^3*x + 7*a^6*b)*sqrt(x))*log(b*sqrt(x) + a) + 420*(7*a*b^6*x^3 + 35*a^3*b^4*x
^2 + 21*a^5*b^2*x + a^7 + (b^7*x^3 + 21*a^2*b^5*x^2 + 35*a^4*b^3*x + 7*a^6*b)*sq
rt(x))*log(sqrt(x)) + 7*(390*a^2*b^5*x^2 + 1595*a^4*b^3*x + 669*a^6*b)*sqrt(x))/
(7*a^9*b^6*x^3 + 35*a^11*b^4*x^2 + 21*a^13*b^2*x + a^15 + (a^8*b^7*x^3 + 21*a^10
*b^5*x^2 + 35*a^12*b^3*x + 7*a^14*b)*sqrt(x))
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Sympy [A] time = 160.334, size = 2684, normalized size = 18.77 \[ \text{result too large to display} \]
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*sqrt(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) + 44
10*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) - 420*a**7*sq
rt(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) + 854*a**7*sqrt(x)/(210*a**15*s
qrt(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) + 7
350*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) + 3038*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 + 14
70*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 4704*a**5*b**2*x**(3/2)/(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 + 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) + 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) + 44
10*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 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) + 2940*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) + 2
10*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) - 14
700*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) + 441
0*a**10*b**5*x**3 + 1470*a**9*b**6*x**(7/2) + 210*a**8*b**7*x**4) - 735*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) + 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) - 2205*
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 + 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) - 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) -
1225*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 + 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) - 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) - 235*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))
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GIAC/XCAS [A] time = 0.278327, size = 138, normalized size = 0.97 \[ -\frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{8}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{8}} + \frac{420 \, a b^{6} x^{3} + 2730 \, a^{2} b^{5} x^{\frac{5}{2}} + 7490 \, a^{3} b^{4} x^{2} + 11165 \, a^{4} b^{3} x^{\frac{3}{2}} + 9639 \, a^{5} b^{2} x + 4683 \, a^{6} b \sqrt{x} + 1089 \, a^{7}}{210 \,{\left (b \sqrt{x} + a\right )}^{7} a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*sqrt(x) + a)^8*x),x, algorithm="giac")
[Out]
-2*ln(abs(b*sqrt(x) + a))/a^8 + ln(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)