3.2 \(\int \frac{d+e x^3}{a-c x^6} \, dx\)

Optimal. Leaf size=323 \[ \frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{2 \sqrt{3} a^{5/6} c^{2/3}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{2 \sqrt{3} a^{5/6} \sqrt [6]{c}} \]

[Out]

-((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[(a^(1/6) - 2*c^(1/6)*x)/(Sqrt[3]*a^(1/6))])/(
2*Sqrt[3]*a^(5/6)*c^(1/6)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(a^(1/6) + 2*c^(1/6
)*x)/(Sqrt[3]*a^(1/6))])/(2*Sqrt[3]*a^(5/6)*c^(2/3)) - ((Sqrt[c]*d + Sqrt[a]*e)*
Log[a^(1/6) - c^(1/6)*x])/(6*a^(5/6)*c^(2/3)) + ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a
^(1/6) + c^(1/6)*x])/(6*a^(5/6)*c^(1/6)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/3
) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(1/6)) + ((Sqrt[c]*d + Sqrt[
a]*e)*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3))

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Rubi [A]  time = 0.415444, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{2 \sqrt{3} a^{5/6} c^{2/3}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{2 \sqrt{3} a^{5/6} \sqrt [6]{c}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^3)/(a - c*x^6),x]

[Out]

-((d - (Sqrt[a]*e)/Sqrt[c])*ArcTan[(a^(1/6) - 2*c^(1/6)*x)/(Sqrt[3]*a^(1/6))])/(
2*Sqrt[3]*a^(5/6)*c^(1/6)) + ((Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(a^(1/6) + 2*c^(1/6
)*x)/(Sqrt[3]*a^(1/6))])/(2*Sqrt[3]*a^(5/6)*c^(2/3)) - ((Sqrt[c]*d + Sqrt[a]*e)*
Log[a^(1/6) - c^(1/6)*x])/(6*a^(5/6)*c^(2/3)) + ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a
^(1/6) + c^(1/6)*x])/(6*a^(5/6)*c^(1/6)) - ((d - (Sqrt[a]*e)/Sqrt[c])*Log[a^(1/3
) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(1/6)) + ((Sqrt[c]*d + Sqrt[
a]*e)*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3))

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Rubi in Sympy [A]  time = 75.1308, size = 311, normalized size = 0.96 \[ - \frac{\left (\sqrt{a} e - \sqrt{c} d\right ) \log{\left (\sqrt [6]{a} + \sqrt [6]{c} x \right )}}{6 a^{\frac{5}{6}} c^{\frac{2}{3}}} + \frac{\left (\sqrt{a} e - \sqrt{c} d\right ) \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{c} x^{2} - \sqrt{a} \sqrt [6]{c} x \right )}}{12 a^{\frac{5}{6}} c^{\frac{2}{3}}} + \frac{\sqrt{3} \left (\sqrt{a} e - \sqrt{c} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [6]{a}}{3} - \frac{2 \sqrt [6]{c} x}{3}\right )}{\sqrt [6]{a}} \right )}}{6 a^{\frac{5}{6}} c^{\frac{2}{3}}} - \frac{\left (\sqrt{a} e + \sqrt{c} d\right ) \log{\left (\sqrt [6]{a} - \sqrt [6]{c} x \right )}}{6 a^{\frac{5}{6}} c^{\frac{2}{3}}} + \frac{\left (\sqrt{a} e + \sqrt{c} d\right ) \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{c} x^{2} + \sqrt{a} \sqrt [6]{c} x \right )}}{12 a^{\frac{5}{6}} c^{\frac{2}{3}}} + \frac{\sqrt{3} \left (\sqrt{a} e + \sqrt{c} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [6]{a}}{3} + \frac{2 \sqrt [6]{c} x}{3}\right )}{\sqrt [6]{a}} \right )}}{6 a^{\frac{5}{6}} c^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**3+d)/(-c*x**6+a),x)

[Out]

