3.1.82 \(\int \frac {x}{(a+b x^4)^2} \, dx\) [82]

Optimal. Leaf size=48 \[ \frac {x^2}{4 a \left (a+b x^4\right )}+\frac {\arctan \left (\sqrt {\frac {b}{a}} x^2\right )}{4 a \sqrt {a b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {281, 205, 211} \begin {gather*} \frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}+\frac {x^2}{4 a \left (a+b x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^4)^2,x]

[Out]

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

Rule 205

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 281

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 {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{4 a \left (a+b x^4\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac {x^2}{4 a \left (a+b x^4\right )}+\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 49, normalized size = 1.02 \begin {gather*} \frac {x^2}{4 a \left (a+b x^4\right )}+\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^4)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 40, normalized size = 0.83

method	result	size
default	\(\frac {x^{2}}{4 a \left (b \,x^{4}+a \right )}+\frac {\arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{4 a \sqrt {a b}}\)	\(40\)
risch	\(\frac {x^{2}}{4 a \left (b \,x^{4}+a \right )}-\frac {\ln \left (x^{2} \sqrt {-a b}-a \right )}{8 \sqrt {-a b}\, a}+\frac {\ln \left (x^{2} \sqrt {-a b}+a \right )}{8 \sqrt {-a b}\, a}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.46, size = 39, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{4 \, {\left (a b x^{4} + a^{2}\right )}} + \frac {\arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.66, size = 129, normalized size = 2.69 \begin {gather*} \left [\frac {2 \, a b x^{2} - {\left (b x^{4} + a\right )} \sqrt {-a b} \log \left (\frac {b x^{4} - 2 \, \sqrt {-a b} x^{2} - a}{b x^{4} + a}\right )}{8 \, {\left (a^{2} b^{2} x^{4} + a^{3} b\right )}}, \frac {a b x^{2} - {\left (b x^{4} + a\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b x^{2}}\right )}{4 \, {\left (a^{2} b^{2} x^{4} + a^{3} b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*a*b*x^2 - (b*x^4 + a)*sqrt(-a*b)*log((b*x^4 - 2*sqrt(-a*b)*x^2 - a)/(b*x^4 + a)))/(a^2*b^2*x^4 + a^3*b
), 1/4*(a*b*x^2 - (b*x^4 + a)*sqrt(a*b)*arctan(sqrt(a*b)/(b*x^2)))/(a^2*b^2*x^4 + a^3*b)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (34) = 68\).
time = 0.14, size = 83, normalized size = 1.73 \begin {gather*} \frac {x^{2}}{4 a^{2} + 4 a b x^{4}} - \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x^{2} \right )}}{8} + \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x^{2} \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(4*a**2 + 4*a*b*x**4) - sqrt(-1/(a**3*b))*log(-a**2*sqrt(-1/(a**3*b)) + x**2)/8 + sqrt(-1/(a**3*b))*log(a
**2*sqrt(-1/(a**3*b)) + x**2)/8

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 39, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{4 \, {\left (b x^{4} + a\right )} a} + \frac {\arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 37, normalized size = 0.77 \begin {gather*} \frac {x^2}{4\,a\,\left (b\,x^4+a\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x^2}{\sqrt {a}}\right )}{4\,a^{3/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^4)^2,x)

[Out]

x^2/(4*a*(a + b*x^4)) + atan((b^(1/2)*x^2)/a^(1/2))/(4*a^(3/2)*b^(1/2))

________________________________________________________________________________________