3.116 \(\int \frac{c+d x}{a+b x^4} \, dx\)

Optimal. Leaf size=219 \[ -\frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}-\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.397686, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{c \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{c \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b}}-\frac{c \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{c \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)/(a + b*x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 58.5732, size = 207, normalized size = 0.95 \[ \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b}} - \frac{\sqrt{2} c \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b}} + \frac{\sqrt{2} c \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x + \sqrt{a} + \sqrt{b} x^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{b}} - \frac{\sqrt{2} c \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b}} + \frac{\sqrt{2} c \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)/(b*x**4+a),x)

[Out]

d*atan(sqrt(b)*x**2/sqrt(a))/(2*sqrt(a)*sqrt(b)) - sqrt(2)*c*log(-sqrt(2)*a**(1/
4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*b**(1/4)) + sqrt(2)*c*log(sq
rt(2)*a**(1/4)*b**(1/4)*x + sqrt(a) + sqrt(b)*x**2)/(8*a**(3/4)*b**(1/4)) - sqrt
(2)*c*atan(1 - sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4)*b**(1/4)) + sqrt(2)*c*at
an(1 + sqrt(2)*b**(1/4)*x/a**(1/4))/(4*a**(3/4)*b**(1/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.140262, size = 184, normalized size = 0.84 \[ \frac{-2 \left (2 \sqrt [4]{a} d+\sqrt{2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt{2} \sqrt [4]{b} c-2 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )+\sqrt{2} \sqrt [4]{b} c \left (\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )\right )}{8 a^{3/4} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)/(a + b*x^4),x]

[Out]

(-2*(Sqrt[2]*b^(1/4)*c + 2*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] +
2*(Sqrt[2]*b^(1/4)*c - 2*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sq
rt[2]*b^(1/4)*c*(-Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Log[S
qrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]))/(8*a^(3/4)*Sqrt[b])

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 151, normalized size = 0.7 \[{\frac{c\sqrt{2}}{8\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{c\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d}{2}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(b*x^4+a),x)

[Out]

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

_______________________________________________________________________________________

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((d*x + c)/(b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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((d*x + c)/(b*x^4 + a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

Sympy [A]  time = 1.35383, size = 124, normalized size = 0.57 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{2} + 32 t^{2} a^{2} b d^{2} - 16 t a b c^{2} d + a d^{4} + b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 128 t^{3} a^{3} b d^{2} - 16 t^{2} a^{2} b c^{2} d - 8 t a^{2} d^{4} - 4 t a b c^{4} + 5 a c^{2} d^{3}}{4 a c d^{4} - b c^{5}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)/(b*x**4+a),x)

[Out]

RootSum(256*_t**4*a**3*b**2 + 32*_t**2*a**2*b*d**2 - 16*_t*a*b*c**2*d + a*d**4 +
b*c**4, Lambda(_t, _t*log(x + (-128*_t**3*a**3*b*d**2 - 16*_t**2*a**2*b*c**2*d
- 8*_t*a**2*d**4 - 4*_t*a*b*c**4 + 5*a*c**2*d**3)/(4*a*c*d**4 - b*c**5))))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216012, size = 288, normalized size = 1.32 \[ \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{a b} b d - \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{2}} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{a b} b d - \left (a b^{3}\right )^{\frac{1}{4}} b c\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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