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

________________________________________________________________________________________

Rubi [A]  time = 0.0252541, antiderivative size = 46, normalized size of antiderivative = 1., 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 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 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]

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*}

Mathematica [A]  time = 0.0572175, size = 69, normalized size = 1.5 \[ \frac{\sqrt{b} \sqrt{\frac{a x^4}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{b}}\right )-\sqrt{a} x^2}{2 a^{3/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]

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)

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 67, normalized size = 1.5 \begin{align*} -{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ({x}^{2}{a}^{{\frac{3}{2}}}-\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) a\sqrt{a{x}^{4}+b} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \end{align*}

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(x^2*a^(1/2)+(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)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.50679, size = 365, normalized size = 7.93 \begin{align*} \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 ] \end{align*}

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)]

________________________________________________________________________________________

Sympy [B]  time = 2.99204, size = 187, normalized size = 4.07 \begin{align*} - \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} \end{align*}

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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a + b/x^4)^(3/2)*x), x)