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

Optimal. Leaf size=31 \[ \frac {\log \left (\sqrt {a x^4+b}+\sqrt {a} x^2\right )}{2 \sqrt {a}} \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 0.97, 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} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[b + a*x^4],x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.97 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4+b}}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[b + a*x^4],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.12, size = 31, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[b + a*x^4],x]

[Out]

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

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fricas [A]  time = 0.50, size = 63, normalized size = 2.03 \begin {gather*} \left [\frac {\log \left (-2 \, a x^{4} - 2 \, \sqrt {a x^{4} + b} \sqrt {a} x^{2} - b\right )}{4 \, \sqrt {a}}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a} x^{2}}{\sqrt {a x^{4} + b}}\right )}{2 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.51, size = 25, normalized size = 0.81 \begin {gather*} -\frac {\log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{2 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 24, normalized size = 0.77

method	result	size
default	\(\frac {\ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )}{2 \sqrt {a}}\)	\(24\)
elliptic	\(\frac {\ln \left (\sqrt {a}\, x^{2}+\sqrt {a \,x^{4}+b}\right )}{2 \sqrt {a}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 45, normalized size = 1.45 \begin {gather*} -\frac {\log \left (-\frac {\sqrt {a} - \frac {\sqrt {a x^{4} + b}}{x^{2}}}{\sqrt {a} + \frac {\sqrt {a x^{4} + b}}{x^{2}}}\right )}{4 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b + a*x^4)^(1/2),x)

[Out]

int(x/(b + a*x^4)^(1/2), x)

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sympy [A]  time = 0.86, size = 20, normalized size = 0.65 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{2 \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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