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

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

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Rubi [A]  time = 0.19, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1417, 200, 31, 634, 617, 204, 628} \begin {gather*} \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}} \end {gather*}

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)) - ((Sqr
t[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])/(1
2*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))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1417

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[-(a/c), 2]}, Dist[(d + e*q)/
2, Int[1/(a + c*q*x^n), x], x] + Dist[(d - e*q)/2, Int[1/(a - c*q*x^n), x], x]] /; FreeQ[{a, c, d, e, n}, x] &
& EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NegQ[a*c] && IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {d+e x^3}{a-c x^6} \, dx &=\frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a+\sqrt {a} \sqrt {c} x^3} \, dx+\frac {1}{2} \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a-\sqrt {a} \sqrt {c} x^3} \, dx\\ &=\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x} \, dx}{6 a^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x}{a^{2/3}-\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{6 a^{2/3}}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x} \, dx}{6 a^{2/3}}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x}{a^{2/3}+\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{6 a^{2/3}}\\ &=-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 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\right )}{6 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \int \frac {\sqrt {a} \sqrt [6]{c}+2 \sqrt [3]{a} \sqrt [3]{c} x}{a^{2/3}+\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{12 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a^{2/3}-\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{4 \sqrt [3]{a}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {-\sqrt {a} \sqrt [6]{c}+2 \sqrt [3]{a} \sqrt [3]{c} x}{a^{2/3}-\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a^{2/3}+\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{4 \sqrt [3]{a}}\\ &=-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 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\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{2 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{2 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}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\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 (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 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\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 337, normalized size = 1.04 \begin {gather*} \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}} \end {gather*}

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] - Sqrt[c]*d*Lo
g[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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^3}{a-c x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.94, size = 3178, normalized size = 9.84

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(1/3*(2*(sqrt(3)*(a^4*c^4*d^2 + a^5*c^3*e^2)*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^
5*c^3)) - 2*sqrt(3)*(a^2*c^3*d^4*e + 3*a^3*c^2*d^2*e^3))*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^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) - 2*(sqrt(3)*(a^4*c^4*d^2 + a^5*c^3*e^2)*x*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3))
- 2*sqrt(3)*(a^2*c^3*d^4*e + 3*a^3*c^2*d^2*e^3)*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))^(2/3) - sqrt(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))/(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)) - 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(1/3*(2*(sqrt(3)*(a^4*c^4*d
^2 + a^5*c^3*e^2)*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 2*sqrt(3)*(a^2*c^3*d^4*e + 3*a^3
*c^2*d^2*e^3))*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*s
qrt((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^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) - 2*(sqrt(3)*(a^4*c^4*d^2 + a^5*c^3
*e^2)*x*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 2*sqrt(3)*(a^2*c^3*d^4*e + 3*a^3*c^2*d^2*e
^3)*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))^(2/
3) + sqrt(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))/(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)) - 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*sq
rt((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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giac [A]  time = 0.38, size = 308, normalized size = 0.95 \begin {gather*} \frac {{\left | c \right |} e \log \left (x^{2} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (-a c^{5}\right )^{\frac {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 (\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 )} \log \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 )} \log \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 386, normalized size = 1.20 \begin {gather*} \frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, d \arctan \left (\frac {2 \sqrt {3}\, x}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, d \arctan \left (\frac {2 \sqrt {3}\, x}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \ln \left (-x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x -\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, e \arctan \left (\frac {2 \sqrt {3}\, x}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, e \arctan \left (\frac {2 \sqrt {3}\, x}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (-x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x -\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}-\frac {d \ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right )}{6 \left (\frac {a}{c}\right )^{\frac {5}{6}} c}+\frac {d \ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right )}{6 \left (\frac {a}{c}\right )^{\frac {5}{6}} c}-\frac {e \ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right )}{6 \left (\frac {a}{c}\right )^{\frac {1}{3}} c}-\frac {e \ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right )}{6 \left (\frac {a}{c}\right )^{\frac {1}{3}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^3+d)/(-c*x^6+a),x)

