3.398 \(\int \frac{1}{(d+e x) \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=416 \[ -\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}+\frac{e^3 \log (d+e x)}{a e^4+c d^4}-\frac{\sqrt{c} d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^4+c d^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.945997, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647 \[ -\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{c} d^2-\sqrt{a} e^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}+\frac{\sqrt [4]{c} d \left (\sqrt{a} e^2+\sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a e^4+c d^4\right )}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}+\frac{e^3 \log (d+e x)}{a e^4+c d^4}-\frac{\sqrt{c} d^2 e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^4+c d^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + c*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 133.876, size = 379, normalized size = 0.91 \[ - \frac{e^{3} \log{\left (a + c x^{4} \right )}}{4 \left (a e^{4} + c d^{4}\right )} + \frac{e^{3} \log{\left (d + e x \right )}}{a e^{4} + c d^{4}} - \frac{\sqrt{c} d^{2} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a e^{4} + c d^{4}\right )} + \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} - \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} - \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} + \frac{\sqrt{2} \sqrt [4]{c} d \left (\sqrt{a} e^{2} + \sqrt{c} d^{2}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} \left (a e^{4} + c d^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**4+a),x)

[Out]

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

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Mathematica [A]  time = 0.285521, size = 404, normalized size = 0.97 \[ \frac{-2 a^{3/4} e^3 \log \left (a+c x^4\right )+8 a^{3/4} e^3 \log (d+e x)-\sqrt{2} c^{3/4} d^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+\sqrt{2} c^{3/4} d^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-2 \sqrt [4]{c} d \left (-2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{c} d \left (2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt{2} \sqrt{a} e^2+\sqrt{2} \sqrt{c} d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+\sqrt{2} \sqrt{a} \sqrt [4]{c} d e^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )-\sqrt{2} \sqrt{a} \sqrt [4]{c} d e^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{8 a^{3/4} \left (a e^4+c d^4\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + c*x^4)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 433, normalized size = 1. \[{\frac{c{d}^{3}\sqrt{2}}{ \left ( 8\,a{e}^{4}+8\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,a{e}^{4}+4\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{c{d}^{3}\sqrt{2}}{ \left ( 4\,a{e}^{4}+4\,c{d}^{4} \right ) a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{{d}^{2}ec}{2\,a{e}^{4}+2\,c{d}^{4}}\arctan \left ({x}^{2}\sqrt{{\frac{c}{a}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{2}d\sqrt{2}}{8\,a{e}^{4}+8\,c{d}^{4}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{2}d\sqrt{2}}{4\,a{e}^{4}+4\,c{d}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{{e}^{2}d\sqrt{2}}{4\,a{e}^{4}+4\,c{d}^{4}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{{e}^{3}\ln \left ( c{x}^{4}+a \right ) }{4\,a{e}^{4}+4\,c{d}^{4}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) }{a{e}^{4}+c{d}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^4+a),x)

[Out]

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

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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(1/((c*x^4 + a)*(e*x + d)),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(1/((c*x^4 + a)*(e*x + d)),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(1/(e*x+d)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.292374, size = 520, normalized size = 1.25 \[ \frac{\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} - 2 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e - \sqrt{2} \sqrt{a c} a c^{2} e^{2}\right )}} + \frac{\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a c^{3} d^{2} + 2 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e - \sqrt{2} \sqrt{a c} a c^{2} e^{2}\right )}} + \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{4} + \sqrt{2} a^{2} c^{2} e^{4}\right )}} - \frac{{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac{3}{4}} d e^{2}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{4 \,{\left (\sqrt{2} a c^{3} d^{4} + \sqrt{2} a^{2} c^{2} e^{4}\right )}} - \frac{e^{3}{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \,{\left (c d^{4} + a e^{4}\right )}} + \frac{e^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{4} e + a e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*(e*x + d)),x, algorithm="giac")

[Out]

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