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)) - ((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))

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Rubi [A]  time = 0.188983, 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, Rules used = {1417, 200, 31, 634, 617, 204, 628} \[ \frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{2 \sqrt{3} a^{5/6} c^{2/3}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac{\left (d-\frac{\sqrt{a} e}{\sqrt{c}}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{2 \sqrt{3} a^{5/6} \sqrt [6]{c}} \]

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)) - ((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 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]

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 31

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

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{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*}

Mathematica [A]  time = 0.110284, 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] - 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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Maple [A]  time = 0.12, size = 386, normalized size = 1.2 \begin{align*}{\frac{e}{12\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ( \sqrt [6]{{\frac{a}{c}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{c}}} \right ) }-{\frac{d}{12\,a}\sqrt [6]{{\frac{a}{c}}}\ln \left ( \sqrt [6]{{\frac{a}{c}}}x-{x}^{2}-\sqrt [3]{{\frac{a}{c}}} \right ) }-{\frac{\sqrt{3}e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ( -{\frac{\sqrt{3}}{3}}+{\frac{2\,x\sqrt{3}}{3}{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}} \right ) }+{\frac{\sqrt{3}d}{6\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ( -{\frac{\sqrt{3}}{3}}+{\frac{2\,x\sqrt{3}}{3}{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}} \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{\sqrt{3}e}{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{\sqrt{3}d}{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 ) } \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*(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*arctan(-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-1/6/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 [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

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Fricas [B]  time = 2.41328, size = 6541, normalized size = 20.25 \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*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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Sympy [A]  time = 2.18282, size = 168, normalized size = 0.52 \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))))

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Giac [A]  time = 1.14158, size = 424, normalized size = 1.31 \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)