3.913 \(\int \frac{1}{a+b+2 a x^2+a x^4} \, dx\)

Optimal. Leaf size=359 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}} \]

[Out]

-ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a
+ b]]]/(2*Sqrt[2]*a^(1/4)*Sqrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) + ArcTan[(Sq
rt[-Sqrt[a] + Sqrt[a + b]] + Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]]/(2*
Sqrt[2]*a^(1/4)*Sqrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) - Log[Sqrt[a + b] - Sqr
t[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2]/(4*Sqrt[2]*a^(1/4)*Sq
rt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) + Log[Sqrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt
[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2]/(4*Sqrt[2]*a^(1/4)*Sqrt[a + b]*Sqrt[-S
qrt[a] + Sqrt[a + b]])

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Rubi [A]  time = 0.690382, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} x \sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{a+b}+\sqrt{a} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}-\sqrt{a}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}-\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a+b}-\sqrt{a}}+\sqrt{2} \sqrt [4]{a} x}{\sqrt{\sqrt{a+b}+\sqrt{a}}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt{a+b} \sqrt{\sqrt{a+b}+\sqrt{a}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b + 2*a*x^2 + a*x^4)^(-1),x]

[Out]

-ArcTan[(Sqrt[-Sqrt[a] + Sqrt[a + b]] - Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a
+ b]]]/(2*Sqrt[2]*a^(1/4)*Sqrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) + ArcTan[(Sq
rt[-Sqrt[a] + Sqrt[a + b]] + Sqrt[2]*a^(1/4)*x)/Sqrt[Sqrt[a] + Sqrt[a + b]]]/(2*
Sqrt[2]*a^(1/4)*Sqrt[a + b]*Sqrt[Sqrt[a] + Sqrt[a + b]]) - Log[Sqrt[a + b] - Sqr
t[2]*a^(1/4)*Sqrt[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2]/(4*Sqrt[2]*a^(1/4)*Sq
rt[a + b]*Sqrt[-Sqrt[a] + Sqrt[a + b]]) + Log[Sqrt[a + b] + Sqrt[2]*a^(1/4)*Sqrt
[-Sqrt[a] + Sqrt[a + b]]*x + Sqrt[a]*x^2]/(4*Sqrt[2]*a^(1/4)*Sqrt[a + b]*Sqrt[-S
qrt[a] + Sqrt[a + b]])

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Rubi in Sympy [A]  time = 88.3596, size = 321, normalized size = 0.89 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt [4]{a} x - \frac{\sqrt{- 2 \sqrt{a} + 2 \sqrt{a + b}}}{2}\right )}{\sqrt{\sqrt{a} + \sqrt{a + b}}} \right )}}{4 \sqrt [4]{a} \sqrt{\sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt [4]{a} x + \frac{\sqrt{- 2 \sqrt{a} + 2 \sqrt{a + b}}}{2}\right )}{\sqrt{\sqrt{a} + \sqrt{a + b}}} \right )}}{4 \sqrt [4]{a} \sqrt{\sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} - \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{a + b}}{\sqrt{a}} - \frac{\sqrt{2} x \sqrt{- \sqrt{a} + \sqrt{a + b}}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} \sqrt{- \sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} + \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{a + b}}{\sqrt{a}} + \frac{\sqrt{2} x \sqrt{- \sqrt{a} + \sqrt{a + b}}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} \sqrt{- \sqrt{a} + \sqrt{a + b}} \sqrt{a + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*x**4+2*a*x**2+a+b),x)

[Out]

sqrt(2)*atan(sqrt(2)*(a**(1/4)*x - sqrt(-2*sqrt(a) + 2*sqrt(a + b))/2)/sqrt(sqrt
(a) + sqrt(a + b)))/(4*a**(1/4)*sqrt(sqrt(a) + sqrt(a + b))*sqrt(a + b)) + sqrt(
2)*atan(sqrt(2)*(a**(1/4)*x + sqrt(-2*sqrt(a) + 2*sqrt(a + b))/2)/sqrt(sqrt(a) +
sqrt(a + b)))/(4*a**(1/4)*sqrt(sqrt(a) + sqrt(a + b))*sqrt(a + b)) - sqrt(2)*lo
g(x**2 + sqrt(a + b)/sqrt(a) - sqrt(2)*x*sqrt(-sqrt(a) + sqrt(a + b))/a**(1/4))/
(8*a**(1/4)*sqrt(-sqrt(a) + sqrt(a + b))*sqrt(a + b)) + sqrt(2)*log(x**2 + sqrt(
a + b)/sqrt(a) + sqrt(2)*x*sqrt(-sqrt(a) + sqrt(a + b))/a**(1/4))/(8*a**(1/4)*sq
rt(-sqrt(a) + sqrt(a + b))*sqrt(a + b))