-(sqrt(a)*e - sqrt(c)*d)*log(a**(1/6) + c**(1/6)*x)/(6*a**(5/6)*c**(2/3)) + (sqr
t(a)*e - sqrt(c)*d)*log(a**(2/3) + a**(1/3)*c**(1/3)*x**2 - sqrt(a)*c**(1/6)*x)/
(12*a**(5/6)*c**(2/3)) + sqrt(3)*(sqrt(a)*e - sqrt(c)*d)*atan(sqrt(3)*(a**(1/6)/
3 - 2*c**(1/6)*x/3)/a**(1/6))/(6*a**(5/6)*c**(2/3)) - (sqrt(a)*e + sqrt(c)*d)*lo
g(a**(1/6) - c**(1/6)*x)/(6*a**(5/6)*c**(2/3)) + (sqrt(a)*e + sqrt(c)*d)*log(a**
(2/3) + a**(1/3)*c**(1/3)*x**2 + sqrt(a)*c**(1/6)*x)/(12*a**(5/6)*c**(2/3)) + sq
rt(3)*(sqrt(a)*e + sqrt(c)*d)*atan(sqrt(3)*(a**(1/6)/3 + 2*c**(1/6)*x/3)/a**(1/6
))/(6*a**(5/6)*c**(2/3))

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Mathematica [A]  time = 0.189456, size = 337, normalized size = 1.04 \[ \frac{-2 \sqrt{3} \left (\sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt{3}}\right )+2 \sqrt{3} \left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+1}{\sqrt{3}}\right )-\sqrt{c} d \log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )+\sqrt{c} d \log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )-2 \sqrt{c} d \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )+2 \sqrt{c} d \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )+\sqrt{a} e \log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )+\sqrt{a} e \log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )-2 \sqrt{a} e \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )-2 \sqrt{a} e \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{12 a^{5/6} c^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^3)/(a - c*x^6),x]

[Out]

(-2*Sqrt[3]*(Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(1 - (2*c^(1/6)*x)/a^(1/6))/Sqrt[3]]
+ 2*Sqrt[3]*(Sqrt[c]*d + Sqrt[a]*e)*ArcTan[(1 + (2*c^(1/6)*x)/a^(1/6))/Sqrt[3]]
- 2*Sqrt[c]*d*Log[a^(1/6) - c^(1/6)*x] - 2*Sqrt[a]*e*Log[a^(1/6) - c^(1/6)*x] +
2*Sqrt[c]*d*Log[a^(1/6) + c^(1/6)*x] - 2*Sqrt[a]*e*Log[a^(1/6) + c^(1/6)*x] - Sq
rt[c]*d*Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + Sqrt[a]*e*Log[a^(1/3) -
a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] + Sqrt[c]*d*Log[a^(1/3) + a^(1/6)*c^(1/6)*x +
c^(1/3)*x^2] + Sqrt[a]*e*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(
5/6)*c^(2/3))

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Maple [A]  time = 0.109, size = 380, normalized size = 1.2 \[{\frac{e}{12\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}-\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }-{\frac{e\sqrt{3}}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{2\,x\sqrt{3}}{3}{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-{\frac{\sqrt{3}}{3}} \right ) }-{\frac{d}{12\,a}\sqrt [6]{{\frac{a}{c}}}\ln \left ({x}^{2}-\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{d\sqrt{3}}{6\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ({\frac{2\,x\sqrt{3}}{3}{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-{\frac{\sqrt{3}}{3}} \right ) }-{\frac{e}{6\,c}\ln \left ( x+\sqrt [6]{{\frac{a}{c}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{c}}}}}}+{\frac{d}{6\,c}\ln \left ( x+\sqrt [6]{{\frac{a}{c}}} \right ) \left ({\frac{a}{c}} \right ) ^{-{\frac{5}{6}}}}-{\frac{e}{6\,c}\ln \left ( -x+\sqrt [6]{{\frac{a}{c}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{c}}}}}}-{\frac{d}{6\,c}\ln \left ( -x+\sqrt [6]{{\frac{a}{c}}} \right ) \left ({\frac{a}{c}} \right ) ^{-{\frac{5}{6}}}}+{\frac{e}{12\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{e\sqrt{3}}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{2\,x\sqrt{3}}{3}{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+{\frac{\sqrt{3}}{3}} \right ) }+{\frac{d}{12\,a}\sqrt [6]{{\frac{a}{c}}}\ln \left ({x}^{2}+\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{d\sqrt{3}}{6\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ({\frac{2\,x\sqrt{3}}{3}{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+{\frac{\sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^3+d)/(-c*x^6+a),x)

