3.1322 \(\int \frac{x^4}{a+b x^6} \, dx\)

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

[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))

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Rubi [A]  time = 0.484503, antiderivative size = 215, normalized size of antiderivative = 1., 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 = {295, 634, 618, 204, 628, 205} \[ \frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} \sqrt [6]{a} b^{5/6}}-\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} \sqrt [6]{a} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}} \]

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 295

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)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
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 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 618

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 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]

Rule 205

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

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{\operatorname{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{\operatorname{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*}

Mathematica [A]  time = 0.019082, size = 154, normalized size = 0.72 \[ \frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{12 \sqrt [6]{a} b^{5/6}} \]

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 = 0.037, size = 159, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{3}}{12\,a} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{6\,b}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{1}{3\,b}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{12\,a} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{6\,b}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/a*3^(1/2)*(1/b*a)^(5/6)*ln(x^2+3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))+1/6/b/(1/b*a)^(1/6)*arctan(2*x/(1/
b*a)^(1/6)+3^(1/2))+1/3/b/(1/b*a)^(1/6)*arctan(x/(1/b*a)^(1/6))+1/12/a*3^(1/2)*(1/b*a)^(5/6)*ln(x^2-3^(1/2)*(1
/b*a)^(1/6)*x+(1/b*a)^(1/3))+1/6/b/(1/b*a)^(1/6)*arctan(2*x/(1/b*a)^(1/6)-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(x^4/(b*x^6+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.79985, size = 927, normalized size = 4.31 \begin{align*} -\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{align*}

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 = 0.158561, size = 26, normalized size = 0.12 \begin{align*} \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{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError