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

Optimal. Leaf size=133 \[ \frac{\left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} e+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b} \]

[Out]

((Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + ((Sq
rt[b]*c + Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + (d*ArcT
anh[(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 = 0.282279, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{\left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} e+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + ((Sq
rt[b]*c + Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + (d*ArcT
anh[(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 [A]  time = 38.7046, size = 119, normalized size = 0.89 \[ - \frac{f \log{\left (a - b x^{4} \right )}}{4 b} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b}} - \frac{\left (\sqrt{a} e - \sqrt{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{3}{4}}} + \frac{\left (\sqrt{a} e + \sqrt{b} c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-f*log(a - b*x**4)/(4*b) + d*atanh(sqrt(b)*x**2/sqrt(a))/(2*sqrt(a)*sqrt(b)) - (
sqrt(a)*e - sqrt(b)*c)*atan(b**(1/4)*x/a**(1/4))/(2*a**(3/4)*b**(3/4)) + (sqrt(a
)*e + sqrt(b)*c)*atanh(b**(1/4)*x/a**(1/4))/(2*a**(3/4)*b**(3/4))

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Mathematica [A]  time = 0.110883, size = 214, normalized size = 1.61 \[ -\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{3/4} e+\sqrt [4]{a} \sqrt{b} c+\sqrt{a} \sqrt [4]{b} d\right )}{4 a b^{3/4}}-\frac{\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-a^{3/4} e-\sqrt [4]{a} \sqrt{b} c+\sqrt{a} \sqrt [4]{b} d\right )}{4 a b^{3/4}}+\frac{\left (\sqrt [4]{a} \sqrt{b} c-a^{3/4} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a b^{3/4}}+\frac{d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{a} \sqrt{b}}-\frac{f \log \left (a-b x^4\right )}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 177, normalized size = 1.3 \[{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d}{4}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{4\,b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{f\ln \left ( b{x}^{4}-a \right ) }{4\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c*(a/b)^(1/4)/a*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/2*c*(a/b)^(1/4)/a*arct
an(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(-(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(-(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 [A]  time = 25.6382, size = 952, normalized size = 7.16 \[ - \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{4} - 256 t^{3} a^{3} b^{3} f + t^{2} \left (96 a^{3} b^{2} f^{2} - 64 a^{2} b^{3} c e - 32 a^{2} b^{3} d^{2}\right ) + t \left (- 16 a^{3} b f^{3} + 32 a^{2} b^{2} c e f + 16 a^{2} b^{2} d^{2} f - 16 a^{2} b^{2} d e^{2} - 16 a b^{3} c^{2} d\right ) + a^{3} f^{4} - 4 a^{2} b c e f^{2} - 2 a^{2} b d^{2} f^{2} + 4 a^{2} b d e^{2} f - a^{2} b e^{4} + 4 a b^{2} c^{2} d f + 2 a b^{2} c^{2} e^{2} - 4 a b^{2} c d^{2} e + a b^{2} d^{4} - b^{3} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{4} b^{3} e^{3} - 64 t^{3} a^{3} b^{4} c^{2} e + 128 t^{3} a^{3} b^{4} c d^{2} + 48 t^{2} a^{4} b^{2} e^{3} f + 48 t^{2} a^{3} b^{3} c^{2} e f - 96 t^{2} a^{3} b^{3} c d^{2} f + 48 t^{2} a^{3} b^{3} c d e^{2} - 32 t^{2} a^{3} b^{3} d^{3} e - 16 t^{2} a^{2} b^{4} c^{3} d - 12 t a^{4} b e^{3} f^{2} - 