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

Optimal. Leaf size=305 \[ -\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{3} \sqrt{c} d\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{c} d-\sqrt{3} \sqrt{a} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]

[Out]

(d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*c^(1/6)) - ((Sqrt[c]*d + Sqrt[3]*Sqrt
[a]*e)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)])/(6*a^(5/6)*c^(2/3)) + ((Sqrt[c]*
d - Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)])/(6*a^(5/6)*c^(2/
3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((Sqrt[3]*Sqrt[c]*d -
Sqrt[a]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*
c^(2/3)) + ((Sqrt[3]*Sqrt[c]*d + Sqrt[a]*e)*Log[a^(1/3) + Sqrt[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 = 0.559384, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{3} \sqrt{c} d\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{c} d-\sqrt{3} \sqrt{a} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^3)/(a + c*x^6),x]

[Out]

(d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*c^(1/6)) - ((Sqrt[c]*d + Sqrt[3]*Sqrt
[a]*e)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)])/(6*a^(5/6)*c^(2/3)) + ((Sqrt[c]*
d - Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)])/(6*a^(5/6)*c^(2/
3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((Sqrt[3]*Sqrt[c]*d -
Sqrt[a]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*
c^(2/3)) + ((Sqrt[3]*Sqrt[c]*d + Sqrt[a]*e)*Log[a^(1/3) + Sqrt[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 = 114.599, size = 309, normalized size = 1.01 \[ - \frac{e \log{\left (\sqrt [3]{a} + \sqrt [3]{c} x^{2} \right )}}{6 \sqrt [3]{a} c^{\frac{2}{3}}} + \frac{\left (\sqrt{a} e - \sqrt{3} \sqrt{c} d\right ) \log{\left (1 + \frac{\sqrt [3]{c} x^{2}}{\sqrt [3]{a}} - \frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}} \right )}}{12 a^{\frac{5}{6}} c^{\frac{2}{3}}} + \frac{\left (\sqrt{a} e + \sqrt{3} \sqrt{c} d\right ) \log{\left (1 + \frac{\sqrt [3]{c} x^{2}}{\sqrt [3]{a}} + \frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}} \right )}}{12 a^{\frac{5}{6}} c^{\frac{2}{3}}} + \frac{\left (- \sqrt{3} \sqrt{a} e + \sqrt{c} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [6]{a} + \frac{2 \sqrt{3} \sqrt [6]{c} x}{3}\right )}{\sqrt [6]{a}} \right )}}{6 a^{\frac{5}{6}} c^{\frac{2}{3}}} - \frac{\left (\sqrt{3} \sqrt{a} e + \sqrt{c} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [6]{a} - \frac{2 \sqrt{3} \sqrt [6]{c} x}{3}\right )}{\sqrt [6]{a}} \right )}}{6 a^{\frac{5}{6}} c^{\frac{2}{3}}} + \frac{d \operatorname{atan}{\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}} \right )}}{3 a^{\frac{5}{6}} \sqrt [6]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.169395, size = 334, normalized size = 1.1 \[ -\frac{\left (\sqrt{3} \sqrt [6]{a} \sqrt{c} d-a^{2/3} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}}-\frac{\left (-a^{2/3} e-\sqrt{3} \sqrt [6]{a} \sqrt{c} d\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}}+\frac{\left (\sqrt{3} a^{2/3} e+\sqrt [6]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x-\sqrt{3} \sqrt [6]{a}}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}+\frac{\left (\sqrt [6]{a} \sqrt{c} d-\sqrt{3} a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.111, size = 329, normalized size = 1.1 \[{\frac{c\sqrt{3}d}{12\,{a}^{2}} \left ({\frac{a}{c}} \right ) ^{{\frac{7}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{e}{12\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{d}{6\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+\sqrt{3} \right ) }-{\frac{\sqrt{3}e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+\sqrt{3} \right ) }-{\frac{e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{d}{3\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}} \right ) }+{\frac{e}{12\,a}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}}-{\frac{\sqrt{3}d}{12\,a}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) \sqrt [6]{{\frac{a}{c}}}}+{\frac{\sqrt{3}e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-\sqrt{3} \right ) }+{\frac{d}{6\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^3+d)/(c*x^6+a),x)

