3.2218 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^5 x} \, dx\)

Optimal. Leaf size=89 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{\log (x)}{a^5}+\frac{2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{1}{a^3 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^2 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a \left (a+b \sqrt{x}\right )^4} \]

[Out]

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

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Rubi [A]  time = 0.120346, antiderivative size = 89, 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 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{\log (x)}{a^5}+\frac{2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{1}{a^3 \left (a+b \sqrt{x}\right )^2}+\frac{2}{3 a^2 \left (a+b \sqrt{x}\right )^3}+\frac{1}{2 a \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^5*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.3848, size = 87, normalized size = 0.98 \[ \frac{1}{2 a \left (a + b \sqrt{x}\right )^{4}} + \frac{2}{3 a^{2} \left (a + b \sqrt{x}\right )^{3}} + \frac{1}{a^{3} \left (a + b \sqrt{x}\right )^{2}} + \frac{2}{a^{4} \left (a + b \sqrt{x}\right )} + \frac{2 \log{\left (\sqrt{x} \right )}}{a^{5}} - \frac{2 \log{\left (a + b \sqrt{x} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b*x**(1/2))**5,x)

[Out]

1/(2*a*(a + b*sqrt(x))**4) + 2/(3*a**2*(a + b*sqrt(x))**3) + 1/(a**3*(a + b*sqrt
(x))**2) + 2/(a**4*(a + b*sqrt(x))) + 2*log(sqrt(x))/a**5 - 2*log(a + b*sqrt(x))
/a**5

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Mathematica [A]  time = 0.10164, size = 71, normalized size = 0.8 \[ \frac{\frac{a \left (25 a^3+52 a^2 b \sqrt{x}+42 a b^2 x+12 b^3 x^{3/2}\right )}{\left (a+b \sqrt{x}\right )^4}-12 \log \left (a+b \sqrt{x}\right )+6 \log (x)}{6 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^5*x),x]

[Out]

((a*(25*a^3 + 52*a^2*b*Sqrt[x] + 42*a*b^2*x + 12*b^3*x^(3/2)))/(a + b*Sqrt[x])^4
- 12*Log[a + b*Sqrt[x]] + 6*Log[x])/(6*a^5)

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Maple [A]  time = 0.014, size = 76, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{{a}^{5}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{5}}}+{\frac{1}{2\,a} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{2}{3\,{a}^{2}} \left ( a+b\sqrt{x} \right ) ^{-3}}+{\frac{1}{{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-2}}+2\,{\frac{1}{{a}^{4} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^(1/2))^5,x)

[Out]

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

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Maxima [A]  time = 1.44198, size = 131, normalized size = 1.47 \[ \frac{12 \, b^{3} x^{\frac{3}{2}} + 42 \, a b^{2} x + 52 \, a^{2} b \sqrt{x} + 25 \, a^{3}}{6 \,{\left (a^{4} b^{4} x^{2} + 4 \, a^{5} b^{3} x^{\frac{3}{2}} + 6 \, a^{6} b^{2} x + 4 \, a^{7} b \sqrt{x} + a^{8}\right )}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{5}} + \frac{\log \left (x\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x),x, algorithm="maxima")

[Out]

1/6*(12*b^3*x^(3/2) + 42*a*b^2*x + 52*a^2*b*sqrt(x) + 25*a^3)/(a^4*b^4*x^2 + 4*a
^5*b^3*x^(3/2) + 6*a^6*b^2*x + 4*a^7*b*sqrt(x) + a^8) - 2*log(b*sqrt(x) + a)/a^5
+ log(x)/a^5

