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

Optimal. Leaf size=114 \[ \frac {\left (\sqrt {a}-\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{5/4} \sqrt {b}}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{5/4} \sqrt {b}}+\frac {x}{a} \]

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Rubi [A]  time = 0.17, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1122, 1166, 205} \begin {gather*} \frac {\left (\sqrt {a}-\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{5/4} \sqrt {b}}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{5/4} \sqrt {b}}+\frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a - b + 2*a*x^2 + a*x^4),x]

[Out]

x/a + ((Sqrt[a] - Sqrt[b])^(3/2)*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(5/4)*Sqrt[b]) - ((Sqrt[a]
+ Sqrt[b])^(3/2)*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(5/4)*Sqrt[b])

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 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 144, normalized size = 1.26 \begin {gather*} \frac {\left (\sqrt {a}-\sqrt {b}\right )^2 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a-\sqrt {a} \sqrt {b}}}\right )}{2 a \sqrt {b} \sqrt {a-\sqrt {a} \sqrt {b}}}-\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{2 a \sqrt {b} \sqrt {\sqrt {a} \sqrt {b}+a}}+\frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/a + ((Sqrt[a] - Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a - Sqrt[a]*Sqrt[b]]])/(2*a*Sqrt[a - Sqrt[a]*Sqrt[b]]*Sqr
t[b]) - ((Sqrt[a] + Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(2*a*Sqrt[a + Sqrt[a]*Sqrt[b]]*S
qrt[b])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{a-b+2 a x^2+a x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.85, size = 603, normalized size = 5.29 \begin {gather*} \frac {a \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b - a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}}\right ) - a \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b - a b^{2}\right )} \sqrt {-\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}}\right ) - a \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x + {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b + a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}}\right ) + a \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x - {\left (a^{4} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b + a b^{2}\right )} \sqrt {\frac {a^{2} b \sqrt {\frac {9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}}\right ) + 4 \, x}{4 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + a + 3*b)/(a^2*b))*log(-(3*a^2 - 2*a*b - b^2)*x + (a^
4*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) - 3*a^2*b - a*b^2)*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) +
a + 3*b)/(a^2*b))) - a*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + a + 3*b)/(a^2*b))*log(-(3*a^2 - 2*a*
b - b^2)*x - (a^4*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) - 3*a^2*b - a*b^2)*sqrt(-(a^2*b*sqrt((9*a^2 + 6*a*b +
b^2)/(a^5*b)) + a + 3*b)/(a^2*b))) - a*sqrt((a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) - a - 3*b)/(a^2*b))*log
(-(3*a^2 - 2*a*b - b^2)*x + (a^4*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + 3*a^2*b + a*b^2)*sqrt((a^2*b*sqrt((9*
a^2 + 6*a*b + b^2)/(a^5*b)) - a - 3*b)/(a^2*b))) + a*sqrt((a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) - a - 3*b
)/(a^2*b))*log(-(3*a^2 - 2*a*b - b^2)*x - (a^4*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) + 3*a^2*b + a*b^2)*sqrt((
a^2*b*sqrt((9*a^2 + 6*a*b + b^2)/(a^5*b)) - a - 3*b)/(a^2*b))) + 4*x)/a

