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

Optimal. Leaf size=30 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 \sqrt {b}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 217, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b*x^4],x]

[Out]

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

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])

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{2 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + b*x^4],x]

[Out]

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

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fricas [A]  time = 0.84, size = 63, normalized size = 2.10 \[ \left [\frac {\log \left (-2 \, b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right )}{4 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{2}}{\sqrt {b x^{4} + a}}\right )}{2 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 25, normalized size = 0.83 \[ -\frac {\log \left ({\left | -\sqrt {b} x^{2} + \sqrt {b x^{4} + a} \right |}\right )}{2 \, \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 24, normalized size = 0.80 \[ \frac {\ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 3.03, size = 45, normalized size = 1.50 \[ -\frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{4} + a}}{x^{2}}}{\sqrt {b} + \frac {\sqrt {b x^{4} + a}}{x^{2}}}\right )}{4 \, \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{\sqrt {b\,x^4+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^4)^(1/2),x)

[Out]

int(x/(a + b*x^4)^(1/2), x)

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sympy [A]  time = 1.94, size = 20, normalized size = 0.67 \[ \frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b))

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