[Out]

1/12*c*(1/c*a)^(7/6)/a^2*ln(x^2+3^(1/2)*(1/c*a)^(1/6)*x+(1/c*a)^(1/3))*3^(1/2)*d
+1/12*(1/c*a)^(2/3)/a*ln(x^2+3^(1/2)*(1/c*a)^(1/6)*x+(1/c*a)^(1/3))*e+1/6*(1/c*a
)^(1/6)/a*arctan(2*x/(1/c*a)^(1/6)+3^(1/2))*d-1/6*(1/c*a)^(2/3)/a*arctan(2*x/(1/
c*a)^(1/6)+3^(1/2))*3^(1/2)*e-1/6*(1/c*a)^(2/3)/a*e*ln(x^2+(1/c*a)^(1/3))+1/3*(1
/c*a)^(1/6)/a*d*arctan(x/(1/c*a)^(1/6))+1/12/a*ln(x^2-3^(1/2)*(1/c*a)^(1/6)*x+(1
/c*a)^(1/3))*(1/c*a)^(2/3)*e-1/12/a*ln(x^2-3^(1/2)*(1/c*a)^(1/6)*x+(1/c*a)^(1/3)
)*3^(1/2)*(1/c*a)^(1/6)*d+1/6/a*(1/c*a)^(2/3)*arctan(2*x/(1/c*a)^(1/6)-3^(1/2))*
3^(1/2)*e+1/6/a*(1/c*a)^(1/6)*arctan(2*x/(1/c*a)^(1/6)-3^(1/2))*d

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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 = 0.358673, size = 4134, normalized size = 13.55 \[ \text{result too large to display} \]

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*sq
rt(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*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)) + 1/6*((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 = 7.3355, size = 165, normalized size = 0.54 \[ \operatorname{RootSum}{\left (46656 t^{6} a^{5} c^{4} + t^{3} \left (432 a^{4} c^{2} e^{3} - 1296 a^{3} c^{3} d^{2} e\right ) + a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}, \left ( t \mapsto t \log{\left (x + \frac{- 1296 t^{4} a^{4} c^{2} e - 6 t a^{3} e^{4} + 36 t a^{2} c d^{2} e^{2} - 6 t a c^{2} d^{4}}{3 a^{2} d e^{4} + 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} \]

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 = 0.29698, size = 397, normalized size = 1.3 \[ \frac{\left (a c^{5}\right )^{\frac{1}{6}} d \arctan \left (\frac{x}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{3 \, a c} - \frac{\left (a c^{5}\right )^{\frac{2}{3}}{\left | c \right |} e{\rm ln}\left (x^{2} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{6 \, a c^{5}} + \frac{{\left (\left (a c^{5}\right )^{\frac{1}{6}} c^{3} d - \sqrt{3} \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{a}{c}\right )^{\frac{1}{6}}}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} + \frac{{\left (\left (a c^{5}\right )^{\frac{1}{6}} c^{3} d + \sqrt{3} \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{a}{c}\right )^{\frac{1}{6}}}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} + \frac{{\left (\sqrt{3} \left (a c^{5}\right )^{\frac{1}{6}} c^{3} d + \left (a c^{5}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} + \sqrt{3} x \left (\frac{a}{c}\right )^{\frac{1}{6}} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} - \frac{{\left (\sqrt{3} \left (a c^{5}\right )^{\frac{1}{6}} c^{3} d - \left (a c^{5}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} - \sqrt{3} x \left (\frac{a}{c}\right )^{\frac{1}{6}} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} \]

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)