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 = 0.337668, 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 \[ \frac{\left (\sqrt{a}-\sqrt{b}\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^{5/4} \sqrt{b}}-\frac{\left (\sqrt{a}+\sqrt{b}\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^{5/4} \sqrt{b}}+\frac{x}{a} \]

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])

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Rubi in Sympy [A]  time = 43.3603, size = 128, normalized size = 1.12 \[ \frac{x}{a} - \frac{\left (2 \sqrt{a} \sqrt{b} + a + b\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a} + \sqrt{b}}} \right )}}{2 a^{\frac{5}{4}} \sqrt{b} \sqrt{\sqrt{a} + \sqrt{b}}} + \frac{\left (- 2 \sqrt{a} \sqrt{b} + a + b\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt{\sqrt{a} - \sqrt{b}}} \right )}}{2 a^{\frac{5}{4}} \sqrt{b} \sqrt{\sqrt{a} - \sqrt{b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a*x**4+2*a*x**2+a-b),x)

[Out]

x/a - (2*sqrt(a)*sqrt(b) + a + b)*atan(a**(1/4)*x/sqrt(sqrt(a) + sqrt(b)))/(2*a*
*(5/4)*sqrt(b)*sqrt(sqrt(a) + sqrt(b))) + (-2*sqrt(a)*sqrt(b) + a + b)*atan(a**(
1/4)*x/sqrt(sqrt(a) - sqrt(b)))/(2*a**(5/4)*sqrt(b)*sqrt(sqrt(a) - sqrt(b)))

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Mathematica [A]  time = 0.150328, 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]]*Sqrt[b]) - ((Sqrt[a] + Sqrt[b])^2*ArcTan[(Sqrt[a]*x)/
Sqrt[a + Sqrt[a]*Sqrt[b]]])/(2*a*Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])

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Maple [B]  time = 0.04, size = 210, normalized size = 1.8 \[{\frac{x}{a}}-{1\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}-{\frac{a}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}-{\frac{b}{2}\arctan \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}}+{1{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}-{\frac{a}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}-{\frac{b}{2}{\it Artanh} \left ({ax{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \]

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

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

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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Fricas [A]  time = 0.285181, size = 814, normalized size = 7.14 \[ \frac{a \sqrt{-\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x +{\left (a^{4} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b - a b^{2}\right )} \sqrt{-\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}}\right ) - a \sqrt{-\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x -{\left (a^{4} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - 3 \, a^{2} b - a b^{2}\right )} \sqrt{-\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + a + 3 \, b}{a^{2} b}}\right ) - a \sqrt{\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x +{\left (a^{4} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b + a b^{2}\right )} \sqrt{\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}}\right ) + a \sqrt{\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}} \log \left (-{\left (3 \, a^{2} - 2 \, a b - b^{2}\right )} x -{\left (a^{4} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} + 3 \, a^{2} b + a b^{2}\right )} \sqrt{\frac{a^{2} b \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{5} b}} - a - 3 \, b}{a^{2} b}}\right ) + 4 \, x}{4 \, a} \]

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*s
qrt((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 = 3.7333, size = 105, normalized size = 0.92 \[ \operatorname{RootSum}{\left (256 t^{4} a^{5} b^{2} + t^{2} \left (32 a^{4} b + 96 a^{3} b^{2}\right ) + a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}, \left ( t \mapsto t \log{\left (x + \frac{64 t^{3} a^{4} b + 4 t a^{3} + 24 t a^{2} b + 4 t a b^{2}}{3 a^{2} - 2 a b - b^{2}} \right )} \right )\right )} + \frac{x}{a} \]

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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: TypeError