3.2092 \(\int \frac {1}{(a+\frac {b}{x^4})^{3/2} x} \, dx\)

Optimal. Leaf size=46 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {1}{2 a \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^4)^(3/2)*x),x]

[Out]

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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

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Mathematica [A]  time = 0.09, size = 69, normalized size = 1.50 \[ \frac {\sqrt {b} \sqrt {\frac {a x^4}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {b}}\right )-\sqrt {a} x^2}{2 a^{3/2} x^2 \sqrt {a+\frac {b}{x^4}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^4)^(3/2)*x),x]

[Out]

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

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fricas [B]  time = 0.82, size = 163, normalized size = 3.54 \[ \left [-\frac {2 \, a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - {\left (a x^{4} + b\right )} \sqrt {a} \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right )}{4 \, {\left (a^{3} x^{4} + a^{2} b\right )}}, -\frac {a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} + {\left (a x^{4} + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{2 \, {\left (a^{3} x^{4} + a^{2} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2)/x,x, algorithm="fricas")

[Out]

[-1/4*(2*a*x^4*sqrt((a*x^4 + b)/x^4) - (a*x^4 + b)*sqrt(a)*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4)
- b))/(a^3*x^4 + a^2*b), -1/2*(a*x^4*sqrt((a*x^4 + b)/x^4) + (a*x^4 + b)*sqrt(-a)*arctan(sqrt(-a)*x^4*sqrt((a*
x^4 + b)/x^4)/(a*x^4 + b)))/(a^3*x^4 + a^2*b)]

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giac [A]  time = 0.22, size = 51, normalized size = 1.11 \[ -\frac {x^{2}}{2 \, \sqrt {a x^{4} + b} a} - \frac {\log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{2 \, a^{\frac {3}{2}}} + \frac {\log \left ({\left | b \right |}\right )}{4 \, a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2)/x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 67, normalized size = 1.46 \[ -\frac {\left (a \,x^{4}+b \right ) \left (a^{\frac {3}{2}} x^{2}-\sqrt {a \,x^{4}+b}\, a \ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )\right )}{2 \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} a^{\frac {5}{2}} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^4)^(3/2)/x,x)

[Out]

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

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maxima [A]  time = 1.93, size = 52, normalized size = 1.13 \[ -\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{4 \, a^{\frac {3}{2}}} - \frac {1}{2 \, \sqrt {a + \frac {b}{x^{4}}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^4)^(3/2)/x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.36, size = 34, normalized size = 0.74 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {1}{2\,a\,\sqrt {a+\frac {b}{x^4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b/x^4)^(3/2)),x)

[Out]

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

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sympy [B]  time = 2.33, size = 187, normalized size = 4.07 \[ - \frac {2 a^{3} x^{4} \sqrt {1 + \frac {b}{a x^{4}}}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} - \frac {a^{3} x^{4} \log {\left (\frac {b}{a x^{4}} \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} + \frac {2 a^{3} x^{4} \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x^{4}} \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x^{4}}} + 1 \right )}}{4 a^{\frac {9}{2}} x^{4} + 4 a^{\frac {7}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**4)**(3/2)/x,x)

[Out]

-2*a**3*x**4*sqrt(1 + b/(a*x**4))/(4*a**(9/2)*x**4 + 4*a**(7/2)*b) - a**3*x**4*log(b/(a*x**4))/(4*a**(9/2)*x**
4 + 4*a**(7/2)*b) + 2*a**3*x**4*log(sqrt(1 + b/(a*x**4)) + 1)/(4*a**(9/2)*x**4 + 4*a**(7/2)*b) - a**2*b*log(b/
(a*x**4))/(4*a**(9/2)*x**4 + 4*a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x**4)) + 1)/(4*a**(9/2)*x**4 + 4*a**(7
/2)*b)

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