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Fricas [A]  time = 0.243501, size = 230, normalized size = 2.58 \[ \frac{42 \, a^{2} b^{2} x + 25 \, a^{4} - 12 \,{\left (b^{4} x^{2} + 6 \, a^{2} b^{2} x + a^{4} + 4 \,{\left (a b^{3} x + a^{3} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 12 \,{\left (b^{4} x^{2} + 6 \, a^{2} b^{2} x + a^{4} + 4 \,{\left (a b^{3} x + a^{3} b\right )} \sqrt{x}\right )} \log \left (\sqrt{x}\right ) + 4 \,{\left (3 \, a b^{3} x + 13 \, a^{3} b\right )} \sqrt{x}}{6 \,{\left (a^{5} b^{4} x^{2} + 6 \, a^{7} b^{2} x + a^{9} + 4 \,{\left (a^{6} b^{3} x + a^{8} b\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x),x, algorithm="fricas")

[Out]

1/6*(42*a^2*b^2*x + 25*a^4 - 12*(b^4*x^2 + 6*a^2*b^2*x + a^4 + 4*(a*b^3*x + a^3*
b)*sqrt(x))*log(b*sqrt(x) + a) + 12*(b^4*x^2 + 6*a^2*b^2*x + a^4 + 4*(a*b^3*x +
a^3*b)*sqrt(x))*log(sqrt(x)) + 4*(3*a*b^3*x + 13*a^3*b)*sqrt(x))/(a^5*b^4*x^2 +
6*a^7*b^2*x + a^9 + 4*(a^6*b^3*x + a^8*b)*sqrt(x))

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Sympy [A]  time = 18.3514, size = 1049, normalized size = 11.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b*x**(1/2))**5,x)

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (log(x)/a**5, Eq(b, 0)), (-2/(5*b
**5*x**(5/2)), Eq(a, 0)), (6*a**4*sqrt(x)*log(x)/(6*a**9*sqrt(x) + 24*a**8*b*x +
36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 12*a**4*sqr
t(x)*log(a/b + sqrt(x))/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) +
24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) + 25*a**4*sqrt(x)/(6*a**9*sqrt(x) + 24
*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) +
24*a**3*b*x*log(x)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a*
*6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 48*a**3*b*x*log(a/b + sqrt(x))/(6*a**9*sq
rt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x*
*(5/2)) + 52*a**3*b*x/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24
*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) + 36*a**2*b**2*x**(3/2)*log(x)/(6*a**9*s
qrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x
**(5/2)) - 72*a**2*b**2*x**(3/2)*log(a/b + sqrt(x))/(6*a**9*sqrt(x) + 24*a**8*b*
x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) + 42*a**2*
b**2*x**(3/2)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b*
*3*x**2 + 6*a**5*b**4*x**(5/2)) + 24*a*b**3*x**2*log(x)/(6*a**9*sqrt(x) + 24*a**
8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 48*a
*b**3*x**2*log(a/b + sqrt(x))/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3
/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) + 12*a*b**3*x**2/(6*a**9*sqrt(x)
+ 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2
)) + 6*b**4*x**(5/2)*log(x)/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2
) + 24*a**6*b**3*x**2 + 6*a**5*b**4*x**(5/2)) - 12*b**4*x**(5/2)*log(a/b + sqrt(
x))/(6*a**9*sqrt(x) + 24*a**8*b*x + 36*a**7*b**2*x**(3/2) + 24*a**6*b**3*x**2 +
6*a**5*b**4*x**(5/2)), True))

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GIAC/XCAS [A]  time = 0.277833, size = 93, normalized size = 1.04 \[ -\frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{5}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, a b^{3} x^{\frac{3}{2}} + 42 \, a^{2} b^{2} x + 52 \, a^{3} b \sqrt{x} + 25 \, a^{4}}{6 \,{\left (b \sqrt{x} + a\right )}^{4} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x),x, algorithm="giac")

[Out]

-2*ln(abs(b*sqrt(x) + a))/a^5 + ln(abs(x))/a^5 + 1/6*(12*a*b^3*x^(3/2) + 42*a^2*
b^2*x + 52*a^3*b*sqrt(x) + 25*a^4)/((b*sqrt(x) + a)^4*a^5)