[Out]

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

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maxima [A]  time = 1.34, size = 313, normalized size = 0.97 \begin {gather*} \frac {\sqrt {3} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} {\left (\sqrt {c} d - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {c} d + \sqrt {a} e\right )} \log \left (x^{2} + x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{12 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (x^{2} - x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{12 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {c} d + \sqrt {a} e\right )} \log \left (x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*(sqrt(c)*d + sqrt(a)*e)*arctan(1/3*sqrt(3)*(2*x + (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)/sqrt(c))^(1/3)
)/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) + 1/6*sqrt(3)*(sqrt(c)*d - sqrt(a)*e)*arctan(1/3*sqrt(3)*(2*x - (sqrt(a)
/sqrt(c))^(1/3))/(sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) + 1/12*(sqrt(c)*d + sqrt(a)*e)*l
og(x^2 + x*(sqrt(a)/sqrt(c))^(1/3) + (sqrt(a)/sqrt(c))^(2/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) - 1/12*(sqrt
(c)*d - sqrt(a)*e)*log(x^2 - x*(sqrt(a)/sqrt(c))^(1/3) + (sqrt(a)/sqrt(c))^(2/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))
^(2/3)) + 1/6*(sqrt(c)*d - sqrt(a)*e)*log(x + (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3)) - 1
/6*(sqrt(c)*d + sqrt(a)*e)*log(x - (sqrt(a)/sqrt(c))^(1/3))/(sqrt(a)*c*(sqrt(a)/sqrt(c))^(2/3))

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mupad [B]  time = 2.97, size = 1293, normalized size = 4.00 \begin {gather*} \ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}+e\,x\,\sqrt {a^5\,c^5}+a^2\,c^3\,d\,x\right )\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}+\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}-e\,x\,\sqrt {a^5\,c^5}+a^2\,c^3\,d\,x\right )\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}-\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}-2\,e\,x\,\sqrt {a^5\,c^5}-2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}+\ln \left (e\,x\,\sqrt {a^5\,c^5}-\frac {a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}}{2}+a^2\,c^3\,d\,x+\frac {\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}+\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}+2\,e\,x\,\sqrt {a^5\,c^5}-2\,a^2\,c^3\,d\,x-\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}-\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}+2\,e\,x\,\sqrt {a^5\,c^5}-2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {a^5\,c^5}+3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^3)/(a - c*x^6),x)

[Out]

log(a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(
1/3) + e*x*(a^5*c^5)^(1/2) + a^2*c^3*d*x)*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2
*(a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e
- 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) - e*x*(a^5*c^5)^(1/2) + a^2*c^3*d*x)*(-(a^4*c^2*e^3 - c*d^3*(a^5
*c^5)^(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) - log(a^3*c^3*(-(a^4*c^2*e^3 +
c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) - 2*e*x*(a^5*c^5)^(1/2)
+ 3^(1/2)*a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*
c^4))^(1/3)*1i - 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 + 1/2)*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*
e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(e*x*(a^5*c^5)^(1/2) - (a^3*c^3*(-(a^4*c^2*e^3 + c*d^
3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3))/2 + (3^(1/2)*a^3*c^3*(-(a^4
*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i)/2 + a^2*c
^3*d*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a^4*c^2*e^3 + c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e + 3*a*d*e^2*(a^5*c^5)^
(1/2))/(216*a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2
*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) + 2*e*x*(a^5*c^5)^(1/2) - 3^(1/2)*a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5)^
(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i - 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 - 1/
2)*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3)
- log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))
^(1/3) + 2*e*x*(a^5*c^5)^(1/2) + 3^(1/2)*a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(a^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e - 3*
a*d*e^2*(a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i - 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 + 1/2)*(-(a^4*c^2*e^3 - c*d^3*(a
^5*c^5)^(1/2) + 3*a^3*c^3*d^2*e - 3*a*d*e^2*(a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3)

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sympy [A]  time = 3.12, size = 168, normalized size = 0.52 \begin {gather*} - \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 )} \end {gather*}

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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