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

Optimal. Leaf size=311 \[ \frac{\left (\sqrt{3} \sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac{\left (\sqrt{3} \sqrt{a} d-\sqrt{c} e\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}+\frac{\left (\sqrt{a} d-\sqrt{3} \sqrt{c} e\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac{\left (\sqrt{a} d+\sqrt{3} \sqrt{c} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac{\sqrt [6]{a} d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac{d x}{c} \]

[Out]

(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((Sqrt[a]*d - Sq
rt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)])/(6*a^(1/3)*c^(7/6)) -
((Sqrt[a]*d + Sqrt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)])/(6*a^(
1/3)*c^(7/6)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) + ((Sqrt[3]*S
qrt[a]*d + Sqrt[c]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(1
2*a^(1/3)*c^(7/6)) - ((Sqrt[3]*Sqrt[a]*d - Sqrt[c]*e)*Log[a^(1/3) + Sqrt[3]*a^(1
/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(1/3)*c^(7/6))

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Rubi [A]  time = 0.639059, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{\left (\sqrt{3} \sqrt{a} d+\sqrt{c} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac{\left (\sqrt{3} \sqrt{a} d-\sqrt{c} e\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}+\frac{\left (\sqrt{a} d-\sqrt{3} \sqrt{c} e\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac{\left (\sqrt{a} d+\sqrt{3} \sqrt{c} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac{\sqrt [6]{a} d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac{d x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e/x^3)/(c + a/x^6),x]

[Out]

(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((Sqrt[a]*d - Sq
rt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)])/(6*a^(1/3)*c^(7/6)) -
((Sqrt[a]*d + Sqrt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)])/(6*a^(
1/3)*c^(7/6)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) + ((Sqrt[3]*S
qrt[a]*d + Sqrt[c]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(1
2*a^(1/3)*c^(7/6)) - ((Sqrt[3]*Sqrt[a]*d - Sqrt[c]*e)*Log[a^(1/3) + Sqrt[3]*a^(1
/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(1/3)*c^(7/6))

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Rubi in Sympy [A]  time = 124.579, size = 314, normalized size = 1.01 \[ - \frac{\sqrt [6]{a} d \operatorname{atan}{\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}} \right )}}{3 c^{\frac{7}{6}}} + \frac{d x}{c} - \frac{e \log{\left (\sqrt [3]{a} + \sqrt [3]{c} x^{2} \right )}}{6 \sqrt [3]{a} c^{\frac{2}{3}}} + \frac{\left (\sqrt{a} d - \sqrt{3} \sqrt{c} e\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [6]{a} - \frac{2 \sqrt{3} \sqrt [6]{c} x}{3}\right )}{\sqrt [6]{a}} \right )}}{6 \sqrt [3]{a} c^{\frac{7}{6}}} - \frac{\left (\sqrt{a} d + \sqrt{3} \sqrt{c} e\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\sqrt [6]{a} + \frac{2 \sqrt{3} \sqrt [6]{c} x}{3}\right )}{\sqrt [6]{a}} \right )}}{6 \sqrt [3]{a} c^{\frac{7}{6}}} + \frac{\left (- \sqrt{3} \sqrt{a} d + \sqrt{c} e\right ) \log{\left (1 + \frac{\sqrt [3]{c} x^{2}}{\sqrt [3]{a}} + \frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}} \right )}}{12 \sqrt [3]{a} c^{\frac{7}{6}}} + \frac{\left (\sqrt{3} \sqrt{a} d + \sqrt{c} e\right ) \log{\left (1 + \frac{\sqrt [3]{c} x^{2}}{\sqrt [3]{a}} - \frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}} \right )}}{12 \sqrt [3]{a} c^{\frac{7}{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/6)*d*atan(c**(1/6)*x/a**(1/6))/(3*c**(7/6)) + d*x/c - e*log(a**(1/3) + c*
*(1/3)*x**2)/(6*a**(1/3)*c**(2/3)) + (sqrt(a)*d - sqrt(3)*sqrt(c)*e)*atan(sqrt(3
)*(a**(1/6) - 2*sqrt(3)*c**(1/6)*x/3)/a**(1/6))/(6*a**(1/3)*c**(7/6)) - (sqrt(a)
*d + sqrt(3)*sqrt(c)*e)*atan(sqrt(3)*(a**(1/6) + 2*sqrt(3)*c**(1/6)*x/3)/a**(1/6
))/(6*a**(1/3)*c**(7/6)) + (-sqrt(3)*sqrt(a)*d + sqrt(c)*e)*log(1 + c**(1/3)*x**
2/a**(1/3) + sqrt(3)*c**(1/6)*x/a**(1/6))/(12*a**(1/3)*c**(7/6)) + (sqrt(3)*sqrt
(a)*d + sqrt(c)*e)*log(1 + c**(1/3)*x**2/a**(1/3) - sqrt(3)*c**(1/6)*x/a**(1/6))
/(12*a**(1/3)*c**(7/6))

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Mathematica [A]  time = 0.210588, size = 346, normalized size = 1.11 \[ -\frac{\left (-\sqrt{3} a^{7/6} \sqrt{c} d-a^{2/3} c e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}}-\frac{\left (\sqrt{3} a^{7/6} \sqrt{c} d-a^{2/3} c e\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}}+\frac{\left (\sqrt{3} a^{2/3} c e-a^{7/6} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x-\sqrt{3} \sqrt [6]{a}}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}+\frac{\left (a^{7/6} \left (-\sqrt{c}\right ) d-\sqrt{3} a^{2/3} c e\right ) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}-\frac{\sqrt [6]{a} d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac{d x}{c} \]

Antiderivative was successfully verified.

