3.2197 \(\int \frac {1}{(a+b \sqrt {x}) x^3} \, dx\)

Optimal. Leaf size=75 \[ -\frac {2 b^4 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {b^4 \log (x)}{a^5}+\frac {2 b^3}{a^4 \sqrt {x}}-\frac {b^2}{a^3 x}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {1}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 75, 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} \[ \frac {2 b^3}{a^4 \sqrt {x}}-\frac {b^2}{a^3 x}-\frac {2 b^4 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {b^4 \log (x)}{a^5}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])*x^3),x]

[Out]

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

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Mathematica [A]  time = 0.07, size = 69, normalized size = 0.92 \[ \frac {\frac {a \left (-3 a^3+4 a^2 b \sqrt {x}-6 a b^2 x+12 b^3 x^{3/2}\right )}{x^2}-12 b^4 \log \left (a+b \sqrt {x}\right )+6 b^4 \log (x)}{6 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])*x^3),x]

[Out]

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

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fricas [A]  time = 0.75, size = 69, normalized size = 0.92 \[ -\frac {12 \, b^{4} x^{2} \log \left (b \sqrt {x} + a\right ) - 12 \, b^{4} x^{2} \log \left (\sqrt {x}\right ) + 6 \, a^{2} b^{2} x + 3 \, a^{4} - 4 \, {\left (3 \, a b^{3} x + a^{3} b\right )} \sqrt {x}}{6 \, a^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(12*b^4*x^2*log(b*sqrt(x) + a) - 12*b^4*x^2*log(sqrt(x)) + 6*a^2*b^2*x + 3*a^4 - 4*(3*a*b^3*x + a^3*b)*sq
rt(x))/(a^5*x^2)

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giac [A]  time = 0.16, size = 69, normalized size = 0.92 \[ -\frac {2 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{5}} + \frac {b^{4} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, a b^{3} x^{\frac {3}{2}} - 6 \, a^{2} b^{2} x + 4 \, a^{3} b \sqrt {x} - 3 \, a^{4}}{6 \, a^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 66, normalized size = 0.88 \[ \frac {b^{4} \ln \relax (x )}{a^{5}}-\frac {2 b^{4} \ln \left (b \sqrt {x}+a \right )}{a^{5}}+\frac {2 b^{3}}{a^{4} \sqrt {x}}-\frac {b^{2}}{a^{3} x}+\frac {2 b}{3 a^{2} x^{\frac {3}{2}}}-\frac {1}{2 a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.85, size = 64, normalized size = 0.85 \[ -\frac {2 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{5}} + \frac {b^{4} \log \relax (x)}{a^{5}} + \frac {12 \, b^{3} x^{\frac {3}{2}} - 6 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - 3 \, a^{3}}{6 \, a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 60, normalized size = 0.80 \[ -\frac {\frac {1}{2\,a}-\frac {2\,b\,\sqrt {x}}{3\,a^2}+\frac {b^2\,x}{a^3}-\frac {2\,b^3\,x^{3/2}}{a^4}}{x^2}-\frac {4\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^(1/2))),x)

[Out]

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

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sympy [A]  time = 4.53, size = 99, normalized size = 1.32 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {1}{2 a x^{2}} & \text {for}\: b = 0 \\- \frac {1}{2 a x^{2}} + \frac {2 b}{3 a^{2} x^{\frac {3}{2}}} - \frac {b^{2}}{a^{3} x} + \frac {2 b^{3}}{a^{4} \sqrt {x}} + \frac {b^{4} \log {\relax (x )}}{a^{5}} - \frac {2 b^{4} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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