3.36 \(\int \frac{d+\frac{e}{x^2}}{c+\frac{a}{x^4}} \, dx\)

Optimal. Leaf size=253 \[ \frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{d x}{c} \]

[Out]

(d*x)/c + ((Sqrt[a]*d - Sqrt[c]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*S
qrt[2]*a^(1/4)*c^(5/4)) - ((Sqrt[a]*d - Sqrt[c]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x
)/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*c^(5/4)) + ((Sqrt[a]*d + Sqrt[c]*e)*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(1/4)*c^(5/4)) - ((Sqrt[
a]*d + Sqrt[c]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqr
t[2]*a^(1/4)*c^(5/4))

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Rubi [A]  time = 0.420356, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d+\sqrt{c} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}-\frac{\left (\sqrt{a} d-\sqrt{c} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} \sqrt [4]{a} c^{5/4}}+\frac{d x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e/x^2)/(c + a/x^4),x]

[Out]

(d*x)/c + ((Sqrt[a]*d - Sqrt[c]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*S
qrt[2]*a^(1/4)*c^(5/4)) - ((Sqrt[a]*d - Sqrt[c]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x
)/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*c^(5/4)) + ((Sqrt[a]*d + Sqrt[c]*e)*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(1/4)*c^(5/4)) - ((Sqrt[
a]*d + Sqrt[c]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqr
t[2]*a^(1/4)*c^(5/4))

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Rubi in Sympy [A]  time = 72.4602, size = 235, normalized size = 0.93 \[ \frac{d x}{c} + \frac{\sqrt{2} \left (\sqrt{a} d - \sqrt{c} e\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} c^{\frac{5}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} d - \sqrt{c} e\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 \sqrt [4]{a} c^{\frac{5}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} d + \sqrt{c} e\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 \sqrt [4]{a} c^{\frac{5}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} d + \sqrt{c} e\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 \sqrt [4]{a} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/c + sqrt(2)*(sqrt(a)*d - sqrt(c)*e)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(4
*a**(1/4)*c**(5/4)) - sqrt(2)*(sqrt(a)*d - sqrt(c)*e)*atan(1 + sqrt(2)*c**(1/4)*
x/a**(1/4))/(4*a**(1/4)*c**(5/4)) + sqrt(2)*(sqrt(a)*d + sqrt(c)*e)*log(-sqrt(2)
*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*x**2)/(8*a**(1/4)*c**(5/4)) - sqrt(2)
*(sqrt(a)*d + sqrt(c)*e)*log(sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*x
**2)/(8*a**(1/4)*c**(5/4))

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Mathematica [A]  time = 0.170371, size = 293, normalized size = 1.16 \[ \frac{\left (a^{5/4} \sqrt{c} d+a^{3/4} c e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a c^{7/4}}-\frac{\left (a^{5/4} \sqrt{c} d+a^{3/4} c e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a c^{7/4}}+\frac{\left (a^{3/4} c e-a^{5/4} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{2 \sqrt [4]{c} x-\sqrt{2} \sqrt [4]{a}}{\sqrt{2} \sqrt [4]{a}}\right )}{2 \sqrt{2} a c^{7/4}}+\frac{\left (a^{3/4} c e-a^{5/4} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a}+2 \sqrt [4]{c} x}{\sqrt{2} \sqrt [4]{a}}\right )}{2 \sqrt{2} a c^{7/4}}+\frac{d x}{c} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e/x^2)/(c + a/x^4),x]

[Out]

(d*x)/c + ((-(a^(5/4)*Sqrt[c]*d) + a^(3/4)*c*e)*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*c
^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2]*a*c^(7/4)) + ((-(a^(5/4)*Sqrt[c]*d) + a
^(3/4)*c*e)*ArcTan[(Sqrt[2]*a^(1/4) + 2*c^(1/4)*x)/(Sqrt[2]*a^(1/4))])/(2*Sqrt[2
]*a*c^(7/4)) + ((a^(5/4)*Sqrt[c]*d + a^(3/4)*c*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a*c^(7/4)) - ((a^(5/4)*Sqrt[c]*d + a^(3/4)*
c*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a*c^(7/4
))