[Out]

1/12/a*(1/c*a)^(2/3)*e*ln(x^2-(1/c*a)^(1/6)*x+(1/c*a)^(1/3))-1/6/a*(1/c*a)^(2/3)
*e*3^(1/2)*arctan(2/3*3^(1/2)/(1/c*a)^(1/6)*x-1/3*3^(1/2))-1/12/a*d*(1/c*a)^(1/6
)*ln(x^2-(1/c*a)^(1/6)*x+(1/c*a)^(1/3))+1/6/a*d*(1/c*a)^(1/6)*3^(1/2)*arctan(2/3
*3^(1/2)/(1/c*a)^(1/6)*x-1/3*3^(1/2))-1/6/c/(1/c*a)^(1/3)*ln(x+(1/c*a)^(1/6))*e+
1/6/c/(1/c*a)^(5/6)*ln(x+(1/c*a)^(1/6))*d-1/6/c/(1/c*a)^(1/3)*ln(-x+(1/c*a)^(1/6
))*e-1/6/c/(1/c*a)^(5/6)*ln(-x+(1/c*a)^(1/6))*d+1/12/a*(1/c*a)^(2/3)*e*ln(x^2+(1
/c*a)^(1/6)*x+(1/c*a)^(1/3))+1/6/a*(1/c*a)^(2/3)*e*3^(1/2)*arctan(2/3*3^(1/2)/(1
/c*a)^(1/6)*x+1/3*3^(1/2))+1/12/a*d*(1/c*a)^(1/6)*ln(x^2+(1/c*a)^(1/6)*x+(1/c*a)
^(1/3))+1/6/a*d*(1/c*a)^(1/6)*3^(1/2)*arctan(2/3*3^(1/2)/(1/c*a)^(1/6)*x+1/3*3^(
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^3 + d)/(c*x^6 - a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.377304, size = 4082, normalized size = 12.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x^3 + d)/(c*x^6 - a),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)
) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)*arctan(-(sqrt(3)*a^4*c^2*e*sqrt((c^2*d^6
+ 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - sqrt(3)*(a*c^2*d^4 + 3*a^2*c*d^2*
e^2))*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c
*d^2*e + a*e^3)/(a^2*c^2))^(1/3)/(2*(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x +
2*(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*sqrt(((c^3*d^7 + a*c^2*d^5*e^2 - 5*a^2
*c*d^3*e^4 + 3*a^3*d*e^6)*x^2 - (2*a^5*c^3*d*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9
*a^2*d^2*e^4)/(a^5*c^3)) - a^2*c^3*d^5 - 4*a^3*c^2*d^3*e^2 - 3*a^4*c*d*e^4)*(-(a
^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a
*e^3)/(a^2*c^2))^(2/3) + ((a^4*c^3*d^2*e - a^5*c^2*e^3)*x*sqrt((c^2*d^6 + 6*a*c*
d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - (a*c^3*d^6 + 2*a^2*c^2*d^4*e^2 - 3*a^3*c*d
^2*e^4)*x)*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3))
+ 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3))/(c^3*d^7 + a*c^2*d^5*e^2 - 5*a^2*c*d^3*e^
4 + 3*a^3*d*e^6)) + (a^4*c^2*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a
^5*c^3)) - a*c^2*d^4 - 3*a^2*c*d^2*e^2)*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2
+ 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3))) + 1/3*sqrt(
3)*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2
*e - a*e^3)/(a^2*c^2))^(1/3)*arctan(-(sqrt(3)*a^4*c^2*e*sqrt((c^2*d^6 + 6*a*c*d^
4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + sqrt(3)*(a*c^2*d^4 + 3*a^2*c*d^2*e^2))*((a^2
*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e - a*e
^3)/(a^2*c^2))^(1/3)/(2*(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 2*(c^2*d^5 +
2*a*c*d^3*e^2 - 3*a^2*d*e^4)*sqrt(((c^3*d^7 + a*c^2*d^5*e^2 - 5*a^2*c*d^3*e^4 +