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Mathematica [C]  time = 0.119604, size = 119, normalized size = 0.33 \[ \frac{i \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{b} \sqrt{a+i \sqrt{a} \sqrt{b}}}-\frac{i \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a-i \sqrt{a} \sqrt{b}}}\right )}{2 \sqrt{b} \sqrt{a-i \sqrt{a} \sqrt{b}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b + 2*a*x^2 + a*x^4)^(-1),x]

[Out]

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

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Maple [B]  time = 0.062, size = 913, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^4+2*a*x^2+a+b),x)

[Out]

-1/8/(a+b)^(1/2)/a/b*ln(-a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x-(a+b)^(1/2)
)*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/8/(a+b)^(1/2)/b*ln(-a^(1/2)*x^
2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x-(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)-1/(
a+b)^(1/2)/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arctan((-2*a^(1/2
)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)
^(1/2))+1/4/(a+b)^(1/2)/a/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*
arctan((-2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*
(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(1/2)-2*a)^(
1/2)*(a^2+a*b)^(1/2)+1/4/(a+b)^(1/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+
2*a)^(1/2)*arctan((-2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^
(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(
1/2)-2*a)^(1/2)+1/8/(a+b)^(1/2)/a/b*ln(a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)
*x+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)+1/8/(a+b)^(1/2)/b*
ln(a^(1/2)*x^2+(2*(a*(a+b))^(1/2)-2*a)^(1/2)*x+(a+b)^(1/2))*(2*(a^2+a*b)^(1/2)-2
*a)^(1/2)+1/(a+b)^(1/2)/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2)*arct
an((2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b)
)^(1/2)+2*a)^(1/2))-1/4/(a+b)^(1/2)/a/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)
+2*a)^(1/2)*arctan((2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1/2)*(a+b)^
(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(a^2+a*b)^(
1/2)-2*a)^(1/2)*(a^2+a*b)^(1/2)-1/4/(a+b)^(1/2)/b/(4*a^(1/2)*(a+b)^(1/2)-2*(a*(a
+b))^(1/2)+2*a)^(1/2)*arctan((2*a^(1/2)*x+(2*(a*(a+b))^(1/2)-2*a)^(1/2))/(4*a^(1
/2)*(a+b)^(1/2)-2*(a*(a+b))^(1/2)+2*a)^(1/2))*(2*(a*(a+b))^(1/2)-2*a)^(1/2)*(2*(
a^2+a*b)^(1/2)-2*a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{a x^{4} + 2 \, a x^{2} + a + b}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x^4 + 2*a*x^2 + a + b),x, algorithm="maxima")

[Out]

integrate(1/(a*x^4 + 2*a*x^2 + a + b), x)

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Fricas [A]  time = 0.282999, size = 765, normalized size = 2.13 \[ \frac{1}{4} \, \sqrt{\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} \log \left ({\left ({\left (a^{2} b + a b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + b\right )} \sqrt{\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} + x\right ) - \frac{1}{4} \, \sqrt{\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} \log \left (-{\left ({\left (a^{2} b + a b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + b\right )} \sqrt{\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} + 1}{a b + b^{2}}} + x\right ) - \frac{1}{4} \, \sqrt{-\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} \log \left ({\left ({\left (a^{2} b + a b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - b\right )} \sqrt{-\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} + x\right ) + \frac{1}{4} \, \sqrt{-\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} \log \left (-{\left ({\left (a^{2} b + a b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - b\right )} \sqrt{-\frac{{\left (a b + b^{2}\right )} \sqrt{-\frac{1}{a^{3} b + 2 \, a^{2} b^{2} + a b^{3}}} - 1}{a b + b^{2}}} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x^4 + 2*a*x^2 + a + b),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.94328, size = 63, normalized size = 0.18 \[ \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} b^{2} + 256 a b^{3}\right ) - 32 t^{2} a b + 1, \left ( t \mapsto t \log{\left (64 t^{3} a^{2} b + 64 t^{3} a b^{2} - 4 t a + 4 t b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x**4+2*a*x**2+a+b),x)

[Out]

RootSum(_t**4*(256*a**2*b**2 + 256*a*b**3) - 32*_t**2*a*b + 1, Lambda(_t, _t*log
(64*_t**3*a**2*b + 64*_t**3*a*b**2 - 4*_t*a + 4*_t*b + x)))

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GIAC/XCAS [A]  time = 0.677847, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x^4 + 2*a*x^2 + a + b),x, algorithm="giac")

[Out]

Done