3.14.20 \(\int \frac {x^6}{a+b x^6} \, dx\) [1320]

Optimal. Leaf size=220 \[ \frac {x}{b}-\frac {\sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac {\sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac {\sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}} \]

[Out]

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

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Rubi [A]
time = 0.30, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {327, 215, 648, 632, 210, 642, 211} \begin {gather*} -\frac {\sqrt [6]{a} \text {ArcTan}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac {\sqrt [6]{a} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac {\sqrt [6]{a} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac {\sqrt [6]{a} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [6]{a} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}+\frac {x}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^6),x]

[Out]

x/b - (a^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(3*b^(7/6)) + (a^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1
/6)])/(6*b^(7/6)) - (a^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(6*b^(7/6)) + (a^(1/6)*Log[a^(1/
3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*b^(7/6)) - (a^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^
(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*b^(7/6))

Rule 210

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

Rule 211

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

Rule 215

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] +
Int[(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/
(r^2 + s^2*x^2), x] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 632

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

Rule 642

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 648

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]

Rubi steps

\begin {align*} \int \frac {x^6}{a+b x^6} \, dx &=\frac {x}{b}-\frac {a \int \frac {1}{a+b x^6} \, dx}{b}\\ &=\frac {x}{b}-\frac {\sqrt [6]{a} \int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{3 b}-\frac {\sqrt [6]{a} \int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{3 b}-\frac {\sqrt [3]{a} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{3 b}\\ &=\frac {x}{b}-\frac {\sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac {\sqrt [6]{a} \int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [6]{a} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [3]{a} \int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 b}-\frac {\sqrt [3]{a} \int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 b}\\ &=\frac {x}{b}-\frac {\sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [6]{a} \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{6 \sqrt {3} b^{7/6}}+\frac {\sqrt [6]{a} \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{6 \sqrt {3} b^{7/6}}\\ &=\frac {x}{b}-\frac {\sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac {\sqrt [6]{a} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac {\sqrt [6]{a} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 182, normalized size = 0.83 \begin {gather*} \frac {12 \sqrt [6]{b} x-4 \sqrt [6]{a} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{a} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{a} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt {3} \sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt {3} \sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 b^{7/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^6),x]

[Out]

(12*b^(1/6)*x - 4*a^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)] + 2*a^(1/6)*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] - 2*
a^(1/6)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + Sqrt[3]*a^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^
(1/3)*x^2] - Sqrt[3]*a^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(12*b^(7/6))

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Maple [A]
time = 0.16, size = 171, normalized size = 0.78

method	result	size
risch	\(\frac {x}{b}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{6 b^{2}}\)	\(34\)
default	\(\frac {x}{b}-\frac {\left (\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}\right ) a}{b}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^6+a),x,method=_RETURNVERBOSE)

[Out]

x/b-(1/12/a*3^(1/2)*(a/b)^(1/6)*ln(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+1/6/a*(a/b)^(1/6)*arctan(2*x/(a/b)^(
1/6)+3^(1/2))-1/12/a*3^(1/2)*(a/b)^(1/6)*ln(x^2-3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+1/6/a*(a/b)^(1/6)*arctan(2*
x/(a/b)^(1/6)-3^(1/2))+1/3/a*(a/b)^(1/6)*arctan(x/(a/b)^(1/6)))*a/b

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Maxima [A]
time = 0.50, size = 194, normalized size = 0.88 \begin {gather*} -\frac {\frac {\sqrt {3} a^{\frac {1}{6}} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{b^{\frac {1}{6}}} - \frac {\sqrt {3} a^{\frac {1}{6}} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{b^{\frac {1}{6}}} + \frac {4 \, a^{\frac {1}{3}} \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, a^{\frac {1}{3}} \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, a^{\frac {1}{3}} \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{12 \, b} + \frac {x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^6+a),x, algorithm="maxima")

[Out]

-1/12*(sqrt(3)*a^(1/6)*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/b^(1/6) - sqrt(3)*a^(1/6)*log(b^
(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/b^(1/6) + 4*a^(1/3)*arctan(b^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/s
qrt(a^(1/3)*b^(1/3)) + 2*a^(1/3)*arctan((2*b^(1/3)*x + sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/sqrt(a^
(1/3)*b^(1/3)) + 2*a^(1/3)*arctan((2*b^(1/3)*x - sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/sqrt(a^(1/3)*
b^(1/3)))/b + x/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (148) = 296\).
time = 0.37, size = 314, normalized size = 1.43 \begin {gather*} -\frac {4 \, \sqrt {3} b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} b^{6} x \left (-\frac {a}{b^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} \sqrt {b^{2} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{3}} + b x \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x^{2}} b^{6} \left (-\frac {a}{b^{7}}\right )^{\frac {5}{6}} - \sqrt {3} a}{3 \, a}\right ) + 4 \, \sqrt {3} b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} b^{6} x \left (-\frac {a}{b^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} \sqrt {b^{2} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{3}} - b x \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x^{2}} b^{6} \left (-\frac {a}{b^{7}}\right )^{\frac {5}{6}} + \sqrt {3} a}{3 \, a}\right ) + b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (b^{2} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{3}} + b x \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) - b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (b^{2} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{3}} - b x \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) + 2 \, b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) - 2 \, b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (-b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) - 12 \, x}{12 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^6+a),x, algorithm="fricas")

