3.170 \(\int \frac{1+x+x^2+x^3}{a+b x^4} \, dx\)

Optimal. Leaf size=277 \[ \frac{\left (\sqrt{a}-\sqrt{b}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{a}-\sqrt{b}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\log \left (a+b x^4\right )}{4 b}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.474812, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{\left (\sqrt{a}-\sqrt{b}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{a}-\sqrt{b}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\log \left (a+b x^4\right )}{4 b}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x + x^2 + x^3)/(a + b*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 73.5883, size = 255, normalized size = 0.92 \[ \frac{\log{\left (a + b x^{4} \right )}}{4 b} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b}} + \frac{\sqrt{2} \left (\sqrt{a} - \sqrt{b}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} - \sqrt{b}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{8 a^{\frac{3}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} + \sqrt{b}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} + \sqrt{b}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+x**2+x+1)/(b*x**4+a),x)

[Out]

log(a + b*x**4)/(4*b) + atan(sqrt(b)*x**2/sqrt(a))/(2*sqrt(a)*sqrt(b)) + sqrt(2)
*(sqrt(a) - sqrt(b))*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2
)/(8*a**(3/4)*b**(3/4)) - sqrt(2)*(sqrt(a) - sqrt(b))*log(sqrt(2)*a**(1/4)*b**(3
/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(8*a**(3/4)*b**(3/4)) - sqrt(2)*(sqrt(a) + sqr
t(b))*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4)*b**(3/4)) + sqrt(2)*(sqr
t(a) + sqrt(b))*atan(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4)*b**(3/4))

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Mathematica [A]  time = 0.448622, size = 283, normalized size = 1.02 \[ \frac{\sqrt{2} \sqrt [4]{b} \left (a^{3/4}-\sqrt [4]{a} \sqrt{b}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+\sqrt{2} \sqrt [4]{b} \left (\sqrt [4]{a} \sqrt{b}-a^{3/4}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )+2 a \log \left (a+b x^4\right )-2 \sqrt [4]{a} \sqrt [4]{b} \left (2 \sqrt [4]{a} \sqrt [4]{b}+\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{b}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{a} \sqrt [4]{b} \left (-2 \sqrt [4]{a} \sqrt [4]{b}+\sqrt{2} \sqrt{a}+\sqrt{2} \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 a b} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x + x^2 + x^3)/(a + b*x^4),x]

[Out]

(-2*a^(1/4)*(Sqrt[2]*Sqrt[a] + 2*a^(1/4)*b^(1/4) + Sqrt[2]*Sqrt[b])*b^(1/4)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*a^(1/4)*(Sqrt[2]*Sqrt[a] - 2*a^(1/4)*b^(
1/4) + Sqrt[2]*Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2
]*(a^(3/4) - a^(1/4)*Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x +
Sqrt[b]*x^2] + Sqrt[2]*(-a^(3/4) + a^(1/4)*Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + 2*a*Log[a + b*x^4])/(8*a*b)

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Maple [A]  time = 0.004, size = 286, normalized size = 1. \[{\frac{\sqrt{2}}{8\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{1}{2}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{\sqrt{2}}{8\,b}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}}{4\,b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}}{4\,b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\ln \left ( b{x}^{4}+a \right ) }{4\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x^2+x+1)/(b*x^4+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + x + 1)/(b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + x + 1)/(b*x^4 + a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 1.90628, size = 187, normalized size = 0.68 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{4} - 256 t^{3} a^{3} b^{3} + t^{2} \left (96 a^{3} b^{2} + 96 a^{2} b^{3}\right ) + t \left (- 16 a^{3} b - 32 a^{2} b^{2} - 16 a b^{3}\right ) + a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}, \left ( t \mapsto t \log{\left (x + \frac{64 t^{3} a^{3} b^{3} - 48 t^{2} a^{3} b^{2} + 16 t^{2} a^{2} b^{3} + 12 t a^{3} b + 16 t a^{2} b^{2} + 4 t a b^{3} - a^{3} - 2 a^{2} b - a b^{2}}{a^{2} b + 2 a b^{2} + b^{3}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x**2+x+1)/(b*x**4+a),x)

[Out]

RootSum(256*_t**4*a**3*b**4 - 256*_t**3*a**3*b**3 + _t**2*(96*a**3*b**2 + 96*a**
2*b**3) + _t*(-16*a**3*b - 32*a**2*b**2 - 16*a*b**3) + a**3 + 3*a**2*b + 3*a*b**
2 + b**3, Lambda(_t, _t*log(x + (64*_t**3*a**3*b**3 - 48*_t**2*a**3*b**2 + 16*_t
**2*a**2*b**3 + 12*_t*a**3*b + 16*_t*a**2*b**2 + 4*_t*a*b**3 - a**3 - 2*a**2*b -
a*b**2)/(a**2*b + 2*a*b**2 + b**3))))

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GIAC/XCAS [A]  time = 0.219624, size = 365, normalized size = 1.32 \[ \frac{{\rm ln}\left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} - \sqrt{2} \sqrt{a b^{3}} b + \left (a b^{3}\right )^{\frac{3}{4}}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} + \sqrt{2} \sqrt{a b^{3}} b + \left (a b^{3}\right )^{\frac{3}{4}}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} - \left (a b^{3}\right )^{\frac{3}{4}}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} - \left (a b^{3}\right )^{\frac{3}{4}}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + x^2 + x + 1)/(b*x^4 + a),x, algorithm="giac")

[Out]

1/4*ln(abs(b*x^4 + a))/b + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^2 - sqrt(2)*sqrt(a*b^3)*
b + (a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(
a*b^3) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^2 + sqrt(2)*sqrt(a*b^3)*b + (a*b^3)^(3/4))
*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^3) + 1/8*sqrt(
2)*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))*ln(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b
))/(a*b^3) - 1/8*sqrt(2)*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))*ln(x^2 - sqrt(2)*x*
(a/b)^(1/4) + sqrt(a/b))/(a*b^3)