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

[Out]

(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((-(a^(7/6)*Sqrt
[c]*d) + Sqrt[3]*a^(2/3)*c*e)*ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*c^(1/6)*x)/a^(1/6)]
)/(6*a*c^(5/3)) + ((-(a^(7/6)*Sqrt[c]*d) - Sqrt[3]*a^(2/3)*c*e)*ArcTan[(Sqrt[3]*
a^(1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(5/3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/
(6*a^(1/3)*c^(2/3)) - ((-(Sqrt[3]*a^(7/6)*Sqrt[c]*d) - a^(2/3)*c*e)*Log[a^(1/3)
- Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(5/3)) - ((Sqrt[3]*a^(7/6)*S
qrt[c]*d - a^(2/3)*c*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/
(12*a*c^(5/3))

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Maple [A]  time = 0.07, size = 334, normalized size = 1.1 \[{\frac{dx}{c}}-{\frac{\sqrt{3}d}{12\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{7}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{e}{12\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }-{\frac{d}{6\,c}\sqrt [6]{{\frac{a}{c}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+\sqrt{3} \right ) }-{\frac{\sqrt{3}e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+\sqrt{3} \right ) }-{\frac{e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{c}}} \right ) }-{\frac{d}{3\,c}\sqrt [6]{{\frac{a}{c}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}} \right ) }+{\frac{e}{12\,a}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}}+{\frac{\sqrt{3}d}{12\,c}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) \sqrt [6]{{\frac{a}{c}}}}+{\frac{\sqrt{3}e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-\sqrt{3} \right ) }-{\frac{d}{6\,c}\sqrt [6]{{\frac{a}{c}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e/x^3)/(c+a/x^6),x)

[Out]

1/c*d*x-1/12*(1/c*a)^(7/6)/a*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/c*
(1/c*a)^(1/6)*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/c*(1/c*a)^(1/6)*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/c*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/c*(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((d + e/x^3)/(c + a/x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.349835, size = 4070, normalized size = 13.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.90937, size = 167, normalized size = 0.54 \[ \operatorname{RootSum}{\left (46656 t^{6} a^{2} c^{7} + t^{3} \left (- 1296 a^{2} c^{4} d^{2} e + 432 a c^{5} e^{3}\right ) + a^{3} d^{6} + 3 a^{2} c d^{4} e^{2} + 3 a c^{2} d^{2} e^{4} + c^{3} e^{6}, \left ( t \mapsto t \log{\left (x + \frac{- 1296 t^{4} a c^{5} e - 6 t a^{2} c d^{4} + 36 t a c^{2} d^{2} e^{2} - 6 t c^{3} e^{4}}{a^{2} d^{5} - 2 a c d^{3} e^{2} - 3 c^{2} d e^{4}} \right )} \right )\right )} + \frac{d x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(46656*_t**6*a**2*c**7 + _t**3*(-1296*a**2*c**4*d**2*e + 432*a*c**5*e**3)
+ a**3*d**6 + 3*a**2*c*d**4*e**2 + 3*a*c**2*d**2*e**4 + c**3*e**6, Lambda(_t, _
t*log(x + (-1296*_t**4*a*c**5*e - 6*_t*a**2*c*d**4 + 36*_t*a*c**2*d**2*e**2 - 6*
_t*c**3*e**4)/(a**2*d**5 - 2*a*c*d**3*e**2 - 3*c**2*d*e**4)))) + d*x/c

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GIAC/XCAS [A]  time = 0.324781, size = 406, normalized size = 1.31 \[ \frac{d x}{c} - \frac{\left (a c^{5}\right )^{\frac{1}{6}} d \arctan \left (\frac{x}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{3 \, c^{2}} - \frac{\left (a c^{5}\right )^{\frac{2}{3}}{\left | c \right |} e{\rm ln}\left (x^{2} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{6 \, a c^{5}} - \frac{{\left (\left (a c^{5}\right )^{\frac{1}{6}} a c^{2} d + \sqrt{3} \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{a}{c}\right )^{\frac{1}{6}}}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} - \frac{{\left (\left (a c^{5}\right )^{\frac{1}{6}} a c^{2} d - \sqrt{3} \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{a}{c}\right )^{\frac{1}{6}}}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} - \frac{{\left (\sqrt{3} \left (a c^{5}\right )^{\frac{1}{6}} a c^{2} d - \left (a c^{5}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} + \sqrt{3} x \left (\frac{a}{c}\right )^{\frac{1}{6}} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} + \frac{{\left (\sqrt{3} \left (a c^{5}\right )^{\frac{1}{6}} a c^{2} d + \left (a c^{5}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} - \sqrt{3} x \left (\frac{a}{c}\right )^{\frac{1}{6}} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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