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

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

[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])

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Rubi [A]  time = 0.165205, antiderivative size = 114, normalized size of antiderivative = 1., 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} \[ \frac{\left (\sqrt{a}-\sqrt{b}\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{5/4} \sqrt{b}}-\frac{\left (\sqrt{a}+\sqrt{b}\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{5/4} \sqrt{b}}+\frac{x}{a} \]

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 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]

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]

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*}

Mathematica [A]  time = 0.0870461, size = 144, normalized size = 1.26 \[ \frac{\left (\sqrt{a}-\sqrt{b}\right )^2 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-\sqrt{a} \sqrt{b}}}\right )}{2 a \sqrt{b} \sqrt{a-\sqrt{a} \sqrt{b}}}-\frac{\left (\sqrt{a}+\sqrt{b}\right )^2 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{2 a \sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{x}{a} \]

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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Maple [B]  time = 0.17, size = 210, normalized size = 1.8 \begin{align*}{\frac{x}{a}}+{{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}-{\frac{a}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}-{\frac{b}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}-{\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}-{\frac{a}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}-{\frac{b}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

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]

Exception raised: AttributeError

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Fricas [B]  time = 1.60055, size = 1287, normalized size = 11.29 \begin{align*} \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{align*}

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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Sympy [A]  time = 1.09527, size = 105, normalized size = 0.92 \begin{align*} \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{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError