3.2061 \(\int \frac{\sqrt{a+\frac{b}{x^4}}}{x^3} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\sqrt{a+\frac{b}{x^4}}}{4 x^2}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{4 \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0360257, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {335, 275, 195, 217, 206} \[ -\frac{\sqrt{a+\frac{b}{x^4}}}{4 x^2}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{4 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^4]/x^3,x]

[Out]

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

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0292741, size = 68, normalized size = 1.36 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (a x^4 \sqrt{\frac{a x^4}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{a x^4}{b}+1}\right )+a x^4+b\right )}{4 x^2 \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^4]/x^3,x]

[Out]

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

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Maple [B]  time = 0.009, size = 90, normalized size = 1.8 \begin{align*} -{\frac{1}{4\,{x}^{2}}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( -a\sqrt{a{x}^{4}+b}{x}^{4}\sqrt{b}+a\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{4}b+ \left ( a{x}^{4}+b \right ) ^{{\frac{3}{2}}}\sqrt{b} \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}{b}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48886, size = 301, normalized size = 6.02 \begin{align*} \left [\frac{a \sqrt{b} x^{2} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ) - 2 \, b \sqrt{\frac{a x^{4} + b}{x^{4}}}}{8 \, b x^{2}}, \frac{a \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{b}\right ) - b \sqrt{\frac{a x^{4} + b}{x^{4}}}}{4 \, b x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(a*sqrt(b)*x^2*log((a*x^4 - 2*sqrt(b)*x^2*sqrt((a*x^4 + b)/x^4) + 2*b)/x^4) - 2*b*sqrt((a*x^4 + b)/x^4))/
(b*x^2), 1/4*(a*sqrt(-b)*x^2*arctan(sqrt(-b)*x^2*sqrt((a*x^4 + b)/x^4)/b) - b*sqrt((a*x^4 + b)/x^4))/(b*x^2)]

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Sympy [A]  time = 3.06728, size = 46, normalized size = 0.92 \begin{align*} - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{4}}}}{4 x^{2}} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{4 \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a)*sqrt(1 + b/(a*x**4))/(4*x**2) - a*asinh(sqrt(b)/(sqrt(a)*x**2))/(4*sqrt(b))

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Giac [A]  time = 1.11637, size = 58, normalized size = 1.16 \begin{align*} \frac{1}{4} \, a{\left (\frac{\arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{a x^{4} + b}}{a x^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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