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

Optimal. Leaf size=109 \[ \frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {b}}-\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {b}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1130, 205} \[ \frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {b}}-\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a - b + 2*a*x^2 + a*x^4),x]

[Out]

-(Sqrt[Sqrt[a] - Sqrt[b]]*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^(3/4)*Sqrt[b]) + (Sqrt[Sqrt[a] + S
qrt[b]]*ArcTan[(a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a^(3/4)*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 1130

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 128, normalized size = 1.17 \[ \frac {\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a-\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a-\sqrt {a} \sqrt {b}}}}{2 \sqrt {a} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.83, size = 267, normalized size = 2.45 \[ \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \log \left (a^{2} b \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \log \left (-a^{2} b \sqrt {-\frac {a b \sqrt {\frac {1}{a^{3} b}} + 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \log \left (a^{2} b \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) + \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \log \left (-a^{2} b \sqrt {\frac {a b \sqrt {\frac {1}{a^{3} b}} - 1}{a b}} \sqrt {\frac {1}{a^{3} b}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.36, size = 199, normalized size = 1.83 \[ \frac {{\left (3 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} a - 4 \, \sqrt {a^{2} + \sqrt {a b} a} \sqrt {a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b - 4 \, a^{3} b^{2}\right )}} - \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} a - 4 \, \sqrt {a^{2} - \sqrt {a b} a} \sqrt {a b} b\right )} {\left | a \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a - b\right )} a + 4 \, a^{2}}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b - 4 \, a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(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)*b)*abs(a)*arctan(2*sqrt(1/2)*
x/sqrt((2*a + sqrt(-4*(a - b)*a + 4*a^2))/a))/(3*a^4*b - 4*a^3*b^2) - 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)*b)*abs(a)*arctan(2*sqrt(1/2)*x/sqrt((2*a - sqrt(-4*(a - b)*a + 4*a^2)
)/a))/(3*a^4*b - 4*a^3*b^2)

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maple [A]  time = 0.01, size = 134, normalized size = 1.23 \[ \frac {a \arctanh \left (\frac {a x}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (-a +\sqrt {a b}\right ) a}}+\frac {a \arctan \left (\frac {a x}{\sqrt {\left (a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {a b}\, \sqrt {\left (a +\sqrt {a b}\right ) a}}-\frac {\arctanh \left (\frac {a x}{\sqrt {\left (-a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {\left (-a +\sqrt {a b}\right ) a}}+\frac {\arctan \left (\frac {a x}{\sqrt {\left (a +\sqrt {a b}\right ) a}}\right )}{2 \sqrt {\left (a +\sqrt {a b}\right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{a x^{4} + 2 \, a x^{2} + a - b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.30, size = 216, normalized size = 1.98 \[ -2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^3+4\,b\,a^2\right )-\frac {4\,a\,x\,\left (\sqrt {a^3\,b^3}+a^2\,b\right )}{b}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}}{2\,a\,b-2\,a^2}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^3}+a^2\,b}{16\,a^3\,b^2}}-2\,\mathrm {atanh}\left (\frac {2\,\left (x\,\left (4\,a^3+4\,b\,a^2\right )+\frac {4\,a\,x\,\left (\sqrt {a^3\,b^3}-a^2\,b\right )}{b}\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}}}{2\,a\,b-2\,a^2}\right )\,\sqrt {\frac {\sqrt {a^3\,b^3}-a^2\,b}{16\,a^3\,b^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a - b + 2*a*x^2 + a*x^4),x)

[Out]

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

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sympy [A]  time = 0.60, size = 44, normalized size = 0.40 \[ \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{2} + 32 t^{2} a^{2} b + a - b, \left (t \mapsto t \log {\left (- 64 t^{3} a^{2} b - 4 t a + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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