3*a^3*d*e^6)*x^2 + (2*a^5*c^3*d*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4
)/(a^5*c^3)) + a^2*c^3*d^5 + 4*a^3*c^2*d^3*e^2 + 3*a^4*c*d*e^4)*((a^2*c^2*sqrt((
c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e - a*e^3)/(a^2*c^
2))^(2/3) - ((a^4*c^3*d^2*e - a^5*c^2*e^3)*x*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a
^2*d^2*e^4)/(a^5*c^3)) + (a*c^3*d^6 + 2*a^2*c^2*d^4*e^2 - 3*a^3*c*d^2*e^4)*x)*((
a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e -
a*e^3)/(a^2*c^2))^(1/3))/(c^3*d^7 + a*c^2*d^5*e^2 - 5*a^2*c*d^3*e^4 + 3*a^3*d*e^
6)) - (a^4*c^2*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + a*c
^2*d^4 + 3*a^2*c*d^2*e^2)*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^
4)/(a^5*c^3)) - 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3))) - 1/12*(-(a^2*c^2*sqrt((c^
2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2)
)^(1/3)*log((c^3*d^7 + a*c^2*d^5*e^2 - 5*a^2*c*d^3*e^4 + 3*a^3*d*e^6)*x^2 - (2*a
^5*c^3*d*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - a^2*c^3*d
^5 - 4*a^3*c^2*d^3*e^2 - 3*a^4*c*d*e^4)*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2
+ 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(2/3) + ((a^4*c^3*d
^2*e - a^5*c^2*e^3)*x*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3))
- (a*c^3*d^6 + 2*a^2*c^2*d^4*e^2 - 3*a^3*c*d^2*e^4)*x)*(-(a^2*c^2*sqrt((c^2*d^6
+ 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3
)) - 1/12*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) -
3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3)*log((c^3*d^7 + a*c^2*d^5*e^2 - 5*a^2*c*d^3*e
^4 + 3*a^3*d*e^6)*x^2 + (2*a^5*c^3*d*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2
*e^4)/(a^5*c^3)) + a^2*c^3*d^5 + 4*a^3*c^2*d^3*e^2 + 3*a^4*c*d*e^4)*((a^2*c^2*sq
rt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e - a*e^3)/(a^
2*c^2))^(2/3) - ((a^4*c^3*d^2*e - a^5*c^2*e^3)*x*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 +
9*a^2*d^2*e^4)/(a^5*c^3)) + (a*c^3*d^6 + 2*a^2*c^2*d^4*e^2 - 3*a^3*c*d^2*e^4)*x
)*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*
e - a*e^3)/(a^2*c^2))^(1/3)) + 1/6*(-(a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*
a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 + 2*
a*c*d^3*e^2 - 3*a^2*d*e^4)*x + (a^4*c^2*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*
d^2*e^4)/(a^5*c^3)) - a*c^2*d^4 - 3*a^2*c*d^2*e^2)*(-(a^2*c^2*sqrt((c^2*d^6 + 6*
a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)) +
1/6*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d
^2*e - a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x -
(a^4*c^2*e*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + a*c^2*d^4
+ 3*a^2*c*d^2*e^2)*((a^2*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^
5*c^3)) - 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3))