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giac [B]  time = 0.36, size = 511, normalized size = 4.48 \begin {gather*} -\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{4} - \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{3} b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{2} b^{2} - 2 \, {\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} b^{2}\right )} a^{2} + {\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} a^{3} b - 7 \, \sqrt {a^{2} + \sqrt {a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} + \sqrt {a b} a} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} + \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b - 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{4} - \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{3} b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{2} b^{2} - 2 \, {\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} b^{2}\right )} a^{2} - {\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} a^{3} b - 7 \, \sqrt {a^{2} - \sqrt {a b} a} a^{2} b^{2} + 4 \, \sqrt {a^{2} - \sqrt {a b} a} a b^{3}\right )} {\left | a \right |}\right )} \arctan \left (\frac {x}{\sqrt {\frac {a^{2} - \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2}}}}\right )}{2 \, {\left (3 \, a^{6} b - 7 \, a^{5} b^{2} + 4 \, a^{4} b^{3}\right )}} + \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^4 - sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^3*b - 4*sqrt(a^2 + sqrt(a*
b)*a)*sqrt(a*b)*a^2*b^2 - 2*(3*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a*b - 4*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*b^2
)*a^2 + (3*sqrt(a^2 + sqrt(a*b)*a)*a^3*b - 7*sqrt(a^2 + sqrt(a*b)*a)*a^2*b^2 + 4*sqrt(a^2 + sqrt(a*b)*a)*a*b^3
)*abs(a))*arctan(x/sqrt((a^2 + sqrt(a^4 - (a^2 - a*b)*a^2))/a^2))/(3*a^6*b - 7*a^5*b^2 + 4*a^4*b^3) + 1/2*(3*s
qrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a^4 - sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a^3*b - 4*sqrt(a^2 - sqrt(a*b)*a)*sqr
t(a*b)*a^2*b^2 - 2*(3*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*a*b - 4*sqrt(a^2 - sqrt(a*b)*a)*sqrt(a*b)*b^2)*a^2 - (
3*sqrt(a^2 - sqrt(a*b)*a)*a^3*b - 7*sqrt(a^2 - sqrt(a*b)*a)*a^2*b^2 + 4*sqrt(a^2 - sqrt(a*b)*a)*a*b^3)*abs(a))
*arctan(x/sqrt((a^2 - sqrt(a^4 - (a^2 - a*b)*a^2))/a^2))/(3*a^6*b - 7*a^5*b^2 + 4*a^4*b^3) + x/a

