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

Optimal. Leaf size=55 \[ -\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}-\frac {x^3}{2 b \left (a+b x^2\right )}+\frac {3 x}{2 b^2} \]

[Out]

3/2*x/b^2-1/2*x^3/b/(b*x^2+a)-3/2*arctan(x*b^(1/2)/a^(1/2))*a^(1/2)/b^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 288, 321, 205} \[ -\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}-\frac {x^3}{2 b \left (a+b x^2\right )}+\frac {3 x}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(3*x)/(2*b^2) - x^3/(2*b*(a + b*x^2)) - (3*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(5/2))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 205

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^4}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {x^4}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {x^3}{2 b \left (a+b x^2\right )}+\frac {3}{2} \int \frac {x^2}{a b+b^2 x^2} \, dx\\ &=\frac {3 x}{2 b^2}-\frac {x^3}{2 b \left (a+b x^2\right )}-\frac {(3 a) \int \frac {1}{a b+b^2 x^2} \, dx}{2 b}\\ &=\frac {3 x}{2 b^2}-\frac {x^3}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 51, normalized size = 0.93 \[ -\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}+\frac {a x}{2 b^2 \left (a+b x^2\right )}+\frac {x}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

x/b^2 + (a*x)/(2*b^2*(a + b*x^2)) - (3*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 136, normalized size = 2.47 \[ \left [\frac {4 \, b x^{3} + 3 \, {\left (b x^{2} + a\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, a x}{4 \, {\left (b^{3} x^{2} + a b^{2}\right )}}, \frac {2 \, b x^{3} - 3 \, {\left (b x^{2} + a\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3 \, a x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 42, normalized size = 0.76 \[ -\frac {3 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {a x}{2 \, {\left (b x^{2} + a\right )} b^{2}} + \frac {x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 43, normalized size = 0.78 \[ \frac {a x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}+\frac {x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 3.08, size = 45, normalized size = 0.82 \[ \frac {a x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} - \frac {3 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.29, size = 43, normalized size = 0.78 \[ \frac {x}{b^2}+\frac {a\,x}{2\,\left (b^3\,x^2+a\,b^2\right )}-\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b^2 + (a*x)/(2*(a*b^2 + b^3*x^2)) - (3*a^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(2*b^(5/2))

________________________________________________________________________________________

sympy [A]  time = 0.27, size = 83, normalized size = 1.51 \[ \frac {a x}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {3 \sqrt {- \frac {a}{b^{5}}} \log {\left (- b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{4} - \frac {3 \sqrt {- \frac {a}{b^{5}}} \log {\left (b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{4} + \frac {x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/(2*a*b**2 + 2*b**3*x**2) + 3*sqrt(-a/b**5)*log(-b**2*sqrt(-a/b**5) + x)/4 - 3*sqrt(-a/b**5)*log(b**2*sqrt(
-a/b**5) + x)/4 + x/b**2

________________________________________________________________________________________