3.171 \(\int \frac{c+d x+e x^2+f x^3+g x^4}{a-b x^4} \, dx\)

Optimal. Leaf size=148 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b}-\frac{g x}{b} \]

[Out]

-((g*x)/b) + ((b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^
(3/4)*b^(5/4)) + ((b*c + Sqrt[a]*Sqrt[b]*e + a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/
(2*a^(3/4)*b^(5/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]) - (
f*Log[a - b*x^4])/(4*b)

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Rubi [A]  time = 0.435726, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b}-\frac{g x}{b} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4),x]

[Out]

-((g*x)/b) + ((b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^
(3/4)*b^(5/4)) + ((b*c + Sqrt[a]*Sqrt[b]*e + a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/
(2*a^(3/4)*b^(5/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]) - (
f*Log[a - b*x^4])/(4*b)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{f \log{\left (a - b x^{4} \right )}}{4 b} - \frac{\int g\, dx}{b} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b}} + \frac{\left (- \sqrt{a} \sqrt{b} e + a g + b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{5}{4}}} + \frac{\left (\sqrt{a} \sqrt{b} e + a g + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)

[Out]

-f*log(a - b*x**4)/(4*b) - Integral(g, x)/b + d*atanh(sqrt(b)*x**2/sqrt(a))/(2*s
qrt(a)*sqrt(b)) + (-sqrt(a)*sqrt(b)*e + a*g + b*c)*atan(b**(1/4)*x/a**(1/4))/(2*
a**(3/4)*b**(5/4)) + (sqrt(a)*sqrt(b)*e + a*g + b*c)*atanh(b**(1/4)*x/a**(1/4))/
(2*a**(3/4)*b**(5/4))

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Mathematica [A]  time = 0.167182, size = 249, normalized size = 1.68 \[ \frac{-a^{3/4} \sqrt [4]{b} f \log \left (a-b x^4\right )-4 a^{3/4} \sqrt [4]{b} g x-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (\sqrt [4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e+a g+b c\right )+\sqrt [4]{a} b^{3/4} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-\sqrt [4]{a} b^{3/4} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e+a g+b c\right )+b c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt{a} \sqrt{b} e \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+a g \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a^{3/4} b^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4),x]

[Out]

(-4*a^(3/4)*b^(1/4)*g*x + 2*(b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a
^(1/4)] - (b*c + a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e + a*g)*Log[a^(1/4) - b^(1
/4)*x] + b*c*Log[a^(1/4) + b^(1/4)*x] - a^(1/4)*b^(3/4)*d*Log[a^(1/4) + b^(1/4)*
x] + Sqrt[a]*Sqrt[b]*e*Log[a^(1/4) + b^(1/4)*x] + a*g*Log[a^(1/4) + b^(1/4)*x] +
a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2] - a^(3/4)*b^(1/4)*f*Log[a - b*x^4]
)/(4*a^(3/4)*b^(5/4))

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Maple [B]  time = 0.006, size = 244, normalized size = 1.7 \[ -{\frac{gx}{b}}+{\frac{g}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{g}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{d}{4}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{4\,b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{f\ln \left ( b{x}^{4}-a \right ) }{4\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x)

[Out]

-g*x/b+1/2/b*(a/b)^(1/4)*arctan(x/(a/b)^(1/4))*g+1/2*c*(a/b)^(1/4)/a*arctan(x/(a
/b)^(1/4))+1/4/b*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*g+1/4*c*(a/b)^(
1/4)/a*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))-1/4*d/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1
/2))/(-a-x^2*(a*b)^(1/2)))-1/2*e/b/(a/b)^(1/4)*arctan(x/(a/b)^(1/4))+1/4*e/b/(a/
b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))-1/4/b*f*ln(b*x^4-a)

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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(-(g*x^4 + f*x^3 + e*x^2 + d*x + c)/(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(-(g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22272, size = 539, normalized size = 3.64 \[ -\frac{g x}{b} - \frac{f{\rm ln}\left ({\left | b x^{4} - a \right |}\right )}{4 \, b} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} a b^{4} d + \left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g + \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} b^{5}} + \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} a b^{4} d + \left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g + \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a^{2} b^{5}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g - \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a^{2} b^{5}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} a b^{4} c + \left (-a b^{3}\right )^{\frac{1}{4}} a^{2} b^{3} g - \left (-a b^{3}\right )^{\frac{3}{4}} a b^{2} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a^{2} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^4 - a),x, algorithm="giac")

[Out]

-g*x/b - 1/4*f*ln(abs(b*x^4 - a))/b + 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*a*b^4*d +
(-a*b^3)^(1/4)*a*b^4*c + (-a*b^3)^(1/4)*a^2*b^3*g + (-a*b^3)^(3/4)*a*b^2*e)*arct
an(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^2*b^5) + 1/4*sqrt(2
)*(sqrt(2)*sqrt(-a*b)*a*b^4*d + (-a*b^3)^(1/4)*a*b^4*c + (-a*b^3)^(1/4)*a^2*b^3*
g + (-a*b^3)^(3/4)*a*b^2*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/
b)^(1/4))/(a^2*b^5) + 1/8*sqrt(2)*((-a*b^3)^(1/4)*a*b^4*c + (-a*b^3)^(1/4)*a^2*b
^3*g - (-a*b^3)^(3/4)*a*b^2*e)*ln(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^
2*b^5) - 1/8*sqrt(2)*((-a*b^3)^(1/4)*a*b^4*c + (-a*b^3)^(1/4)*a^2*b^3*g - (-a*b^
3)^(3/4)*a*b^2*e)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^5)