3.2068 \(\int \frac{(a+\frac{b}{x^4})^{3/2}}{x} \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0340354, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^4)^(3/2)/x,x]

[Out]

-(a*Sqrt[a + b/x^4])/2 - (a + b/x^4)^(3/2)/6 + (a^(3/2)*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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{\left (a+\frac{b}{x^4}\right )^{3/2}}{x} \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^4}}\right )}{2 b}\\ &=-\frac{1}{2} a \sqrt{a+\frac{b}{x^4}}-\frac{1}{6} \left (a+\frac{b}{x^4}\right )^{3/2}+\frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0173859, size = 52, normalized size = 0.88 \[ -\frac{b \sqrt{a+\frac{b}{x^4}} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{a x^4}{b}\right )}{6 x^4 \sqrt{\frac{a x^4}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^4)^(3/2)/x,x]

[Out]

-(b*Sqrt[a + b/x^4]*Hypergeometric2F1[-3/2, -3/2, -1/2, -((a*x^4)/b)])/(6*x^4*Sqrt[1 + (a*x^4)/b])

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Maple [A]  time = 0.015, size = 79, normalized size = 1.3 \begin{align*}{\frac{1}{6} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{3/2}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{6}-4\,a{x}^{4}\sqrt{a{x}^{4}+b}-b\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56579, size = 335, normalized size = 5.68 \begin{align*} \left [\frac{3 \, a^{\frac{3}{2}} x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) - 2 \,{\left (4 \, a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{12 \, x^{4}}, -\frac{3 \, \sqrt{-a} a x^{4} \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right ) +{\left (4 \, a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{6 \, x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*a^(3/2)*x^4*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b) - 2*(4*a*x^4 + b)*sqrt((a*x^4 + b
)/x^4))/x^4, -1/6*(3*sqrt(-a)*a*x^4*arctan(sqrt(-a)*x^4*sqrt((a*x^4 + b)/x^4)/(a*x^4 + b)) + (4*a*x^4 + b)*sqr
t((a*x^4 + b)/x^4))/x^4]

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Sympy [A]  time = 3.3766, size = 80, normalized size = 1.36 \begin{align*} - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{3} - \frac{a^{\frac{3}{2}} \log{\left (\frac{b}{a x^{4}} \right )}}{4} + \frac{a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{2} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{4}}}}{6 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a**(3/2)*sqrt(1 + b/(a*x**4))/3 - a**(3/2)*log(b/(a*x**4))/4 + a**(3/2)*log(sqrt(1 + b/(a*x**4)) + 1)/2 - s
qrt(a)*b*sqrt(1 + b/(a*x**4))/(6*x**4)

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Giac [A]  time = 1.12247, size = 68, normalized size = 1.15 \begin{align*} -\frac{a^{2} \arctan \left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} - \frac{1}{6} \,{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{a + \frac{b}{x^{4}}} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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