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))

________________________________________________________________________________________

Rubi [A]  time = 0.248568, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {1416, 635, 203, 260, 634, 617, 204, 628} \[ -\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{3} \sqrt{c} d\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{c} d-\sqrt{3} \sqrt{a} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]

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))

Rule 1416

Int[((d_) + (e_.)*(x_)^3)/((a_) + (c_.)*(x_)^6), x_Symbol] :> With[{q = Rt[c/a, 6]}, Dist[1/(3*a*q^2), Int[(q^
2*d - e*x)/(1 + q^2*x^2), x], x] + (Dist[1/(6*a*q^2), Int[(2*q^2*d - (Sqrt[3]*q^3*d - e)*x)/(1 - Sqrt[3]*q*x +
q^2*x^2), x], x] + Dist[1/(6*a*q^2), Int[(2*q^2*d + (Sqrt[3]*q^3*d + e)*x)/(1 + Sqrt[3]*q*x + q^2*x^2), x], x
])] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a]

Rule 635

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

Rule 203

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

Rule 260

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

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{d+e x^3}{a+c x^6} \, dx &=\frac{\int \frac{\frac{2 \sqrt [3]{c} d}{\sqrt [3]{a}}-\left (\frac{\sqrt{3} \sqrt{c} d}{\sqrt{a}}-e\right ) x}{1-\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} \sqrt [3]{c}}+\frac{\int \frac{\frac{2 \sqrt [3]{c} d}{\sqrt [3]{a}}+\left (\frac{\sqrt{3} \sqrt{c} d}{\sqrt{a}}+e\right ) x}{1+\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} \sqrt [3]{c}}+\frac{\int \frac{\frac{\sqrt [3]{c} d}{\sqrt [3]{a}}-e x}{1+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} \sqrt [3]{c}}\\ &=\frac{d \int \frac{1}{1+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a}-\frac{e \int \frac{x}{1+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} \sqrt [3]{c}}-\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \int \frac{-\frac{\sqrt{3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac{2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1-\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+\sqrt{a} e\right ) \int \frac{\frac{\sqrt{3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac{2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1+\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a^{5/6} c^{2/3}}+\frac{\left (d-\frac{\sqrt{3} \sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{1+\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a}+\frac{\left (d+\frac{\sqrt{3} \sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{1-\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a}\\ &=\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}}-\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+\sqrt{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{c} d-3 \sqrt{a} e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{18 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+3 \sqrt{a} e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{18 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{\left (\sqrt{c} d+\sqrt{3} \sqrt{a} e\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 (\sqrt{3}+\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+\sqrt{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0981448, 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))

________________________________________________________________________________________

Maple [A]  time = 0.076, size = 329, normalized size = 1.1 \begin{align*}{\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}{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 ) }-{\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 ) } \end{align*}

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 2.60776, size = 6589, normalized size = 21.6 \begin{align*} \text{result too large to display} \end{align*}

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*sqr
t(-(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)) - 1/12*((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))

________________________________________________________________________________________

Sympy [A]  time = 2.13879, size = 165, normalized size = 0.54 \begin{align*} \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{align*}

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))))

________________________________________________________________________________________

Giac [A]  time = 1.12413, size = 397, normalized size = 1.3 \begin{align*} \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 \log \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 )} \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{align*}

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*log(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)*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)