3.2220 \(\int \frac {1}{(a+b \sqrt {x})^5 x} \, dx\)

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]

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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Rubi [A]  time = 0.05, antiderivative size = 89, 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 = {266, 44} \[ \frac {2}{a^4 \left (a+b \sqrt {x}\right )}+\frac {1}{a^3 \left (a+b \sqrt {x}\right )^2}+\frac {2}{3 a^2 \left (a+b \sqrt {x}\right )^3}-\frac {2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {\log (x)}{a^5}+\frac {1}{2 a \left (a+b \sqrt {x}\right )^4} \]

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

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])

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]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^5} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{a^5 x}-\frac {b}{a (a+b x)^5}-\frac {b}{a^2 (a+b x)^4}-\frac {b}{a^3 (a+b x)^3}-\frac {b}{a^4 (a+b x)^2}-\frac {b}{a^5 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\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 (a+b \sqrt {x}\right )}{a^5}+\frac {\log (x)}{a^5}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 71, normalized size = 0.80 \[ \frac {\frac {a \left (25 a^3+52 a^2 b \sqrt {x}+42 a b^2 x+12 b^3 x^{3/2}\right )}{\left (a+b \sqrt {x}\right )^4}-12 \log \left (a+b \sqrt {x}\right )+6 \log (x)}{6 a^5} \]

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*L
og[x])/(6*a^5)

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fricas [B]  time = 1.05, size = 227, normalized size = 2.55 \[ -\frac {6 \, a^{2} b^{6} x^{3} - 21 \, a^{4} b^{4} x^{2} + 16 \, a^{6} b^{2} x - 25 \, a^{8} + 12 \, {\left (b^{8} x^{4} - 4 \, a^{2} b^{6} x^{3} + 6 \, a^{4} b^{4} x^{2} - 4 \, a^{6} b^{2} x + a^{8}\right )} \log \left (b \sqrt {x} + a\right ) - 12 \, {\left (b^{8} x^{4} - 4 \, a^{2} b^{6} x^{3} + 6 \, a^{4} b^{4} x^{2} - 4 \, a^{6} b^{2} x + a^{8}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (3 \, a b^{7} x^{3} - 11 \, a^{3} b^{5} x^{2} + 14 \, a^{5} b^{3} x - 12 \, a^{7} b\right )} \sqrt {x}}{6 \, {\left (a^{5} b^{8} x^{4} - 4 \, a^{7} b^{6} x^{3} + 6 \, a^{9} b^{4} x^{2} - 4 \, a^{11} b^{2} x + a^{13}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

-1/6*(6*a^2*b^6*x^3 - 21*a^4*b^4*x^2 + 16*a^6*b^2*x - 25*a^8 + 12*(b^8*x^4 - 4*a^2*b^6*x^3 + 6*a^4*b^4*x^2 - 4
*a^6*b^2*x + a^8)*log(b*sqrt(x) + a) - 12*(b^8*x^4 - 4*a^2*b^6*x^3 + 6*a^4*b^4*x^2 - 4*a^6*b^2*x + a^8)*log(sq
rt(x)) - 4*(3*a*b^7*x^3 - 11*a^3*b^5*x^2 + 14*a^5*b^3*x - 12*a^7*b)*sqrt(x))/(a^5*b^8*x^4 - 4*a^7*b^6*x^3 + 6*
a^9*b^4*x^2 - 4*a^11*b^2*x + a^13)

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giac [A]  time = 0.16, size = 69, normalized size = 0.78 \[ -\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{5}} + \frac {\log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, a b^{3} x^{\frac {3}{2}} + 42 \, a^{2} b^{2} x + 52 \, a^{3} b \sqrt {x} + 25 \, a^{4}}{6 \, {\left (b \sqrt {x} + a\right )}^{4} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

-2*log(abs(b*sqrt(x) + a))/a^5 + log(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) + 2
5*a^4)/((b*sqrt(x) + a)^4*a^5)

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maple [A]  time = 0.01, size = 76, normalized size = 0.85 \[ \frac {1}{2 \left (b \sqrt {x}+a \right )^{4} a}+\frac {2}{3 \left (b \sqrt {x}+a \right )^{3} a^{2}}+\frac {1}{\left (b \sqrt {x}+a \right )^{2} a^{3}}+\frac {2}{\left (b \sqrt {x}+a \right ) a^{4}}+\frac {\ln \relax (x )}{a^{5}}-\frac {2 \ln \left (b \sqrt {x}+a \right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^(1/2)+a)^5,x)

[Out]

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

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maxima [A]  time = 0.93, size = 97, normalized size = 1.09 \[ \frac {12 \, b^{3} x^{\frac {3}{2}} + 42 \, a b^{2} x + 52 \, a^{2} b \sqrt {x} + 25 \, a^{3}}{6 \, {\left (a^{4} b^{4} x^{2} + 4 \, a^{5} b^{3} x^{\frac {3}{2}} + 6 \, a^{6} b^{2} x + 4 \, a^{7} b \sqrt {x} + a^{8}\right )}} - \frac {2 \, \log \left (b \sqrt {x} + a\right )}{a^{5}} + \frac {\log \relax (x)}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a+b*x^(1/2))^5,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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mupad [B]  time = 0.08, size = 94, normalized size = 1.06 \[ \frac {\frac {25}{6\,a}+\frac {26\,b\,\sqrt {x}}{3\,a^2}+\frac {7\,b^2\,x}{a^3}+\frac {2\,b^3\,x^{3/2}}{a^4}}{a^4+b^4\,x^2+6\,a^2\,b^2\,x+4\,a^3\,b\,\sqrt {x}+4\,a\,b^3\,x^{3/2}}-\frac {4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^(1/2))^5),x)

[Out]

(25/(6*a) + (26*b*x^(1/2))/(3*a^2) + (7*b^2*x)/a^3 + (2*b^3*x^(3/2))/a^4)/(a^4 + b^4*x^2 + 6*a^2*b^2*x + 4*a^3
*b*x^(1/2) + 4*a*b^3*x^(3/2)) - (4*atanh((2*b*x^(1/2))/a + 1))/a^5

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sympy [A]  time = 5.83, size = 1049, normalized size = 11.79 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\relax (x )}}{a^{5}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{5} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {6 a^{4} \sqrt {x} \log {\relax (x )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} - \frac {12 a^{4} \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {25 a^{4} \sqrt {x}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {24 a^{3} b x \log {\relax (x )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} - \frac {48 a^{3} b x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {52 a^{3} b x}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {36 a^{2} b^{2} x^{\frac {3}{2}} \log {\relax (x )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} - \frac {72 a^{2} b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {42 a^{2} b^{2} x^{\frac {3}{2}}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {24 a b^{3} x^{2} \log {\relax (x )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} - \frac {48 a b^{3} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {12 a b^{3} x^{2}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} + \frac {6 b^{4} x^{\frac {5}{2}} \log {\relax (x )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} - \frac {12 b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{9} \sqrt {x} + 24 a^{8} b x + 36 a^{7} b^{2} x^{\frac {3}{2}} + 24 a^{6} b^{3} x^{2} + 6 a^{5} b^{4} x^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

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*sqrt(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*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)) + 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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