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maple [B]  time = 0.03, size = 210, normalized size = 1.84 \begin {gather*} -\frac {a \arctanh \left (\frac {a x}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (-a +\sqrt {a b}\right ) a}}-\frac {a \arctan \left (\frac {a x}{\sqrt {\left (a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (a +\sqrt {a b}\right ) a}}-\frac {b \arctanh \left (\frac {a x}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (-a +\sqrt {a b}\right ) a}}-\frac {b \arctan \left (\frac {a x}{\sqrt {\left (a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (a +\sqrt {a b}\right ) a}}+\frac {\arctanh \left (\frac {a x}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}\right )}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}-\frac {\arctan \left (\frac {a x}{\sqrt {\left (a +\sqrt {a b}\right ) a}}\right )}{\sqrt {\left (a +\sqrt {a b}\right ) a}}+\frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x}{a} - \frac {\frac {{\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} a^{2} b - 4 \, \sqrt {a^{2} + \sqrt {a b} a} a b^{2} + 3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a^{2} - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a b\right )} a \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b - 4 \, a^{3} b^{2}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} a^{2} b - 4 \, \sqrt {a^{2} - \sqrt {a b} a} a b^{2} - 3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a^{2} + 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a b\right )} a \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b - 4 \, a^{3} b^{2}\right )}}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.79, size = 1097, normalized size = 9.62 \begin {gather*} \frac {x}{a}-2\,\mathrm {atanh}\left (\frac {24\,x\,\sqrt {a^5\,b^3}\,\sqrt {-\frac {3}{16\,a^2}-\frac {1}{16\,a\,b}-\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b^2-\frac {6\,\sqrt {a^5\,b^3}}{a}-6\,a^2\,b+2\,b^3+\frac {2\,b^2\,\sqrt {a^5\,b^3}}{a^3}+\frac {4\,b\,\sqrt {a^5\,b^3}}{a^2}}+\frac {8\,x\,\sqrt {a^5\,b^3}\,\sqrt {-\frac {3}{16\,a^2}-\frac {1}{16\,a\,b}-\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{\frac {4\,\sqrt {a^5\,b^3}}{a}-\frac {6\,\sqrt {a^5\,b^3}}{b}+2\,a\,b^2+4\,a^2\,b-6\,a^3+\frac {2\,b\,\sqrt {a^5\,b^3}}{a^2}}-\frac {8\,a\,b^2\,x\,\sqrt {-\frac {3}{16\,a^2}-\frac {1}{16\,a\,b}-\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b+\frac {4\,\sqrt {a^5\,b^3}}{a^2}-6\,a^2+2\,b^2-\frac {6\,\sqrt {a^5\,b^3}}{a\,b}+\frac {2\,b\,\sqrt {a^5\,b^3}}{a^3}}-\frac {24\,a^2\,b\,x\,\sqrt {-\frac {3}{16\,a^2}-\frac {1}{16\,a\,b}-\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b+\frac {4\,\sqrt {a^5\,b^3}}{a^2}-6\,a^2+2\,b^2-\frac {6\,\sqrt {a^5\,b^3}}{a\,b}+\frac {2\,b\,\sqrt {a^5\,b^3}}{a^3}}\right )\,\sqrt {-\frac {3\,a\,\sqrt {a^5\,b^3}+b\,\sqrt {a^5\,b^3}+a^4\,b+3\,a^3\,b^2}{16\,a^5\,b^2}}+2\,\mathrm {atanh}\left (\frac {24\,x\,\sqrt {a^5\,b^3}\,\sqrt {\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{\frac {6\,\sqrt {a^5\,b^3}}{a}+4\,a\,b^2-6\,a^2\,b+2\,b^3-\frac {2\,b^2\,\sqrt {a^5\,b^3}}{a^3}-\frac {4\,b\,\sqrt {a^5\,b^3}}{a^2}}-\frac {8\,x\,\sqrt {a^5\,b^3}\,\sqrt {\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{\frac {4\,\sqrt {a^5\,b^3}}{a}-\frac {6\,\sqrt {a^5\,b^3}}{b}-2\,a\,b^2-4\,a^2\,b+6\,a^3+\frac {2\,b\,\sqrt {a^5\,b^3}}{a^2}}+\frac {8\,a\,b^2\,x\,\sqrt {\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b-\frac {4\,\sqrt {a^5\,b^3}}{a^2}-6\,a^2+2\,b^2+\frac {6\,\sqrt {a^5\,b^3}}{a\,b}-\frac {2\,b\,\sqrt {a^5\,b^3}}{a^3}}+\frac {24\,a^2\,b\,x\,\sqrt {\frac {3\,\sqrt {a^5\,b^3}}{16\,a^4\,b^2}-\frac {1}{16\,a\,b}-\frac {3}{16\,a^2}+\frac {\sqrt {a^5\,b^3}}{16\,a^5\,b}}}{4\,a\,b-\frac {4\,\sqrt {a^5\,b^3}}{a^2}-6\,a^2+2\,b^2+\frac {6\,\sqrt {a^5\,b^3}}{a\,b}-\frac {2\,b\,\sqrt {a^5\,b^3}}{a^3}}\right )\,\sqrt {\frac {3\,a\,\sqrt {a^5\,b^3}+b\,\sqrt {a^5\,b^3}-a^4\,b-3\,a^3\,b^2}{16\,a^5\,b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a - b + 2*a*x^2 + a*x^4),x)

[Out]

