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

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

[Out]

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

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Rubi [A]  time = 0.35954, 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 \[ -\frac{d \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{c}}+\frac{d \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{c}}-\frac{d \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{c}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{c}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + c*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 60.1606, size = 207, normalized size = 0.95 \[ \frac{e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{c}} - \frac{\sqrt{2} d \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{c}} + \frac{\sqrt{2} d \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{8 a^{\frac{3}{4}} \sqrt [4]{c}} - \frac{\sqrt{2} d \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{c}} + \frac{\sqrt{2} d \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101186, size = 184, normalized size = 0.84 \[ \frac{-2 \left (2 \sqrt [4]{a} e+\sqrt{2} \sqrt [4]{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt{2} \sqrt [4]{c} d-2 \sqrt [4]{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt{2} \sqrt [4]{c} d \left (\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )\right )}{8 a^{3/4} \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 151, normalized size = 0.7 \[{\frac{d\sqrt{2}}{8\,a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{d\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{d\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{e}{2}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^4+a),x)

[Out]

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

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.91482, size = 124, normalized size = 0.57 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} c^{2} + 32 t^{2} a^{2} c e^{2} - 16 t a c d^{2} e + a e^{4} + c d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 128 t^{3} a^{3} c e^{2} - 16 t^{2} a^{2} c d^{2} e - 8 t a^{2} e^{4} - 4 t a c d^{4} + 5 a d^{2} e^{3}}{4 a d e^{4} - c d^{5}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*x**4+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.268959, size = 290, normalized size = 1.32 \[ \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} d{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c} - \frac{\sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} d{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{a c} c e - \left (a c^{3}\right )^{\frac{1}{4}} c d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{2}} - \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{a c} c e - \left (a c^{3}\right )^{\frac{1}{4}} c d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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