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

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

[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 = 0.0518065, 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, 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 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 )^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*}

Mathematica [A]  time = 0.0762018, size = 71, normalized size = 0.8 \[ \frac{\frac{a \left (52 a^2 b \sqrt{x}+25 a^3+42 a b^2 x+12 b^3 x^{3/2}\right )}{\left (a+b \sqrt{x}\right )^4}-12 \log \left (a+b \sqrt{x}\right )+6 \log (x)}{6 a^5} \]

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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Maple [A]  time = 0.01, size = 76, normalized size = 0.9 \begin{align*}{\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 ) }} \end{align*}

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 = 1.04872, size = 131, normalized size = 1.47 \begin{align*} \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}} \end{align*}

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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Fricas [B]  time = 1.28093, size = 494, normalized size = 5.55 \begin{align*} -\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 )}} \end{align*}

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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Sympy [A]  time = 6.75743, size = 1049, normalized size = 11.79 \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))**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)) + 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) + 2
4*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)) - 48*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*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)) - 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)) - 108*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)) - 88*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)) - 25*b**4*x**
(5/2)/(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)), True)
)

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Giac [A]  time = 1.10275, size = 93, normalized size = 1.04 \begin{align*} -\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}} \end{align*}

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)