3.2212 \(\int \frac {1}{(a+b \sqrt {x})^3 x^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac {12 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5}+\frac {6 b^2}{a^4 \left (a+b \sqrt {x}\right )}+\frac {6 b}{a^4 \sqrt {x}}+\frac {b^2}{a^3 \left (a+b \sqrt {x}\right )^2}-\frac {1}{a^3 x} \]

[Out]

-1/a^3/x+6*b^2*ln(x)/a^5-12*b^2*ln(a+b*x^(1/2))/a^5+6*b/a^4/x^(1/2)+b^2/a^3/(a+b*x^(1/2))^2+6*b^2/a^4/(a+b*x^(
1/2))

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Rubi [A]  time = 0.06, antiderivative size = 85, 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} \[ \frac {6 b^2}{a^4 \left (a+b \sqrt {x}\right )}+\frac {b^2}{a^3 \left (a+b \sqrt {x}\right )^2}-\frac {12 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5}+\frac {6 b}{a^4 \sqrt {x}}-\frac {1}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^3*x^2),x]

[Out]

b^2/(a^3*(a + b*Sqrt[x])^2) + (6*b^2)/(a^4*(a + b*Sqrt[x])) - 1/(a^3*x) + (6*b)/(a^4*Sqrt[x]) - (12*b^2*Log[a
+ b*Sqrt[x]])/a^5 + (6*b^2*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 )^3 x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^3}-\frac {3 b}{a^4 x^2}+\frac {6 b^2}{a^5 x}-\frac {b^3}{a^3 (a+b x)^3}-\frac {3 b^3}{a^4 (a+b x)^2}-\frac {6 b^3}{a^5 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {b^2}{a^3 \left (a+b \sqrt {x}\right )^2}+\frac {6 b^2}{a^4 \left (a+b \sqrt {x}\right )}-\frac {1}{a^3 x}+\frac {6 b}{a^4 \sqrt {x}}-\frac {12 b^2 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {6 b^2 \log (x)}{a^5}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 77, normalized size = 0.91 \[ \frac {\frac {a \left (-a^3+4 a^2 b \sqrt {x}+18 a b^2 x+12 b^3 x^{3/2}\right )}{x \left (a+b \sqrt {x}\right )^2}-12 b^2 \log \left (a+b \sqrt {x}\right )+6 b^2 \log (x)}{a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^3*x^2),x]

[Out]

((a*(-a^3 + 4*a^2*b*Sqrt[x] + 18*a*b^2*x + 12*b^3*x^(3/2)))/((a + b*Sqrt[x])^2*x) - 12*b^2*Log[a + b*Sqrt[x]]
+ 6*b^2*Log[x])/a^5

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fricas [B]  time = 0.80, size = 155, normalized size = 1.82 \[ -\frac {6 \, a^{2} b^{4} x^{2} - 9 \, a^{4} b^{2} x + a^{6} + 12 \, {\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (b \sqrt {x} + a\right ) - 12 \, {\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (6 \, a b^{5} x^{2} - 10 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt {x}}{a^{5} b^{4} x^{3} - 2 \, a^{7} b^{2} x^{2} + a^{9} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^(1/2))^3,x, algorithm="fricas")

[Out]

-(6*a^2*b^4*x^2 - 9*a^4*b^2*x + a^6 + 12*(b^6*x^3 - 2*a^2*b^4*x^2 + a^4*b^2*x)*log(b*sqrt(x) + a) - 12*(b^6*x^
3 - 2*a^2*b^4*x^2 + a^4*b^2*x)*log(sqrt(x)) - 2*(6*a*b^5*x^2 - 10*a^3*b^3*x + 3*a^5*b)*sqrt(x))/(a^5*b^4*x^3 -
2*a^7*b^2*x^2 + a^9*x)

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giac [A]  time = 0.16, size = 74, normalized size = 0.87 \[ -\frac {12 \, b^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{5}} + \frac {6 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, b^{3} x^{\frac {3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - a^{3}}{{\left (b x + a \sqrt {x}\right )}^{2} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^(1/2))^3,x, algorithm="giac")

[Out]