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Maple [A]  time = 0.007, size = 266, normalized size = 1.1 \[{\frac{dx}{c}}-{\frac{d\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{d\sqrt{2}}{4\,c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{d\sqrt{2}}{8\,c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{e\sqrt{2}}{8\,c}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{e\sqrt{2}}{4\,c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{e\sqrt{2}}{4\,c}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e/x^2)/(c+a/x^4),x)

[Out]

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

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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((d + e/x^2)/(c + a/x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270875, size = 1018, normalized size = 4.02 \[ \frac{c \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x +{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x -{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt{\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x +{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + c \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x -{\left (a c^{4} e \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt{-\frac{c^{2} \sqrt{-\frac{a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + 4 \, d x}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d + e/x^2)/(c + a/x^4),x, algorithm="fricas")

[Out]

1/4*(c*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d*e)/c^2
)*log(-(a^2*d^4 - c^2*e^4)*x + (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4
)/(a*c^5)) + a^2*c*d^3 - a*c^2*d*e^2)*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 +
c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)) - c*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 +
c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x - (a*c^4*e*sqrt(-(a^
2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + a^2*c*d^3 - a*c^2*d*e^2)*sqrt((c^2*s
qrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)) - c*sqrt(-(c^2*
sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^2)*log(-(a^2*d^4 -
c^2*e^4)*x + (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - a^2*
c*d^3 + a*c^2*d*e^2)*sqrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5
)) - 2*d*e)/c^2)) + c*sqrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^
5)) - 2*d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x - (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d
^2*e^2 + c^2*e^4)/(a*c^5)) - a^2*c*d^3 + a*c^2*d*e^2)*sqrt(-(c^2*sqrt(-(a^2*d^4
- 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^2)) + 4*d*x)/c

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Sympy [A]  time = 3.11912, size = 109, normalized size = 0.43 \[ \operatorname{RootSum}{\left (256 t^{4} a c^{5} - 64 t^{2} a c^{3} d e + a^{2} d^{4} + 2 a c d^{2} e^{2} + c^{2} e^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a c^{4} e - 4 t a^{2} c d^{3} + 12 t a c^{2} d e^{2}}{a^{2} d^{4} - c^{2} e^{4}} \right )} \right )\right )} + \frac{d x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(256*_t**4*a*c**5 - 64*_t**2*a*c**3*d*e + a**2*d**4 + 2*a*c*d**2*e**2 + c
**2*e**4, Lambda(_t, _t*log(x + (-64*_t**3*a*c**4*e - 4*_t*a**2*c*d**3 + 12*_t*a
*c**2*d*e**2)/(a**2*d**4 - c**2*e**4)))) + d*x/c

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GIAC/XCAS [A]  time = 0.272603, size = 347, normalized size = 1.37 \[ \frac{d x}{c} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c d - \left (a c^{3}\right )^{\frac{3}{4}} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c d + \left (a c^{3}\right )^{\frac{3}{4}} e\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{3}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{3} d - \left (a c^{3}\right )^{\frac{3}{4}} c^{2} e\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{5}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} a c^{3} d + \left (a c^{3}\right )^{\frac{3}{4}} c^{2} e\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/c - 1/4*sqrt(2)*((a*c^3)^(1/4)*a*c*d - (a*c^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(
2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3) + 1/8*sqrt(2)*((a*c^3)^(1/4)*a*c
*d + (a*c^3)^(3/4)*e)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 1/4*
sqrt(2)*((a*c^3)^(1/4)*a*c^3*d - (a*c^3)^(3/4)*c^2*e)*arctan(1/2*sqrt(2)*(2*x +
sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^5) - 1/8*sqrt(2)*((a*c^3)^(1/4)*a*c^3*d +
(a*c^3)^(3/4)*c^2*e)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^5)