12 t a^{3} b^{2} c^{2} e f^{2} + 24 t a^{3} b^{2} c d^{2} f^{2} - 24 t a^{3} b^{2} c d e^{2} f + 12 t a^{3} b^{2} c e^{4} + 16 t a^{3} b^{2} d^{3} e f + 12 t a^{3} b^{2} d^{2} e^{3} + 8 t a^{2} b^{3} c^{3} d f + 16 t a^{2} b^{3} c^{3} e^{2} - 36 t a^{2} b^{3} c^{2} d^{2} e - 8 t a^{2} b^{3} c d^{4} + 4 t a b^{4} c^{5} + a^{4} e^{3} f^{3} + a^{3} b c^{2} e f^{3} - 2 a^{3} b c d^{2} f^{3} + 3 a^{3} b c d e^{2} f^{2} - 3 a^{3} b c e^{4} f - 2 a^{3} b d^{3} e f^{2} - 3 a^{3} b d^{2} e^{3} f + 3 a^{3} b d e^{5} - a^{2} b^{2} c^{3} d f^{2} - 4 a^{2} b^{2} c^{3} e^{2} f + 9 a^{2} b^{2} c^{2} d^{2} e f + 2 a^{2} b^{2} c d^{4} f - 5 a^{2} b^{2} c d^{3} e^{2} + 2 a^{2} b^{2} d^{5} e - a b^{3} c^{5} f + 5 a b^{3} c^{4} d e - 5 a b^{3} c^{3} d^{3}}{a^{3} b e^{6} + a^{2} b^{2} c^{2} e^{4} - 8 a^{2} b^{2} c d^{2} e^{3} + 4 a^{2} b^{2} d^{4} e^{2} - a b^{3} c^{4} e^{2} + 8 a b^{3} c^{3} d^{2} e - 4 a b^{3} c^{2} d^{4} - b^{4} c^{6}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(256*_t**4*a**3*b**4 - 256*_t**3*a**3*b**3*f + _t**2*(96*a**3*b**2*f**2
- 64*a**2*b**3*c*e - 32*a**2*b**3*d**2) + _t*(-16*a**3*b*f**3 + 32*a**2*b**2*c*e
*f + 16*a**2*b**2*d**2*f - 16*a**2*b**2*d*e**2 - 16*a*b**3*c**2*d) + a**3*f**4 -
4*a**2*b*c*e*f**2 - 2*a**2*b*d**2*f**2 + 4*a**2*b*d*e**2*f - a**2*b*e**4 + 4*a*
b**2*c**2*d*f + 2*a*b**2*c**2*e**2 - 4*a*b**2*c*d**2*e + a*b**2*d**4 - b**3*c**4
, Lambda(_t, _t*log(x + (-64*_t**3*a**4*b**3*e**3 - 64*_t**3*a**3*b**4*c**2*e +
128*_t**3*a**3*b**4*c*d**2 + 48*_t**2*a**4*b**2*e**3*f + 48*_t**2*a**3*b**3*c**2
*e*f - 96*_t**2*a**3*b**3*c*d**2*f + 48*_t**2*a**3*b**3*c*d*e**2 - 32*_t**2*a**3
*b**3*d**3*e - 16*_t**2*a**2*b**4*c**3*d - 12*_t*a**4*b*e**3*f**2 - 12*_t*a**3*b
**2*c**2*e*f**2 + 24*_t*a**3*b**2*c*d**2*f**2 - 24*_t*a**3*b**2*c*d*e**2*f + 12*
_t*a**3*b**2*c*e**4 + 16*_t*a**3*b**2*d**3*e*f + 12*_t*a**3*b**2*d**2*e**3 + 8*_
t*a**2*b**3*c**3*d*f + 16*_t*a**2*b**3*c**3*e**2 - 36*_t*a**2*b**3*c**2*d**2*e -
8*_t*a**2*b**3*c*d**4 + 4*_t*a*b**4*c**5 + a**4*e**3*f**3 + a**3*b*c**2*e*f**3
- 2*a**3*b*c*d**2*f**3 + 3*a**3*b*c*d*e**2*f**2 - 3*a**3*b*c*e**4*f - 2*a**3*b*d
**3*e*f**2 - 3*a**3*b*d**2*e**3*f + 3*a**3*b*d*e**5 - a**2*b**2*c**3*d*f**2 - 4*
a**2*b**2*c**3*e**2*f + 9*a**2*b**2*c**2*d**2*e*f + 2*a**2*b**2*c*d**4*f - 5*a**
2*b**2*c*d**3*e**2 + 2*a**2*b**2*d**5*e - a*b**3*c**5*f + 5*a*b**3*c**4*d*e - 5*
a*b**3*c**3*d**3)/(a**3*b*e**6 + a**2*b**2*c**2*e**4 - 8*a**2*b**2*c*d**2*e**3 +
4*a**2*b**2*d**4*e**2 - a*b**3*c**4*e**2 + 8*a*b**3*c**3*d**2*e - 4*a*b**3*c**2
*d**4 - b**4*c**6))))

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GIAC/XCAS [A]  time = 0.242442, size = 419, normalized size = 3.15 \[ -\frac{f{\rm ln}\left ({\left | b x^{4} - a \right |}\right )}{4 \, b} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} d - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} d - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*f*ln(abs(b*x^4 - a))/b - 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*d - (-a*b^3)^(
1/4)*b^2*c - (-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(
-a/b)^(1/4))/(a*b^3) - 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*d - (-a*b^3)^(1/4)*b^
2*c - (-a*b^3)^(3/4)*e)*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*c - (-a*b^3)^(3/4)*e)*ln(x^2 + s
qrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a*b^3) - 1/8*sqrt(2)*((-a*b^3)^(1/4)*b^2*c
- (-a*b^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a*b^3)