[Out]

-1/12*(4*sqrt(3)*b*(-a/b^7)^(1/6)*arctan(-1/3*(2*sqrt(3)*b^6*x*(-a/b^7)^(5/6) - 2*sqrt(3)*sqrt(b^2*(-a/b^7)^(1
/3) + b*x*(-a/b^7)^(1/6) + x^2)*b^6*(-a/b^7)^(5/6) - sqrt(3)*a)/a) + 4*sqrt(3)*b*(-a/b^7)^(1/6)*arctan(-1/3*(2
*sqrt(3)*b^6*x*(-a/b^7)^(5/6) - 2*sqrt(3)*sqrt(b^2*(-a/b^7)^(1/3) - b*x*(-a/b^7)^(1/6) + x^2)*b^6*(-a/b^7)^(5/
6) + sqrt(3)*a)/a) + b*(-a/b^7)^(1/6)*log(b^2*(-a/b^7)^(1/3) + b*x*(-a/b^7)^(1/6) + x^2) - b*(-a/b^7)^(1/6)*lo
g(b^2*(-a/b^7)^(1/3) - b*x*(-a/b^7)^(1/6) + x^2) + 2*b*(-a/b^7)^(1/6)*log(b*(-a/b^7)^(1/6) + x) - 2*b*(-a/b^7)
^(1/6)*log(-b*(-a/b^7)^(1/6) + x) - 12*x)/b

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Sympy [A]
time = 0.07, size = 22, normalized size = 0.10 \begin {gather*} \operatorname {RootSum} {\left (46656 t^{6} b^{7} + a, \left ( t \mapsto t \log {\left (- 6 t b + x \right )} \right )\right )} + \frac {x}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*b**7 + a, Lambda(_t, _t*log(-6*_t*b + x))) + x/b

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Giac [A]
time = 1.39, size = 180, normalized size = 0.82 \begin {gather*} \frac {x}{b} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, b^{2}} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, b^{2}} - \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, b^{2}} - \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, b^{2}} - \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^6+a),x, algorithm="giac")

[Out]

x/b - 1/12*sqrt(3)*(a*b^5)^(1/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/b^2 + 1/12*sqrt(3)*(a*b^5)^(1/
6)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/b^2 - 1/6*(a*b^5)^(1/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(
a/b)^(1/6))/b^2 - 1/6*(a*b^5)^(1/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/b^2 - 1/3*(a*b^5)^(1/6)*ar
ctan(x/(a/b)^(1/6))/b^2

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Mupad [B]
time = 1.17, size = 227, normalized size = 1.03 \begin {gather*} \frac {x}{b}+\frac {{\left (-a\right )}^{1/6}\,\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,1{}\mathrm {i}}{3\,b^{7/6}}+\frac {{\left (-a\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{25/6}\,x\,1{}\mathrm {i}}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}+\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}+\frac {\sqrt {3}\,{\left (-a\right )}^{25/6}\,x}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}+\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,b^{7/6}}-\frac {{\left (-a\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{25/6}\,x\,1{}\mathrm {i}}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}-\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}-\frac {\sqrt {3}\,{\left (-a\right )}^{25/6}\,x}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}-\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,b^{7/6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b*x^6),x)

[Out]

x/b + ((-a)^(1/6)*atan((b^(1/6)*x*1i)/(-a)^(1/6))*1i)/(3*b^(7/6)) + ((-a)^(1/6)*atan(((-a)^(25/6)*x*1i)/(b^(1/
6)*((-a)^(13/3)/b^(1/3) + (3^(1/2)*(-a)^(13/3)*1i)/b^(1/3))) + (3^(1/2)*(-a)^(25/6)*x)/(b^(1/6)*((-a)^(13/3)/b
^(1/3) + (3^(1/2)*(-a)^(13/3)*1i)/b^(1/3))))*((3^(1/2)*1i)/2 - 1/2)*1i)/(3*b^(7/6)) - ((-a)^(1/6)*atan(((-a)^(
25/6)*x*1i)/(b^(1/6)*((-a)^(13/3)/b^(1/3) - (3^(1/2)*(-a)^(13/3)*1i)/b^(1/3))) - (3^(1/2)*(-a)^(25/6)*x)/(b^(1
/6)*((-a)^(13/3)/b^(1/3) - (3^(1/2)*(-a)^(13/3)*1i)/b^(1/3))))*((3^(1/2)*1i)/2 + 1/2)*1i)/(3*b^(7/6))

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