-12*b^2*log(abs(b*sqrt(x) + a))/a^5 + 6*b^2*log(abs(x))/a^5 + (12*b^3*x^(3/2) + 18*a*b^2*x + 4*a^2*b*sqrt(x) -
a^3)/((b*x + a*sqrt(x))^2*a^4)

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maple [A]  time = 0.01, size = 78, normalized size = 0.92 \[ \frac {b^{2}}{\left (b \sqrt {x}+a \right )^{2} a^{3}}+\frac {6 b^{2}}{\left (b \sqrt {x}+a \right ) a^{4}}+\frac {6 b^{2} \ln \relax (x )}{a^{5}}-\frac {12 b^{2} \ln \left (b \sqrt {x}+a \right )}{a^{5}}+\frac {6 b}{a^{4} \sqrt {x}}-\frac {1}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x^(1/2)+a)^3,x)

[Out]

-1/a^3/x+6*b^2*ln(x)/a^5-12*b^2*ln(b*x^(1/2)+a)/a^5+6*b/a^4/x^(1/2)+b^2/a^3/(b*x^(1/2)+a)^2+6*b^2/a^4/(b*x^(1/
2)+a)

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maxima [A]  time = 0.89, size = 85, normalized size = 1.00 \[ \frac {12 \, b^{3} x^{\frac {3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - a^{3}}{a^{4} b^{2} x^{2} + 2 \, a^{5} b x^{\frac {3}{2}} + a^{6} x} - \frac {12 \, b^{2} \log \left (b \sqrt {x} + a\right )}{a^{5}} + \frac {6 \, b^{2} \log \relax (x)}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^(1/2))^3,x, algorithm="maxima")

[Out]

(12*b^3*x^(3/2) + 18*a*b^2*x + 4*a^2*b*sqrt(x) - a^3)/(a^4*b^2*x^2 + 2*a^5*b*x^(3/2) + a^6*x) - 12*b^2*log(b*s
qrt(x) + a)/a^5 + 6*b^2*log(x)/a^5

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mupad [B]  time = 1.17, size = 79, normalized size = 0.93 \[ \frac {\frac {4\,b\,\sqrt {x}}{a^2}-\frac {1}{a}+\frac {18\,b^2\,x}{a^3}+\frac {12\,b^3\,x^{3/2}}{a^4}}{a^2\,x+b^2\,x^2+2\,a\,b\,x^{3/2}}-\frac {24\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^(1/2))^3),x)

[Out]

((4*b*x^(1/2))/a^2 - 1/a + (18*b^2*x)/a^3 + (12*b^3*x^(3/2))/a^4)/(a^2*x + b^2*x^2 + 2*a*b*x^(3/2)) - (24*b^2*
atanh((2*b*x^(1/2))/a + 1))/a^5

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sympy [A]  time = 4.98, size = 481, normalized size = 5.66 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {a^{4} \sqrt {x}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {4 a^{3} b x}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {6 a^{2} b^{2} x^{\frac {3}{2}} \log {\relax (x )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {12 a^{2} b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {18 a^{2} b^{2} x^{\frac {3}{2}}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {12 a b^{3} x^{2} \log {\relax (x )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {24 a b^{3} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {12 a b^{3} x^{2}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} + \frac {6 b^{4} x^{\frac {5}{2}} \log {\relax (x )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} - \frac {12 b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7} x^{\frac {3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*x**(1/2))**3,x)

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-1/(a**3*x), Eq(b, 0)), (-2/(5*b**3*x**(5/2)), Eq(a, 0)), (-a*
*4*sqrt(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 4*a**3*b*x/(a**7*x**(3/2) + 2*a**6*b*x**2 +
a**5*b**2*x**(5/2)) + 6*a**2*b**2*x**(3/2)*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 12*a*
*2*b**2*x**(3/2)*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 18*a**2*b**2*x**(3/
2)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 12*a*b**3*x**2*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2
+ a**5*b**2*x**(5/2)) - 24*a*b**3*x**2*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)
) + 12*a*b**3*x**2/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 6*b**4*x**(5/2)*log(x)/(a**7*x**(3/2
) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 12*b**4*x**(5/2)*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 +
a**5*b**2*x**(5/2)), True))

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