x/a - 2*atanh((24*x*(a^5*b^3)^(1/2)*(- 3/(16*a^2) - 1/(16*a*b) - (3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - (a^5*b^3)^
(1/2)/(16*a^5*b))^(1/2))/(4*a*b^2 - (6*(a^5*b^3)^(1/2))/a - 6*a^2*b + 2*b^3 + (2*b^2*(a^5*b^3)^(1/2))/a^3 + (4
*b*(a^5*b^3)^(1/2))/a^2) + (8*x*(a^5*b^3)^(1/2)*(- 3/(16*a^2) - 1/(16*a*b) - (3*(a^5*b^3)^(1/2))/(16*a^4*b^2)
- (a^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((4*(a^5*b^3)^(1/2))/a - (6*(a^5*b^3)^(1/2))/b + 2*a*b^2 + 4*a^2*b - 6*a^
3 + (2*b*(a^5*b^3)^(1/2))/a^2) - (8*a*b^2*x*(- 3/(16*a^2) - 1/(16*a*b) - (3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - (a
^5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b + (4*(a^5*b^3)^(1/2))/a^2 - 6*a^2 + 2*b^2 - (6*(a^5*b^3)^(1/2))/(a*b)
+ (2*b*(a^5*b^3)^(1/2))/a^3) - (24*a^2*b*x*(- 3/(16*a^2) - 1/(16*a*b) - (3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - (a^
5*b^3)^(1/2)/(16*a^5*b))^(1/2))/(4*a*b + (4*(a^5*b^3)^(1/2))/a^2 - 6*a^2 + 2*b^2 - (6*(a^5*b^3)^(1/2))/(a*b) +
(2*b*(a^5*b^3)^(1/2))/a^3))*(-(3*a*(a^5*b^3)^(1/2) + b*(a^5*b^3)^(1/2) + a^4*b + 3*a^3*b^2)/(16*a^5*b^2))^(1/
2) + 2*atanh((24*x*(a^5*b^3)^(1/2)*((3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - 1/(16*a*b) - 3/(16*a^2) + (a^5*b^3)^(1/
2)/(16*a^5*b))^(1/2))/((6*(a^5*b^3)^(1/2))/a + 4*a*b^2 - 6*a^2*b + 2*b^3 - (2*b^2*(a^5*b^3)^(1/2))/a^3 - (4*b*
(a^5*b^3)^(1/2))/a^2) - (8*x*(a^5*b^3)^(1/2)*((3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - 1/(16*a*b) - 3/(16*a^2) + (a^
5*b^3)^(1/2)/(16*a^5*b))^(1/2))/((4*(a^5*b^3)^(1/2))/a - (6*(a^5*b^3)^(1/2))/b - 2*a*b^2 - 4*a^2*b + 6*a^3 + (
2*b*(a^5*b^3)^(1/2))/a^2) + (8*a*b^2*x*((3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - 1/(16*a*b) - 3/(16*a^2) + (a^5*b^3)
^(1/2)/(16*a^5*b))^(1/2))/(4*a*b - (4*(a^5*b^3)^(1/2))/a^2 - 6*a^2 + 2*b^2 + (6*(a^5*b^3)^(1/2))/(a*b) - (2*b*
(a^5*b^3)^(1/2))/a^3) + (24*a^2*b*x*((3*(a^5*b^3)^(1/2))/(16*a^4*b^2) - 1/(16*a*b) - 3/(16*a^2) + (a^5*b^3)^(1
/2)/(16*a^5*b))^(1/2))/(4*a*b - (4*(a^5*b^3)^(1/2))/a^2 - 6*a^2 + 2*b^2 + (6*(a^5*b^3)^(1/2))/(a*b) - (2*b*(a^
5*b^3)^(1/2))/a^3))*((3*a*(a^5*b^3)^(1/2) + b*(a^5*b^3)^(1/2) - a^4*b - 3*a^3*b^2)/(16*a^5*b^2))^(1/2)

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sympy [A]  time = 2.02, size = 105, normalized size = 0.92 \begin {gather*} \operatorname {RootSum} {\left (256 t^{4} a^{5} b^{2} + t^{2} \left (32 a^{4} b + 96 a^{3} b^{2}\right ) + a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}, \left (t \mapsto t \log {\left (x + \frac {64 t^{3} a^{4} b + 4 t a^{3} + 24 t a^{2} b + 4 t a b^{2}}{3 a^{2} - 2 a b - b^{2}} \right )} \right )\right )} + \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a**5*b**2 + _t**2*(32*a**4*b + 96*a**3*b**2) + a**3 - 3*a**2*b + 3*a*b**2 - b**3, Lambda(_t,
_t*log(x + (64*_t**3*a**4*b + 4*_t*a**3 + 24*_t*a**2*b + 4*_t*a*b**2)/(3*a**2 - 2*a*b - b**2)))) + x/a

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