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Sympy [A]  time = 7.56856, size = 168, normalized size = 0.52 \[ - \operatorname{RootSum}{\left (46656 t^{6} a^{5} c^{4} + t^{3} \left (- 432 a^{4} c^{2} e^{3} - 1296 a^{3} c^{3} d^{2} e\right ) + a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}, \left ( t \mapsto t \log{\left (x + \frac{- 1296 t^{4} a^{4} c^{2} e + 6 t a^{3} e^{4} + 36 t a^{2} c d^{2} e^{2} + 6 t a c^{2} d^{4}}{3 a^{2} d e^{4} - 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**3+d)/(-c*x**6+a),x)

[Out]

-RootSum(46656*_t**6*a**5*c**4 + _t**3*(-432*a**4*c**2*e**3 - 1296*a**3*c**3*d**
2*e) + a**3*e**6 - 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 - c**3*d**6, Lambda(_
t, _t*log(x + (-1296*_t**4*a**4*c**2*e + 6*_t*a**3*e**4 + 36*_t*a**2*c*d**2*e**2
+ 6*_t*a*c**2*d**4)/(3*a**2*d*e**4 - 2*a*c*d**3*e**2 - c**2*d**5))))

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GIAC/XCAS [A]  time = 0.281019, size = 424, normalized size = 1.31 \[ \frac{\left (-a c^{5}\right )^{\frac{1}{6}} d \arctan \left (\frac{x}{\left (-\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{3 \, a c} - \frac{\left (-a c^{5}\right )^{\frac{2}{3}}{\left | c \right |} e{\rm ln}\left (x^{2} + \left (-\frac{a}{c}\right )^{\frac{1}{3}}\right )}{6 \, a c^{5}} + \frac{{\left (\left (-a c^{5}\right )^{\frac{1}{6}} c^{3} d - \sqrt{3} \left (-a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x + \sqrt{3} \left (-\frac{a}{c}\right )^{\frac{1}{6}}}{\left (-\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} + \frac{{\left (\left (-a c^{5}\right )^{\frac{1}{6}} c^{3} d + \sqrt{3} \left (-a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x - \sqrt{3} \left (-\frac{a}{c}\right )^{\frac{1}{6}}}{\left (-\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} + \frac{{\left (\sqrt{3} \left (-a c^{5}\right )^{\frac{1}{6}} c^{3} d + \left (-a c^{5}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} + \sqrt{3} x \left (-\frac{a}{c}\right )^{\frac{1}{6}} + \left (-\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} - \frac{{\left (\sqrt{3} \left (-a c^{5}\right )^{\frac{1}{6}} c^{3} d - \left (-a c^{5}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} - \sqrt{3} x \left (-\frac{a}{c}\right )^{\frac{1}{6}} + \left (-\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x^3 + d)/(c*x^6 - a),x, algorithm="giac")

[Out]

1/3*(-a*c^5)^(1/6)*d*arctan(x/(-a/c)^(1/6))/(a*c) - 1/6*(-a*c^5)^(2/3)*abs(c)*e*
ln(x^2 + (-a/c)^(1/3))/(a*c^5) + 1/6*((-a*c^5)^(1/6)*c^3*d - sqrt(3)*(-a*c^5)^(2
/3)*e)*arctan((2*x + sqrt(3)*(-a/c)^(1/6))/(-a/c)^(1/6))/(a*c^4) + 1/6*((-a*c^5)
^(1/6)*c^3*d + sqrt(3)*(-a*c^5)^(2/3)*e)*arctan((2*x - sqrt(3)*(-a/c)^(1/6))/(-a
/c)^(1/6))/(a*c^4) + 1/12*(sqrt(3)*(-a*c^5)^(1/6)*c^3*d + (-a*c^5)^(2/3)*e)*ln(x
^2 + sqrt(3)*x*(-a/c)^(1/6) + (-a/c)^(1/3))/(a*c^4) - 1/12*(sqrt(3)*(-a*c^5)^(1/
6)*c^3*d - (-a*c^5)^(2/3)*e)*ln(x^2 - sqrt(3)*x*(-a/c)^(1/6) + (-a/c)^(1/3))/(a*
c^4)