\(\int \frac {x^4}{(a+b x^6)^2} \, dx\) [1335]

Optimal result

Integrand size = 13, antiderivative size = 234 \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}} \]

[Out]

1/6*x^5/a/(b*x^6+a)+1/18*arctan(b^(1/6)*x/a^(1/6))/a^(7/6)/b^(5/6)-1/36*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/
a^(1/6))/a^(7/6)/b^(5/6)+1/36*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(7/6)/b^(5/6)+1/72*ln(a^(1/3)+b^
(1/3)*x^2-a^(1/6)*b^(1/6)*x*3^(1/2))/a^(7/6)/b^(5/6)*3^(1/2)-1/72*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3^(
1/2))/a^(7/6)/b^(5/6)*3^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {296, 301, 648, 632, 210, 642, 211} \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}+\frac {x^5}{6 a \left (a+b x^6\right )} \]

[In]

Int[x^4/(a + b*x^6)^2,x]

[Out]

x^5/(6*a*(a + b*x^6)) + ArcTan[(b^(1/6)*x)/a^(1/6)]/(18*a^(7/6)*b^(5/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)
*x)/a^(1/6)]/(36*a^(7/6)*b^(5/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(36*a^(7/6)*b^(5/6)) + Log
[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(7/6)*b^(5/6)) - Log[a^(1/3) + Sqrt[3]*a^(1/
6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(7/6)*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 296

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

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*} \text {integral}& = \frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\int \frac {x^4}{a+b x^6} \, dx}{6 a} \\ & = \frac {x^5}{6 a \left (a+b x^6\right )}+\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}{18 a^{7/6} 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}{18 a^{7/6} b^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a b^{2/3}} \\ & = \frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} 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}{24 \sqrt {3} a^{7/6} 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}{24 \sqrt {3} a^{7/6} 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}{72 a 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}{72 a b^{2/3}} \\ & = \frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} 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 )}{36 \sqrt {3} a^{7/6} 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 )}{36 \sqrt {3} a^{7/6} b^{5/6}} \\ & = \frac {x^5}{6 a \left (a+b x^6\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{5/6}}-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{7/6} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{7/6} b^{5/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.82 \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {\frac {12 \sqrt [6]{a} x^5}{a+b x^6}+\frac {4 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac {2 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac {2 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac {\sqrt {3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{b^{5/6}}-\frac {\sqrt {3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{b^{5/6}}}{72 a^{7/6}} \]

[In]

Integrate[x^4/(a + b*x^6)^2,x]

[Out]

((12*a^(1/6)*x^5)/(a + b*x^6) + (4*ArcTan[(b^(1/6)*x)/a^(1/6)])/b^(5/6) - (2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^
(1/6)])/b^(5/6) + (2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/b^(5/6) + (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)
*b^(1/6)*x + b^(1/3)*x^2])/b^(5/6) - (Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/b^(5/6))
/(72*a^(7/6))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.46 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.21

method	result	size
risch	\(\frac {x^{5}}{6 a \left (b \,x^{6}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{36 b a}\)	\(48\)
default	\(\frac {x}{18 b a \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b a \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {x}{18 b a \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}}}{36 b a \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {\left (\frac {a}{b}\right )^{\frac {5}{6}} \sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 b a \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {x}{18 b a \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}}}{36 b a \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\left (\frac {a}{b}\right )^{\frac {5}{6}} \sqrt {3}\, \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 b a \left (\frac {a}{b}\right )^{\frac {1}{6}}}\)	\(346\)

[In]

int(x^4/(b*x^6+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/6*x^5/a/(b*x^6+a)+1/36/b/a*sum(1/_R*ln(x-_R),_R=RootOf(_Z^6*b+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (160) = 320\).

Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.64 \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {12 \, x^{5} + 2 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (a^{6} b^{4} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right ) - 2 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (-a^{6} b^{4} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right ) + {\left (a b x^{6} + a^{2} - \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{6} b^{4} + a^{6} b^{4}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right ) - {\left (a b x^{6} + a^{2} - \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{6} b^{4} + a^{6} b^{4}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right ) - {\left (a b x^{6} + a^{2} + \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{6} b^{4} - a^{6} b^{4}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right ) + {\left (a b x^{6} + a^{2} + \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{6} b^{4} - a^{6} b^{4}\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {5}{6}} + x\right )}{72 \, {\left (a b x^{6} + a^{2}\right )}} \]

[In]

integrate(x^4/(b*x^6+a)^2,x, algorithm="fricas")

[Out]

