3.14.22 \(\int \frac {x^4}{a+b x^6} \, dx\) [1322]

Optimal. Leaf size=215 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}} \]

[Out]

1/3*arctan(b^(1/6)*x/a^(1/6))/a^(1/6)/b^(5/6)-1/6*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(1/6)/b^(5/
6)+1/6*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(1/6)/b^(5/6)+1/12*ln(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/
6)*x*3^(1/2))/a^(1/6)/b^(5/6)*3^(1/2)-1/12*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3^(1/2))/a^(1/6)/b^(5/6)*3
^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {301, 648, 632, 210, 642, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}}-\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x^6),x]

[Out]

ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(1/6)*b^(5/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)]/(6*a^(1/6)*b
^(5/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(6*a^(1/6)*b^(5/6)) + Log[a^(1/3) - Sqrt[3]*a^(1/6)*
b^(1/6)*x + b^(1/3)*x^2]/(4*Sqrt[3]*a^(1/6)*b^(5/6)) - Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/
(4*Sqrt[3]*a^(1/6)*b^(5/6))

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 301

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

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

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Mathematica [A]
time = 0.02, size = 154, normalized size = 0.72 \begin {gather*} \frac {4 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt {3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt {3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 \sqrt [6]{a} b^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x^6),x]

[Out]

(4*ArcTan[(b^(1/6)*x)/a^(1/6)] - 2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/
a^(1/6)] + Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] - Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1
/6)*b^(1/6)*x + b^(1/3)*x^2])/(12*a^(1/6)*b^(5/6))

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Maple [A]
time = 0.16, size = 159, normalized size = 0.74

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{6 b}\)	\(27\)
default	\(-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\)	\(159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^6+a),x,method=_RETURNVERBOSE)

[Out]

-1/12/a*3^(1/2)*(a/b)^(5/6)*ln(x^2+3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan(2*x/(a/b)^(1/6)
+3^(1/2))+1/12/a*3^(1/2)*(a/b)^(5/6)*ln(x^2-3^(1/2)*(a/b)^(1/6)*x+(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan(2*x/(a
/b)^(1/6)-3^(1/2))+1/3/b/(a/b)^(1/6)*arctan(x/(a/b)^(1/6))

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Maxima [A]
time = 0.50, size = 184, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {1}{6}} b^{\frac {5}{6}}} + \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {1}{6}} b^{\frac {5}{6}}} + \frac {\arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{3 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\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 )}{6 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\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 )}{6 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (143) = 286\).
time = 0.37, size = 348, normalized size = 1.62 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} b x \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a b^{4} x \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} - a b^{3} \left (-\frac {1}{a b^{5}}\right )^{\frac {2}{3}} + x^{2}} b \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{3} \, \sqrt {3} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} b x \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {-a b^{4} x \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} - a b^{3} \left (-\frac {1}{a b^{5}}\right )^{\frac {2}{3}} + x^{2}} b \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{12} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (a b^{4} x \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} - a b^{3} \left (-\frac {1}{a b^{5}}\right )^{\frac {2}{3}} + x^{2}\right ) - \frac {1}{12} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-a b^{4} x \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} - a b^{3} \left (-\frac {1}{a b^{5}}\right )^{\frac {2}{3}} + x^{2}\right ) + \frac {1}{6} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (a b^{4} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) - \frac {1}{6} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-a b^{4} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.07, size = 26, normalized size = 0.12 \begin {gather*} \operatorname {RootSum} {\left (46656 t^{6} a b^{5} + 1, \left ( t \mapsto t \log {\left (7776 t^{5} a b^{4} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*a*b**5 + 1, Lambda(_t, _t*log(7776*_t**5*a*b**4 + x)))

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Giac [A]
time = 2.83, size = 190, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{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 \, a b^{5}} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{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 \, a b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{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 \, a b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{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 \, a b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, a b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^6+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.14, size = 134, normalized size = 0.62 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {b^{1/6}\,x}{{\left (-a\right )}^{1/6}}\right )}{3\,{\left (-a\right )}^{1/6}\,b^{5/6}}-\frac {\mathrm {atanh}\left (\frac {2\,{\left (-a\right )}^{5/2}\,b^{3/2}\,x}{{\left (-a\right )}^{8/3}\,b^{4/3}-\sqrt {3}\,{\left (-a\right )}^{8/3}\,b^{4/3}\,1{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a\right )}^{1/6}\,b^{5/6}}+\frac {\mathrm {atanh}\left (\frac {2\,{\left (-a\right )}^{5/2}\,b^{3/2}\,x}{{\left (-a\right )}^{8/3}\,b^{4/3}+\sqrt {3}\,{\left (-a\right )}^{8/3}\,b^{4/3}\,1{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a\right )}^{1/6}\,b^{5/6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x^6),x)

[Out]

(atanh((2*(-a)^(5/2)*b^(3/2)*x)/((-a)^(8/3)*b^(4/3) + 3^(1/2)*(-a)^(8/3)*b^(4/3)*1i))*(3^(1/2)*1i - 1))/(6*(-a
)^(1/6)*b^(5/6)) - (atanh((2*(-a)^(5/2)*b^(3/2)*x)/((-a)^(8/3)*b^(4/3) - 3^(1/2)*(-a)^(8/3)*b^(4/3)*1i))*(3^(1
/2)*1i + 1))/(6*(-a)^(1/6)*b^(5/6)) - atanh((b^(1/6)*x)/(-a)^(1/6))/(3*(-a)^(1/6)*b^(5/6))

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