1/72*(12*x^5 + 2*(a*b*x^6 + a^2)*(-1/(a^7*b^5))^(1/6)*log(a^6*b^4*(-1/(a^7*b^5))^(5/6) + x) - 2*(a*b*x^6 + a^2
)*(-1/(a^7*b^5))^(1/6)*log(-a^6*b^4*(-1/(a^7*b^5))^(5/6) + x) + (a*b*x^6 + a^2 - sqrt(-3)*(a*b*x^6 + a^2))*(-1
/(a^7*b^5))^(1/6)*log(1/2*(sqrt(-3)*a^6*b^4 + a^6*b^4)*(-1/(a^7*b^5))^(5/6) + x) - (a*b*x^6 + a^2 - sqrt(-3)*(
a*b*x^6 + a^2))*(-1/(a^7*b^5))^(1/6)*log(-1/2*(sqrt(-3)*a^6*b^4 + a^6*b^4)*(-1/(a^7*b^5))^(5/6) + x) - (a*b*x^
6 + a^2 + sqrt(-3)*(a*b*x^6 + a^2))*(-1/(a^7*b^5))^(1/6)*log(1/2*(sqrt(-3)*a^6*b^4 - a^6*b^4)*(-1/(a^7*b^5))^(
5/6) + x) + (a*b*x^6 + a^2 + sqrt(-3)*(a*b*x^6 + a^2))*(-1/(a^7*b^5))^(1/6)*log(-1/2*(sqrt(-3)*a^6*b^4 - a^6*b
^4)*(-1/(a^7*b^5))^(5/6) + x))/(a*b*x^6 + a^2)

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.20 \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {x^{5}}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{7} b^{5} + 1, \left ( t \mapsto t \log {\left (60466176 t^{5} a^{6} b^{4} + x \right )} \right )\right )} \]

[In]

integrate(x**4/(b*x**6+a)**2,x)

[Out]

x**5/(6*a**2 + 6*a*b*x**6) + RootSum(2176782336*_t**6*a**7*b**5 + 1, Lambda(_t, _t*log(60466176*_t**5*a**6*b**
4 + x)))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {x^{5}}{6 \, {\left (a b x^{6} + a^{2}\right )}} - \frac {\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{72 \, a} \]

[In]

integrate(x^4/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

1/6*x^5/(a*b*x^6 + a^2) - 1/72*(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
)) - 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)) - 4*arctan(b^(1/3)*x/sqr
t(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 2*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))) - 2*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))))/a

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {x^{5}}{6 \, {\left (b x^{6} + a\right )} a} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b^{5}} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a^{2} b^{5}} \]

[In]

integrate(x^4/(b*x^6+a)^2,x, algorithm="giac")

[Out]

1/6*x^5/((b*x^6 + a)*a) - 1/72*sqrt(3)*(a*b^5)^(5/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^2*b^5)
+ 1/72*sqrt(3)*(a*b^5)^(5/6)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^2*b^5) + 1/36*(a*b^5)^(5/6)*arc
tan((2*x + sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^2*b^5) + 1/36*(a*b^5)^(5/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6))
/(a/b)^(1/6))/(a^2*b^5) + 1/18*(a*b^5)^(5/6)*arctan(x/(a/b)^(1/6))/(a^2*b^5)

Mupad [B] (verification not implemented)

Time = 5.77 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\left (a+b x^6\right )^2} \, dx=\frac {x^5}{6\,a\,\left (b\,x^6+a\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,1{}\mathrm {i}}{18\,{\left (-a\right )}^{7/6}\,b^{5/6}}-\frac {\mathrm {atan}\left (\frac {b^{3/2}\,x\,1{}\mathrm {i}}{3888\,{\left (-a\right )}^{5/2}\,\left (\frac {b^{4/3}}{7776\,{\left (-a\right )}^{7/3}}-\frac {\sqrt {3}\,b^{4/3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{7/3}}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{7/6}\,b^{5/6}}+\frac {\mathrm {atan}\left (\frac {b^{3/2}\,x\,1{}\mathrm {i}}{3888\,{\left (-a\right )}^{5/2}\,\left (\frac {b^{4/3}}{7776\,{\left (-a\right )}^{7/3}}+\frac {\sqrt {3}\,b^{4/3}\,1{}\mathrm {i}}{7776\,{\left (-a\right )}^{7/3}}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{7/6}\,b^{5/6}} \]

[In]

int(x^4/(a + b*x^6)^2,x)

[Out]

x^5/(6*a*(a + b*x^6)) - (atan((b^(1/6)*x*1i)/(-a)^(1/6))*1i)/(18*(-a)^(7/6)*b^(5/6)) - (atan((b^(3/2)*x*1i)/(3
888*(-a)^(5/2)*(b^(4/3)/(7776*(-a)^(7/3)) - (3^(1/2)*b^(4/3)*1i)/(7776*(-a)^(7/3)))))*(3^(1/2)*1i + 1)*1i)/(36
*(-a)^(7/6)*b^(5/6)) + (atan((b^(3/2)*x*1i)/(3888*(-a)^(5/2)*(b^(4/3)/(7776*(-a)^(7/3)) + (3^(1/2)*b^(4/3)*1i)
/(7776*(-a)^(7/3)))))*(3^(1/2)*1i - 1)*1i)/(36*(-a)^(7/6)*b^(5/6))